Projective-space bases of Lagrangian fibrations

Proof. Relative ampleness and projectivity of B make X Moishezon; as X is Kähler, it is projective. Equivalently, a sufficiently positive twist of a relative polarization by a base polarization is ample. The structure statements in (1) are the fibration theorems of Matsushita [35, 37]. The pullback of H is nef and nonzero. Its (2n)-th power is zero because \dim B=n, so Fujiki’s relation gives q(e)=0.

For (2), Ou’s Proposition 3.6 gives the reflexive cotangent identifications for the given projective equidimensional Lagrangian fibration, extending the symplectic differential identification of Matsushita [38, 46]. Ou’s Theorem 1.3 is the separate local reflexivity result, and Theorem 1.4 gives the vanishing [46]. The target hypotheses hold because klt spaces are Cohen–Macaulay [31, Theorem 11.1(2), Corollaries 11.9 and 11.14] and K_B is \Q-Cartier. The cited reflexivity theorem has arbitrary-dimensional scope. The higher-cohomology and fiber-restriction statements in (3) are [52, Theorem 0.4]. Statement (4) is Hwang’s theorem [26, Theorem 1.2].□

Proof. On the smooth locus, contraction identifies T_B with \Omega_B^{n-1}\otimes\cO_B(-K_B). Both sides extend reflexively, so

Apply Proposition 2.1(2), since n-1>0.□

Proof. Let h be an ample class. Then (h,e)>0 and q(e)=0, so \langle h,e\rangle_{\R} is a hyperbolic plane. It is perpendicular to the positive real plane generated by \operatorname{Re}\sigma and \operatorname{Im}\sigma. These four independent real directions give the assertion.□

Proof. Fix b\in B. Choose sufficiently positive very ample divisors on the smooth projective variety X. By Bertini, a general intersection of n of them is smooth of dimension n. They may simultaneously be chosen so that their intersection with each component of f^{-1}(b) is zero-dimensional and nonempty: the fiber has finitely many components, all of dimension n, and successive general hyperplanes cut each of them properly. Denote the resulting smooth intersection by Y. The proper map Y\to B is quasi-finite over b, and hence finite over a neighborhood of b. Its image contains that neighborhood, by properness and the dimension theorem. A local branch at a point of Y over b gives the asserted finite map of analytic germs. If an individual algebraic branch is required, split the finite algebra over the henselization and descend the corresponding idempotents to an étale neighborhood.

Let \pi:(Y,y)\to (B,b) be a finite dominating smooth germ, of degree m, and let D be a Weil divisor on (B,b). Pullback of Weil divisors under a finite map is defined by the ramification indices of the discrete valuations at prime divisors. The divisor \pi^*D is principal on the smooth local germ, say \operatorname{div}(g). The valuation formula for the field norm gives

Thus mD is Cartier. The argument is analytic and therefore proves the stronger local assertion needed later, rather than merely algebraic \Q-factoriality.□

Proof. We explain the trace at a branch point. For a finite surjection \pi:Y\to U from a smooth n-fold to a normal connected neighborhood, let m_y be the local generic degree of the branch at y. A section a of \pi_*\Q_Y is locally constant on Y. Set

This is locally constant. Indeed, choose disjoint neighborhoods of the finitely many points above b. After shrinking downstairs, the sum of the local generic degrees in each of these neighborhoods is constant: it equals the degree over the common unbranched dense open of the locally irreducible normal base. The value of a is constant on each chosen neighborhood. This proves the assertion and shows that the trace is a morphism of sheaves. Its composite with the constant-section inclusion \Q_U\to\pi_*\Q_Y is multiplication by \deg\pi. Therefore \Q_U is a direct summand of \pi_*\Q_Y.

Proper base change and the finiteness of the fibers give R^j\pi_*\Q_Y=0 for j>0. Applying the point costalk at b to the preceding direct summand, and using proper base change for costalks, exhibits

The latter group is zero unless k=2n. The top local group on the normal, hence locally irreducible, complex analytic germ U is one-dimensional: the complex orientation gives its fundamental class, and the top fundamental cycles of a pure analytic germ are indexed by its local irreducible components [10, Section 1.4, Theorem 3.2, Section 3.3 and Lemma 4.3]. Here fundamental classes are taken in Borel–Moore homology of representatives and passed to local groups. Equivalently, this follows from an analytic triangulation and the oriented top strata. Thus the costalk of \Q_U is exactly \Q[-2n] at every point, with the compatible complex orientation. This is the rational homology-manifold assertion and identifies the intersection complex with the shifted constant sheaf [34, Theorem 1.1(1)\Leftrightarrow(4)]. The final claim now follows from the base-cohomology theorem recalled in Proposition 2.1; the passage from intersection to ordinary cohomology has been justified before using it.□

Proof. Work first over the strict henselization of the algebraic germ and choose a finite smooth slice as in Lemma 3.1. Choose a connected local factor of its finite algebra, with fraction field L over the base field K. If K'/K defines a finite quasi-étale normal cover, the normalization of its pullback to this slice is unramified outside a set of codimension at least two. Finiteness preserves that codimension bound. The source of the normalized map is normal and the target is regular, so purity of the branch locus [56, Tag 0BMB] makes the map étale everywhere. A connected finite étale cover of a strictly henselian regular local scheme is trivial. A component of the split pullback therefore embeds K' into L over K.

Let M/K be a finite Galois closure of L/K, let W be the normalization of the base in M, and write \Gamma for its Galois group. Normalizations are finite by excellence. Let H\lhd\Gamma be the normal subgroup generated by the inertia groups of all prime divisors of W. The finite quotient C=W/H is unramified over every prime divisor of the normal base and hence is quasi-étale. Conversely, an intermediate extension which is quasi-étale has trivial divisorial inertia; its defining subgroup contains every conjugate of each inertia group, and hence contains H. Every quasi-étale cover is consequently dominated by C. A further quasi-étale cover of C would give one of the base and is therefore trivial. Applying the same split-pullback argument to C shows that it too is finitely dominated by the smooth slice.

All finite algebras, group actions, quotient maps and their identities descend from the strict henselization to an étale neighborhood. Shrinking removes branch divisors not meeting the marked fiber. This construction also gives the analytic maximum. Indeed, the normalized pullback of any finite analytic quasi-étale cover to the smooth germ is étale by analytic purity and splits because a smooth germ is simply connected. One may also obtain this directly: a small smooth ball minus an analytic set of complex codimension at least two is simply connected, by general position for disks against a stratification; its finite cover splits there, and normalization extends that splitting uniquely. The analytic cover is thus an intermediate cover of the analytification of the same finite Galois closure. Finite normalizations and finite-group quotients commute here with analytification; the local normal components and their degrees agree. The divisorial inertia groups through the marked point are therefore the same in both constructions. Finally C is klt. Indeed, writing \pi:C\to B, we have K_C=\pi^*K_B in codimension one, and for a divisorial valuation E over C restricting with ramification index r to a divisorial valuation F over B, the canonical ramification formula on normalized birational models gives A_C(E)=rA_B(F). The latter is positive because B is klt.

We prove the source assertions, including connectedness. The connected fibers of f and normality of B give f_*\cO_X=\cO_B by Stein factorization. This equality persists after the flat base change to U. Let K=\C(U), L=\C(C), and let F be the generic fiber of X_U\to U. Generic smoothness makes F smooth. Proper flat field base change gives

Thus F_{\overline K} is connected. A connected smooth scheme over an algebraically closed field is integral, since its irreducible components are disjoint. In this setting it is an abelian variety after an origin is chosen, by Proposition 2.1. In particular F is geometrically integral, so F_L is integral and normal and has H^0(F_L,\cO_{F_L})=L. This proves uniqueness of the component dominating C and identifies the generic fiber of its normalization Z with F_L.

The normalization is finite by excellence, so h is projective. Its Stein algebra h_*\cO_Z is finite over \cO_C and embeds in the generic algebra L: a section on the integral space Z is determined by its restriction to the generic fiber. Every such section is integral over \cO_C; normality of C therefore gives h_*\cO_Z=\cO_C. Stein factorization now gives connected fibers. This part of the argument used only that C\to U is finite and dominant, with C normal and integral.

Let D\subset U have codimension at least two and contain the branch locus of \pi. Equidimensionality of f gives

The map p is finite over the smooth space X_U and is étale away from f_U^{-1}(D). Purity makes it étale everywhere, so Z is smooth and carries the pulled-back symplectic form. Each component of a fiber of h has dimension at most n, because its finite image lies in a fiber of f_U. The fiber dimension theorem gives the reverse inequality. Hence h is equidimensional of relative dimension n; the same finite images show that its fiber components are Lagrangian. The symplectic form trivializes the canonical bundle of Z.

If \pi is Galois, its deck action lifts to Z. Over the dense unbranched locus the map p is a deck torsor. Both sides of the torsor map \Gamma\times Z\to Z\times_{X_U}Z are finite étale over Z, so their isomorphism on that dense locus extends over Z. This proves freeness of the action also over special fibers. Subsequent étale refinements of the object chart use ordinary base change of h and preserve its projectivity, equidimensionality, connected fibers and smooth symplectic source. The torsor descent is retained from the finite chart before that refinement.□

Proof. Cyclicity of trace proves stability under composition. Stability under tensor product follows by taking a partial trace: for a map a:M\to N and a test map b:N\widehat\otimes P\to M\widehat\otimes P, the trace of b(a\widehat\otimes\id_P) is the trace of a composed with the partial trace of b. These formulas hold for bundles in codimension one and therefore over R. Duality is compatible with the same ideal.

Write A=A_M=\End_R(M) and J=\rad A. The algebra A is finite over the henselian local ring R, so \mathfrak m A\subset J, A/J is a finite-dimensional semisimple \C-algebra, and J^q\subset\mathfrak m A for some q. We recall why idempotents lift in this setting. First lift an idempotent through the nilpotent radical of A/\mathfrak m A. For an ensuing lift a\in A of an idempotent modulo \mathfrak m A, the commutative finite algebra R[a] splits into the factors supported at 0 and 1 in its closed fiber by henselianity. The resulting idempotent lifts the given one. Induction in complementary corners lifts an entire orthogonal system. Thus primitive idempotents of A/J lift to actual direct summands of M.

For x\in J, all sufficiently large powers x^j belong to \mathfrak m A, so their traces vanish modulo \mathfrak m. The generic characteristic polynomial of x has coefficients in R: this is true at every height-one localization, where M is free, and R is the intersection of those valuation rings. If \lambda_1,\ldots,\lambda_r are the roots of its reduction over \C, Newton identities give \sum_i\lambda_i^j=0 for every sufficiently large j. Grouping the distinct nonzero roots, a Vandermonde matrix applied to consecutive such powers forces all their multiplicities to be zero. There are consequently no nonzero roots, and \tr(x)=0 modulo \mathfrak m. The same holds for yx for any y\in A, because J is an ideal. Hence J\subset\mathcal R(M,M).

The residual trace now descends to A/J. On each simple block \operatorname{Mat}_t(\C) it is a scalar multiple of the ordinary matrix trace, by cyclicity. Evaluate it on a primitive diagonal idempotent. Its lift projects onto a nonzero reflexive direct summand M_0, so the scalar is \rk(M_0)>0. The trace pairing is therefore nondegenerate on every block. This proves the reverse inclusion and the equality with J.

Apply the argument to finite direct sums of objects. Their endomorphism algebras in the quotient are semisimple matrix algebras, and their primitive projectors lift. In particular every object is a finite sum of simple objects, distinct simple objects have no morphisms, and the endomorphisms of each simple object are \C. This also proves that the quotient is abelian and that all its exact sequences split. The ideal contains \mathfrak m\Hom_R(M,N), so its Hom spaces are finite-dimensional. Finally the trace of the identity is \rk M, as asserted.□

Proof. Let a:M\to N and lift a left inverse in the quotient to b:N\to M. Then ba=\id_M+r with r\in\rad\End_R(M), so ba is invertible. Replacing b by (ba)^{-1}b gives an actual left inverse. If the quotient map is an isomorphism, apply the same argument to both composites of lifts of it and its inverse; the lift is an isomorphism. Apply this to an isomorphism with R^r for the final statement.□

Proof. Lemma 3.4 gives an abelian rigid symmetric \C-linear category with \End({\bf1})=\C. Tensor product is exact because every exact sequence splits. For an object of rank r, its (r+1)st exterior power is zero: it is zero on the common bundle locus and hence its reflexive extension is zero. Deligne’s criterion [17, Theorem 7.1] therefore applies. The category is tensor-generated by one object; Deligne’s Corollary 6.20 supplies a fiber functor over a finite extension of \C, hence over \C itself. Tensor reconstruction gives the stated group. A tensor generator makes it an algebraic subgroup of its general linear group, and semisimplicity of the representation category makes it reductive in characteristic zero. The fiber dimension equals the categorical dimension, namely \rk V=n.□

Proof. Pull a connection on M_U back to a finite smooth dominating germ \pi:(Y,y)\to(C,c). The complement of \pi^{-1}(U) has codimension at least two. On Y let F=(\pi^*M)^{**}. In local smooth coordinates each coefficient of the connection extends across that complement by the Hartogs property of F. Its target F\otimes\Omega_Y^1 is reflexive because \Omega_Y^1 is locally free. Thus the connection extends to F and still satisfies Leibniz’s rule.

Here is the elementary freeness argument, which does not require a flat connection. For a coordinate derivation \partial, choose a presentation of F and lift the operator \nabla_{\partial} to the free module of generators. If A is the corresponding matrix on generators and P a matrix of relations, stability of the relations gives a matrix Q with

The derivative of every minor of P is therefore a linear combination of minors of the same size. Each Fitting ideal of F is stable under every coordinate derivation. A nonzero ideal in a smooth characteristic-zero local ring with this property is the unit ideal: repeatedly differentiating a nonzero element of minimal order produces an element with nonzero constant term. If r is the generic rank of F, its rth Fitting ideal is nonzero and hence is the unit ideal. Thus F is generated by r elements. The resulting surjection from a free module of rank r has a torsion kernel, which is zero because it is a submodule of a free module over a domain. This proves that F is free.

Take a normal Galois closure W\to C of Y\to C, with group \Gamma. Its normal domain is local over the henselian germ, so the whole group fixes the unique marked point. The reflexive pullback P of M to W is free: it agrees in codimension one with the pullback of the free module F, and both reflexive extensions agree. It has its natural semilinear \Gamma-action. At a prime divisor, divisorial inertia acts trivially on its restriction to that divisor, because there P is the ordinary pullback of a bundle from C. The same inertia therefore acts trivially on the fiber of P at the marked point. To see the last specialization explicitly, for an inertia element of order m, restrict P to its fixed locus and apply the idempotents

Their ranks are locally constant. On the divisor all nonidentity eigenvalues have rank zero, hence they have rank zero at its specialization to the marked point as well.

Since C has no further quasi-étale cover, these inertia groups normally generate \Gamma, by the construction in Proposition 3.3. The action on the marked fiber of P is consequently trivial. Lift a basis of that fiber to sections of P and average over \Gamma. The averaged sections still lift the basis and hence, by Nakayama, form a basis of the free module P. This basis is \Gamma-invariant. The invariant basis identifies P^\Gamma with R^r, because the invariants of the normal cover ring are R. Thus P^\Gamma is reflexive. The natural map M\to P^\Gamma is an isomorphism in codimension one by ordinary finite descent for the pullback of a bundle. Both modules are reflexive, so this isomorphism extends over the normal germ. Hence M=P^\Gamma is free, as claimed.□

Proof. The natural tensor action defining \Phi_M is a bundle map on U; its reflexive extension is the displayed morphism in \mathcal C. Choose local connections \nabla_\alpha on M_U. Their differences represent a(M_U). On T_j(M_U) the corresponding differences are the infinitesimal tensor actions of \nabla_\alpha-\nabla_\beta. Evaluating on the global section t_j gives the coboundary of the local sections \nabla_\alpha t_j. This proves the asserted vanishing after restricting (3.1) to bundles on U and taking ordinary H^1.

An injection in the semisimple quotient has a left inverse there. Lemma 3.5 lifts it to an actual left inverse between reflexive coefficient modules. Restriction to U gives a split map of bundles, which induces an injection on ordinary H^1. Its kernel contains a(M_U), so that class is zero. The argument uses only that the individual t_j are actual sections. No algebraic identities among their images in the quotient need hold for the chosen lifts.□

Proof. For a reductive subgroup of a general linear group, the invariant mixed tensors have that subgroup as their simultaneous pointwise stabilizer. Indeed, projectors onto direct summands and all maps between tensor constructions are themselves such tensors. Preserving them is precisely the condition to give a tensor automorphism of the forgetful functor, and tensor reconstruction identifies those automorphisms with the group. Differentiating, its Lie algebra is the common kernel of the infinitesimal tensor-action maps. Since the endomorphism space is finite-dimensional, finitely many tensors suffice for this last kernel.

Suppose a representation N of G has finite image. Realize it as a direct summand of tensor constructions in V_{\mathrm{fib}}, and lift its projector to obtain an actual module M. Choose finitely many invariant mixed tensors of N whose infinitesimal stabilizer is the zero Lie algebra. By the definition of the quotient Hom spaces these tensors lift to actual maps from R, hence to actual sections on the chart. The map \omega(\overline{\Phi_M}) of Lemma 3.8 has as its kernel this infinitesimal stabilizer inside the full space \End_{\C}(N), and is therefore injective. Thus \overline{\Phi_M} is injective. Tensoring it with \overline V preserves injectivity, so \overline{\Psi_M} is injective and that lemma and Lemma 3.7 make M free. Therefore N is trivial. The regular representation of the finite component group G/G^\circ is one such representation. It is trivial only if that component group is trivial. Hence G is connected.

A character of G corresponds to a rank-one reflexive module on the chart. By analytic \Q-factoriality some positive reflexive tensor power of that module is free. Thus the character has finite order. A connected algebraic group has no nontrivial finite-image character. The character group of G is therefore zero. For a connected reductive group its quotient by the derived group is a torus, whose characters detect it; that torus must be trivial. Thus G is semisimple.□

Proof. Local coordinate connections on \Omega_U^1 are torsion-free. Their differences show that a(V_U) belongs to the direct summand

On the other hand, choose local connections preserving the actual splitting V_U=\bigoplus_i(V_i)_U. The off-diagonal endomorphism components of the same cohomology class are zero. These two conditions imply

Here is the index check. Block preservation allows only coefficients V_a\otimes V_b\otimes V_b^\vee. Symmetry interchanges the first two covariant slots. If a\ne b, its image is an off-diagonal coefficient V_b\otimes V_a\otimes V_b^\vee, whose class is zero. Because these are actual direct sums of bundles, the original mixed component is also zero. This leaves precisely the terms in (3.2). In particular a_i is the class a((V_i)_U) under the indicated inclusion of its coefficients.

Choose lifts t_j of finitely many G-invariant tensors in natural mixed powers of V_{\mathrm{fib}} cutting out \mathfrak g infinitesimally. Use them to define Q and \Phi_V as in Lemma 3.8. For the ith block set

The symmetric inclusion in the first two slots, followed by the diagonal inclusion of the ith endomorphism block, gives

The required one-block coefficient map is

On U the source of this map is \Sym^2(V_i)_U\otimes(V_i)_U^\vee. Projecting the first slot of (\Psi_{V,U})_*a(V_U)=0 onto (V_i)_U and using (3.2) gives (\kappa_{i,U})_*a_i=0 in ordinary H^1.

Write L=V_{\mathrm{fib}} and L_i=V_{i,\mathrm{fib}}. The kernel of \omega(\overline{\Phi_V}) is \mathfrak g inside \End_{\C}(L). Hence the fiber kernel of (3.3) is the explicit intersection

Partial evaluation of the first covariant slot by x\in T_i=L_i^\vee is supported on the ith endomorphism block. It belongs to \mathfrak g precisely when it belongs to \mathfrak h_i. The symmetry of the first two slots is the defining symmetry in \mathfrak h_i^{(1)} on T_i. Passing from the action on L_i to the dual action on T_i transposes and changes sign, which leaves this linear kernel unchanged. Thus (3.4) identifies with \mathfrak h_i^{(1)}.

If the kernel is zero, faithfulness of \omega makes \overline{\kappa_i} injective, hence split injective in the semisimple quotient. Lemma 3.5 lifts a left inverse for \kappa_i between the reflexive coefficient modules. Restricting that left inverse to U makes the induced ordinary H^1 map injective. Thus a_i=0 in ordinary H^1, and Lemma 3.7 makes V_i free. A free irreducible object is the tensor unit. This proves the assertion.□

Proof. Write a prolongation tensor as A(x,y)=A_x(y) with A_x\in\mathfrak h. Fix x=u\otimes w_1 with w_1\ne0 and choose w_2 independent of w_1. For every v\in U, symmetry gives

The two indicated tensor subspaces have zero intersection, so A_xv=0 for every v. Thus A_x=0 for every pure tensor x, and hence for every x. No basis of W invariant under any larger Lie algebra has been assumed.□

Proof. Use homogeneous polynomial vector fields on the affine space T. Let \mathfrak p_{-1}=T be the translations, \mathfrak p_0=\mathfrak h, and for k\ge1 set

These spaces form a graded Lie algebra \mathfrak p=\bigoplus_{k\ge-1}\mathfrak p_k under the vector-field bracket. Indeed, differentiating a bracket gives the sum of brackets with derivatives. Induction on the total degree, starting with closure of \mathfrak h under brackets, proves [\mathfrak p_i,\mathfrak p_j]\subset\mathfrak p_{i+j}. The representation also gives [\mathfrak h,\mathfrak p_k]\subset\mathfrak p_k directly.

Suppose first that \mathfrak p is finite-dimensional. Its solvable radical is graded, because it is preserved by the scaling automorphisms of this grading. From any nonzero homogeneous polynomial vector field in that radical, successive brackets with translations produce a nonzero constant field. Irreducibility under \mathfrak h then puts all of T in the radical. Since \mathfrak p_1\ne0, the bracket [T,\mathfrak p_1] is nonzero. It lies in the intersection of the radical with \mathfrak h, a solvable ideal of a semisimple algebra, which is impossible. Thus the radical is zero.

The grading derivation D, acting as k\id on \mathfrak p_k, must now be inner. Recall a short proof of this fact. For any derivation D of a semisimple algebra \mathfrak a, nondegeneracy of the Killing form gives z\in\mathfrak a such that E=D-\operatorname{ad}(z) satisfies \tr(E\operatorname{ad}(x))=0 for all x. Then

so E=0. Apply this to \mathfrak a=\mathfrak p and write D=\operatorname{ad}(z). Comparing homogeneous degrees and using the zero center shows z\in\mathfrak p_0=\mathfrak h. But D|_T=-\id_T. This would give a nonzero scalar in the semisimple subalgebra \mathfrak h\subset\mathfrak{sl}(T), again impossible. Hence \mathfrak p is infinite-dimensional.

We make the symbol argument for this last alternative explicit. Let S=\Sym T, and let \mathfrak h^\perp\subset T\otimes T^* be the annihilator of \mathfrak h\subset T^*\otimes T. Form the finitely generated graded S-module

where the relations have degree one. Perfect pairing with homogeneous polynomial vector fields identifies \mathcal M_d^*=\mathfrak p_{d-1} for d\ge0: annihilating a relation multiplied by d-1 linear factors is exactly the displayed partial-evaluation condition. Since \mathfrak p is infinite-dimensional, \mathcal M is not a finite-length module supported at the origin. Its support thus contains a nonzero point \xi\in T^*=\Spec(S)(\C).

The fiber at \xi is

and is nonzero. Choose a nonzero vector u\in T annihilating its relation space. Then \langle r,u\otimes\xi\rangle=0 for every r\in\mathfrak h^\perp. Hence u\otimes\xi\in\mathfrak h. Since both u and \xi are nonzero, this is the required rank-one endomorphism.□

Proof. We give the weight argument, including the exceptional short-root possibilities. Consider in \PP(\mathfrak h) the closure of the orbit of a rank-one operator. It consists of rank-one operators, since the rank-at-most-one locus is closed and the zero operator does not occur in projective space. A Borel subgroup has a fixed point on this closure. In this particular linear setting one can see the fixed point directly: triangularize the solvable Borel action, choose the smallest term of an invariant complete flag whose projectivization meets the closure, and intersect there. This projective intersection avoids the preceding hyperplane and is therefore also affine. It is finite; a connected group fixes each of its points. In the adjoint representation a Borel-fixed line is the highest-root line of one simple ideal. The highest root is long. We have therefore found a highest-long-root operator of rank one in one simple ideal.

An irreducible representation of a direct sum of simple Lie algebras is an external tensor product of their irreducible representations. The rank of this root operator on T is its rank on the indicated factor multiplied by the dimensions of all other factors. Rank one forces those other factors to have dimension one. They act trivially, and faithfulness eliminates them. Thus \mathfrak h is simple.

All long roots are Weyl-conjugate. For every long root \alpha, the raising operator e_\alpha has rank one. Restriction to the root \mathfrak{sl}_2 is a direct sum of irreducibles; a summand of highest weight m contributes rank m to e_\alpha. There is consequently exactly one two-dimensional standard summand and otherwise only trivial summands. The Cartan subalgebra commuting with this root \mathfrak{sl}_2 acts on the unique standard summand by a single character. Thus the root reflection s_\alpha interchanges exactly two full Cartan weights, each of multiplicity one and differing by \alpha, and fixes every other full weight. This assertion does not yet exclude a zero weight of higher multiplicity.

Consider one irreducible component of the subsystem of long roots. Its simple reflections act by distinct transpositions on the set of weights. Adjacent simple roots have a product of reflections of order three, so their transpositions share exactly one vertex. Orthogonal simple roots give commuting transpositions. Two different simple roots cannot give the same transposition, since the difference of the transposed weights would be both roots up to sign. Connectedness of the diagram therefore gives a single connected moving set of vertices. The transpositions generate the full symmetric group on this set.

Map the augmentation representation on these vertices to the weight space by sending a vertex difference to the corresponding weight difference. The map is nonzero and equivariant. The augmentation representation of a symmetric group is irreducible, so the map is injective. Its image is exactly the span of the root component: consecutive vertex differences are its simple roots, and all differences lie in that span. Conjugating a simple-root transposition shows that every pair difference is a root. Conversely every root in this component arises this way, by conjugacy of its root reflections. The component is therefore of type A, with the moving weights the vertices of its standard simplex, up to their common centroid.

Distinct long-root components have disjoint moving sets. Their transpositions commute, so an overlap could only be an identical pair; the difference of that pair would then belong to two orthogonal root spans, which is impossible. The long roots span the entire Cartan dual. Hence the centroid of a moving simplex, being fixed by its component and by every other component, is zero. A weight moved by no long-root component is zero.

The ordinary classification of reduced irreducible root systems [12, Theorem 3.10] now leaves very few cases. For completeness, their long-root subsystems are

Here B_2=C_2 and the rank-one cases are A_1. For the non-simply-laced rows, this table follows directly from the coordinate roots: the long roots of B_\ell and F_4 are \pm\epsilon_i\pm\epsilon_j, those of C_\ell are \pm2\epsilon_i, and the long roots of G_2 form a regular hexagon. In the simply-laced cases every root is long. Thus only A, C, B_3 (where D_3=A_3), and G_2 are possible. The Weyl groups of B_3 and G_2 contain -\id. In those cases the nonzero weights would be respectively a tetrahedron or a triangle with centroid zero. Such a simplex is not preserved by -\id; adjoining zero weights cannot make it centrally symmetric. These two cases are excluded.

For type A_\ell, the remaining simplex weights are the standard weights or their negatives. The highest weight is therefore the first or last fundamental weight. Highest-weight uniqueness [12, Theorem 4.3] identifies T with the standard module or its dual. For type C_\ell, the long roots are \pm2\epsilon_j; each centered two-vertex simplex is \{\epsilon_j,-\epsilon_j\}. The highest weight is \epsilon_1, so highest-weight uniqueness gives the standard symplectic module. In both cases this last uniqueness statement excludes any additional zero-weight multiplicity. The root-system and highest-weight facts used here are the usual finite-dimensional characteristic-zero theorems; the rank-one and simplex deductions have been proved above.□

Proof. By Proposition 3.9, \mathfrak g is semisimple. The one-block algebra \mathfrak h_i of Lemma 3.10 is an ideal of \mathfrak g, hence semisimple. Split off its complementary ideal. On the irreducible dual of the ith block the representation is U\otimes W, with \mathfrak h_i acting on U. If \dim W>1, Lemma 3.11 makes the first prolongation zero, contrary to Lemma 3.10 for a nontrivial block. Therefore \mathfrak h_i acts irreducibly there. It acts faithfully, since its elements act by zero on every other block and the total representation is faithful. Lemmas 3.12 and 3.13 identify it with the full special linear or symplectic algebra in its standard module.

The ideals so obtained for distinct blocks commute, since each is zero on all the other blocks. A remaining complementary semisimple ideal would commute with these irreducible standard actions. By Schur’s lemma it would act by scalars on each block; a semisimple Lie algebra has no scalar character. It also acts trivially on the trivial blocks, so faithfulness eliminates it. If two nontrivial blocks were isomorphic as \mathfrak g-modules, an element acting trivially on one would act trivially on the other; their one-block ideals would be zero. This again contradicts Lemma 3.10.

It remains to check the group, rather than just its Lie algebra. Take the simply connected covering group of the connected semisimple group G. Its factors are the indicated \operatorname{SL} and \operatorname{Sp} groups. Each factor acts faithfully by its standard representation on its own block, and trivially on the others. Thus the kernel on the full representation is trivial, including any proposed central or diagonal central subgroup. The covering map is therefore an isomorphism and its image is the blockwise product in (3.5). Finally a trivial summand in the trace quotient is an actual free summand by Lemma 3.5, and conversely. This identifies s as claimed.□

Proof. For a standard \operatorname{SL}_m block with m>2, select its degree-one representation. The exterior powers in degrees 1,\ldots,m-1 are irreducible with distinct highest weights, while degrees 0 and m are trivial. The selected representation therefore occurs once and in degree one only.

For a standard \operatorname{Sp}_{2m} block L_i, select

where the arrow is symplectic contraction and negative exterior powers are zero. The primitive decomposition

has irreducible, mutually distinct primitive terms P_i^s of highest weight \varpi_s [20, Theorem 17.5]. Thus P_i^m occurs once and only for b=m. This includes \operatorname{SL}_2=\operatorname{Sp}_2.

Take the external tensor product of the selected representations. It is simple for the product group and occurs in \Lambda^\bullet W once, in a single degree j_0. In \Lambda^{j_0}V_{\mathrm{fib}} it still has multiplicity one: a positive exterior degree from the trivial summand would require its occurrence in a smaller degree of W. The factorwise rule is preserved by permutations. In the symplectic case the kernel defining P_i^m is also unchanged when the symplectic form is rescaled, so the selection has the asserted intrinsic meaning.□

Proof. We first prove the last assertion, which is stronger than the analogous statement for ordinary differentials. A derivation has a local holomorphic flow: after embedding the germ in a polydisc, lift its coordinate values to an ambient vector field and solve the holomorphic differential equation. Preservation of the defining ideal makes the flow preserve the embedded analytic space. If the derivation evaluates to zero, its flow \phi_z fixes v.

Choose a finite smooth cover a:(Y,y)\to(V,v) and consider H(z,w)=\phi_z(a(w)). The map (z,w)\mapsto(z,H(z,w)) is finite: it is \id_\Delta\times a followed by the flow automorphism of \Delta\times V. Thus the inverse image of \Delta\times V_{\mathrm{sing}} has codimension at least two. The pullback of \beta from V_{\mathrm{reg}} consequently extends to a holomorphic one-form \alpha on the smooth space \Delta\times Y.

We claim that \alpha restricts to zero on C=\Delta\times\{y\}. To check this, pull a fixed finite smooth cover a_0:Y_0\to V back by H, normalize a component dominating \Delta\times Y, and resolve it. The resulting diagram has a proper dominant map q:

The pullback of \beta extends to a holomorphic form \beta_0 on Y_0. On \widetilde W the forms q^*\alpha and g^*\beta_0 agree, since they agree on a dense open. An irreducible subvariety of q^{-1}(C) dominating C maps under g to a point of the finite fiber a_0^{-1}(v). Restriction to a resolution of that subvariety therefore annihilates q^*\alpha. Pullback of one-forms from the smooth curve C under a dominant map is injective in characteristic zero, proving the claim. Finally, the definition of the flow gives the identity

It holds first over the regular locus and then everywhere by holomorphic extension. Evaluation at y, using \alpha|_C=0, yields \beta(\xi)(v)=0.

Suppose that \Omega_V^{[1]}{}_v has a free summand with basis \beta_1,\ldots,\beta_s and dual vector fields \xi_1,\ldots,\xi_s. A linear combination of their evaluations that vanishes pairs to zero with every \beta_j by the preceding paragraph. The identities \beta_j(\xi_i)=\delta_{ij} imply independence of the evaluations. Thus s\leq r(v).

Conversely choose vector fields with r=r(v) independent evaluations and regular functions z_1,\ldots,z_r with \xi_i(z_j)(v)=\delta_{ij}. Cut a transverse germ by z_1=\cdots=z_r=0. Successive flows of the chosen fields define a map from a polydisc times this transverse germ to V. To invert it, apply the inverse flows and solve for their times by the analytic implicit function theorem; its Jacobian in the time variables is the identity at v. This argument works in an ambient embedding and hence also on the embedded germ. The resulting product has r free cotangent summands. The transverse factor cannot have a vector field with nonzero evaluation, since such a field would give an additional independent evaluation on the product.□

Proof. Coherence of the sheaf of derivations, followed by evaluation on finitely many ambient coordinates, expresses the condition by minors of a matrix of holomorphic functions. For the dimension assertion we use the following trace observation. If \pi:(Y,y)\to(V,v) is a finite map from a smooth germ, with this single point over v, and m is its degree, then a vector field \eta on Y induces the derivation

The trace is holomorphic by integrality and normality, and the Leibniz rule follows from linearity of trace over \cO_V. At v, the trace of a holomorphic function on this local branch is m times its value at y. Consequently the evaluation of (4.2) is exactly d\pi_y(\eta_y). A local branch can be isolated analytically even if the original finite cover has several points over v. We obtain

Let Z be an irreducible subvariety of the rank locus. A component of its reduced inverse image under a finite smooth cover dominates Z. At general smooth points of this component and of Z, the restricted finite map has differential rank \dim Z, by characteristic zero. Equation (4.3) therefore gives \dim Z\leq a.□

Proof. Smooth factors persist under further finite local quasi-étale covers. Indeed such a cover is étale over the smooth locus by purity, and a cover of \Delta^r\times T_{\mathrm{reg}} is pulled back from T_{\mathrm{reg}}, since the polydisc is simply connected. Normalization extends the product across the omitted codimension-two set. More precisely, if c' is any nearby point on a finite quasi-étale chart over b', its pointed local branch is dominated by a maximal cover of (B,b'). Each of its r(c') smooth factors persists on that cover, so r(c')\leq s(b'). A product with r smooth directions therefore implies s\geq r throughout its image. Each pointed finite branch dominates its base germ, so the image of a sufficiently small neighborhood contains a neighborhood downstairs. Along the smooth factor through its marked point, the transverse germ is unchanged; the same description of covers shows that its maximal-cover property is unchanged as well. Hence s is lower semicontinuous in the analytic topology and is constant along that factor.

At a minimum point choose the product in Lemma 4.1. The preimage of the minimum locus lies in the evaluation-rank locus r\leq d: any extra smooth factor at another point of this chart would persist on its maximal cover. The rank locus on the full chart is the product of \Delta^d with the transverse rank-zero locus Z. Apply Lemma 4.2 to this full chart, which is finitely dominated by a smooth germ. It gives d+\dim Z\leq d, so the germ of Z at the marked point is just that point. Conversely the whole smooth factor has the original maximal transverse type. This proves (4.4) and also proves that the minimum locus is locally analytic, rather than merely topologically closed.

Its closedness and projectivity of B make it algebraic by Chow’s theorem. Locally it is a finite quotient of the smooth orbit by its deck stabilizer, and hence is normal. Distinct irreducible components of a normal variety are disjoint, so a connected component is irreducible. If d=n, Lemma 4.1 makes every maximal chart smooth, which is exactly the quotient-singularity assertion. The construction of the stack, with its specified stabilizers, is included in Proposition 4.5 below.□

Proof. Choose algebraic functions z_1,\ldots,z_d whose restrictions give independent cotangents on S at v, and define T_0 by their vanishing. In the given analytic product these functions may replace the polydisc coordinates: their Jacobian in those coordinates is invertible. The analytic implicit function theorem, with the transverse germ retained, identifies the resulting transversal with T and gives a formal isomorphism of the algebraic pointed pairs in (4.5).

Here is why approximation can retain the marked ideal exactly. Write finite algebra presentations for the target product and generators a_1,\ldots,a_h and b_1,\ldots,b_l for its marked ideal and the ideal of S in V, respectively. Besides the target’s defining equations, include unknown coefficients in the finitely many equations

The formal pair isomorphism supplies a solution. Artin approximation over the henselization of the algebraic local ring produces a solution in an étale neighborhood, arbitrarily close to that formal solution; this is the polynomial-equation form of [56, Tags 0CAU and 0CAV]. Denominators in local presentations can be included by equations for their inverses. The displayed equations give the two ideal inclusions exactly.

Choose approximation modulo the square of the maximal ideal. Composing the resulting map on completions with the original formal inverse gives an endomorphism that is the identity on residue field and cotangent space. It is the identity on the associated graded ring, which is generated in degree one. Induction on the powers of the maximal ideal and completeness show that it is an isomorphism. Thus the second map in (4.5) induces an isomorphism of completed local rings and is étale at the marked point; shrink to its étale locus. This verifies the inverse-map issue even for a singular transversal.

The restriction S_U\to\mathbb A^d is étale. Both U_k and S_U\times\Spec A_k are étale over \mathbb A^d\times\Spec A_k and have the same reduction over \mathbb A^d. Uniqueness of lifting morphisms between étale schemes through a nilpotent thickening gives (4.6) [56, Tag 025H]. The same uniqueness makes the isomorphisms compatible as k varies. Notice that this last argument uses the equality of the two ideals, not closeness of those ideals to some finite order.□

Proof. Start with a chart maximal at one point, as in Proposition 3.3. By (4.4) and Lemma 4.4, its transverse analytic type is constant along the part of S_\alpha covered by one common étale product neighborhood. The image is open in S_\alpha, because an étale map is open. Every point there still has a maximal germ, since the transverse germ and its covers have not changed. Take the union of translates of this open neighborhood around the finite deck orbit over the base point. The complement in S_\alpha has closed image under the finite chart map. Removing that image downstairs arranges maximality at every point over the remaining stratum. Quasi-compactness of S_0 gives the finite list in (1). Applying Lemma 4.4 at its inverse images and using quasi-compactness gives a finite family of product refinements as in (3).

We construct the groupoids while keeping the full finite normalizations. Put V=\coprod_\alpha V_\alpha. Choose a common dense open B^\circ\subset B_{\mathrm{reg}} over which all the charts are étale, and write V^\circ=V\times_BB^\circ. For a=2,3,4, let W_a be the reduced union of the irreducible components of V^{\times_B a} meeting (V^\circ)^{\times_{B^\circ}a}, and let

be its full normalization. It is finite over the closed subscheme W_a\subset V^{\times_Ba}. Its restriction over B^\circ is the ordinary product, which is étale over the smooth space B^\circ and therefore normal. Write \overline R=\overline V_2, with projections s,t:\overline R\to V.

The projections of these normalizations are étale at every point over S_0. To see this locally, pass to a common étale base neighborhood for the chosen object charts. A projection is there a finite quasi-étale map followed by an étale map: away from the codimension-two branch loci it is a base change of the original charts, and finiteness preserves that codimension bound. Each pointed target chart over S_0 is maximal, so every connected local finite quasi-étale factor is trivial. This proves the asserted étaleness.

We will also need an arrow through every pair of object points over a point of S_0. Pass to the common strict henselian base germ at that point and select the pointed local factors of the two finite charts. By the algebraic–analytic comparison in Proposition 3.3, the maximality at these points also gives maximality of the selected strict henselian factors. They are therefore isomorphic over this germ. The graph of such an isomorphism is a main branch of their product and passes through the chosen pair. Normalization commutes with this strict henselian base change: the base change of a finite normalization is normal under étale localization and has the same generic normalization property, and the assertion passes to the filtered strict henselian limit. The graph branch therefore supplies a point of \overline R over the chosen pair. This proves the assertion for the original algebraic charts.

Let R\subset\overline R be the open where both s and t are étale. It contains the entire inverse image of S_0. Let I_\alpha be the reduced ideal of S_\alpha and let I be their union on V. On the full normalization define the extended ideals

Their zero sets are both the inverse image of S_0. On R they are equal: the two étale pullbacks are reduced and have that same support. Let V_k\subset V be defined by I^k, and define R_k to be the closed subscheme of the full \overline R defined by J_s^k. Its entire support lies in R, so

In particular its two maps to V_k are étale. The map to V\times_BV factors through V_k\times_BV_k, because both extended ideals vanish to order k on R_k. The resulting map R_k\to V_k\times_BV_k is finite: it is a closed quotient of the full finite normalization, with a factorization through that closed subscheme. This is the finiteness used below.

Here are the normalization comparisons needed for composition. Put

It is normal, since it is étale over the normal object charts. Every irreducible component of P meets the open over B^\circ: its image under an étale projection to an object component is a nonempty open, and therefore meets the dense V^\circ. Over B^\circ, the scheme P^\circ is an open subscheme of the ordinary triple product (V^\circ)^{\times_{B^\circ}3}. Hence each component of P maps dominantly onto a main component of W_3. Its generic map lifts to that component of \overline V_3, and the lift extends over the normal component of P. Indeed, each element of the finite normalization is integral over the local target ring; its generic pullback is integral over a normal local ring of P and so belongs to that ring. This constructs P\to\overline V_3.

The three pair-forgetting maps \overline V_3\to W_2 lift in the same way to maps \rho_{01},\rho_{12},\rho_{02}:\overline V_3\to \overline R. Componentwise dominance here follows on B^\circ: each projection of the ordinary triple product is étale, and its nonempty image is open in the corresponding ordinary pair component. Let

It contains the entire inverse image of S_0, and its two adjacent pairs give a map O_3\to P. Conversely, the adjacent pair maps of P\to\overline V_3 agree with the two projections P\to R on the dense unbranched open, hence on P; the lift therefore lands in O_3. This map and O_3\to P are inverse between O_3 and P. Their composites are the identity on the componentwise dense ordinary unbranched products; equality extends because the sources are reduced and the targets separated. Restrict further to O'_3=O_3\cap\rho_{02}^{-1}(R), which still contains the entire closed support. The map \rho_{02} gives composition there.

On P the three extended ideals from the object coordinates are equal, by the equality J_s=J_t on each arrow. Call their common ideal J. Base change by the closed immersions therefore gives the scheme equality

Its support is the whole inverse image of S_0 in P, hence lies in the domain corresponding to O'_3. The three ideals agree there also along \rho_{02}, so composition restricts to R_k\times_{V_k}R_k\to R_k for every k.

For associativity use R\times_VR\times_VR and \overline V_4. The same argument applies component by component: the former is normal, and its unbranched part is open in the ordinary quadruple product and meets every component. Each component therefore dominates a main component of W_4, while all pair-forgetting projections are dominant onto their main pair components there. It identifies with the open of \overline V_4 where the adjacent pairs 01,12,23 land in R. Restrict to the further open where 02,13,03 also land in R. Every one of these opens contains the entire inverse image of S_0. Both parenthesizations are defined on this last open and agree on its componentwise dense unbranched product, hence agree everywhere. Their restrictions agree on every quotient by the common kth power of the stratum ideal. Interchanging the two coordinates lifts to the normalization \overline R and gives inversion on the open where both the original and interchanged arrow land in R. The diagonal of the ordinary unbranched pair product is open and closed; its main branch lifts the diagonal map from the normal V to \overline R. Restricting to the inverse image of R gives the unit. These domains contain the entire closed support, by the étaleness of all pair points over S_0. The unit and inverse identities follow on their support-containing domains by the same dense-open comparison and then restrict to all k. Thus R_k\rightrightarrows V_k is an étale groupoid. Four factors suffice because associativity is the only law involving three composable arrows.

Since B is separated, V_k\times_BV_k is closed in V_k\times V_k. The finiteness proved above therefore makes the diagonal of [V_k/R_k] finite. This quotient is a separated Deligne–Mumford stack with finite inertia. Its local finite quotient presentation can be read after base change to B_\alpha. The diagonal B_\alpha\to B_\alpha\times_BB_\alpha is open and closed, since B_\alpha\to B is separated and étale. On that component the self-overlap is the normalization of V_\alpha\times_{B_\alpha}V_\alpha, namely \coprod_{\gamma\in\Gamma_\alpha}V_\alpha with its distinct deck labels. The existence of a main arrow through every pair of closed object points, proved above, makes the other object charts étale refinements of this presentation: the relevant arrow projections are étale and surjective on the closed support, which is the underlying support of every V_k. Consequently over B_\alpha the stack is

Its coarse algebra is precisely

Here the intersection is the contracted ideal in the finite chart algebra; exactness of finite-group invariants in characteristic zero gives the equality. These finite algebras glue étale locally. They give a coarse space finite over the projective variety B, and therefore projective. A quotient by a finite group is proper over its coarse space, so the stack is proper. At k=1 the coarse space is S_0 and the object charts are smooth, giving \mathcal S_1=\cS.

The construction keeps the label of every \gamma\in\Gamma_\alpha even when it fixes the whole smooth factor. Stabilizers may increase on closed subsets of S_\alpha, and remain subgroups of these fixed finite deck groups. The ambient groupoid laws and their restrictions therefore retain the full stabilizers on \cS.

For (4), write X_\alpha\to V_\alpha for the normalized source chart. Proposition 3.3 makes X_\alpha\to X\times_BB_\alpha a finite étale \Gamma_\alpha-torsor. The normalized overlaps give its ordinary descent identifications. The ideals I_\alpha^k\cO_{X_\alpha} descend to a coherent ideal near f^{-1}(S_0); gluing it with the unit ideal off this closed set gives a closed subscheme X_k\subset X. This is the descended ideal quotient, with no identification with a fiber product over the coarse thickening required. Put X_{\alpha,k}=X_\alpha/I_\alpha^k\cO_{X_\alpha}. The local diagram

is 2-Cartesian. Its lower left quotient is the descended scheme X_k\times_BB_\alpha, because the source action is a torsor. These diagrams glue to a map f_k:X_k\to\mathcal S_k. For a coherent sheaf M on \mathcal S_k, the ordinary sheaves \cO_{X_{\alpha,k}}\otimes_{\cO_{(V_\alpha)_k}}M_\alpha descend to a coherent sheaf on X_k, and then on X. Increasing k does not change this pullback, denoted f^*M. The displayed Cartesian diagram is also the reason that ordinary chart base change and the canonical higher-direct-image coefficient maps descend. After the product refinements in (3), these constructions are their ordinary étale pullbacks from the finite charts.

All schemes, full finite normalizations, group actions, product maps, marked ideals and maps in the diagrams with at most four factors are of finite type. Their identities, ideal inclusions, and the disjointness of each omitted closed complement from its closed support are finitely many algebraic conditions. Individual marked pairs were approximated in Lemma 4.4; their transition maps are the normalized algebraic maps just constructed. This proves the asserted finite nature of the compatible data.□

Proof. In our applications the transversal T_0 is normal near t_0. Its analytic germ is the transverse factor of a normal product, hence normal. Its completion is normal by excellence and is also the completion of the algebraic local ring; faithful flatness of completion then descends normality to that ring. Shrink T_0 accordingly. On a product \mathbb A^d\times T_0, restriction to the smooth locus of the normal transversal gives

Both sides are reflexive, so the equality extends across codimension two. It remains valid under the common étale pullbacks. Equation (4.6) gives the claimed description on each whole thickened chart. For finite modules, flat base change commutes with Hom, as follows from a finite presentation; it also commutes with kernels and images. It therefore commutes with double duals and the fixed reflexive tensor constructions here. Composition and contraction are preserved, and trace can be checked on the dense locally free locus. The trace pairing on the reduced smooth factor is thus the scalar extension of a pairing of finite-dimensional complex vector spaces. Its kernel and quotient are vector bundles there. This proves the claim about barred Hom.

We make explicit how the residual tensor category of the full maximal chart is used in these transverse calculations. At a product point put

The completion of the full chart cotangent module is E\otimes_RS. For finite reflexive tensor objects P,Q generated by E, flat base change gives

and also commutes with their reflexive tensor operations, composition and trace. Reduction modulo \mathfrak m_S=(\mathfrak m_R,t_1,\ldots,t_d) gives the same trace pairing as reduction over R modulo \mathfrak m_R. Its kernels and residual Hom spaces therefore agree. The same flat Hom calculation compares the analytic local ring of the full chart with its completion. Idempotent lifting over the henselian rings, as in Lemmas 3.4 and 3.5, identifies their idempotent-complete residual tensor categories. Thus the residual category of the full maximal chart is transported by scalar extension to the category computed from the transverse module E. Theorem 3.14 and Corollary 3.15 are applied on the full chart and carried to the transverse calculation by this identification.

One may now choose an exponent on each of the finitely many transversals and take their maximum. Equalities of coherent maps, surjectivity, and specified powers annihilating a kernel descend from faithfully flat completions. This proves precisely the asserted uniformity for the fixed finite diagrams.

For the last assertion, the stratum ideal in S is \mathfrak m_RS. If t'_1,\ldots,t'_d restrict to another coordinate system on S/\mathfrak m_RS, substitution t_i\mapsto t'_i, fixing R, is an automorphism of this complete ring by the formal inverse function theorem. It preserves the stratum ideal. Completed differentials and their reflexive tensor maps are functorial for this automorphism, so it transports the completed Hom, trace and ideal data as well.□

Proof. The restriction C^\circ\to B_{\mathrm{reg}} is étale. Indeed it factors through the smooth U\times_BB_{\mathrm{reg}}, and purity makes the finite quasi-étale factor étale there. Its complement in C has codimension at least two. Over C^\circ the normalized source is the ordinary base change X\times_BC^\circ: that base change is smooth and normal, all its components dominate C^\circ by equidimensionality, and it has the integral generic fiber used in Proposition 3.3. Proper flat base change and étale pullback of differentials therefore pull \phi_j to

This open includes the entire inverse image of the discriminant over the regular part of B; only the singular locus of B has been removed.

The discrepancy argument in Proposition 3.3 applies to any such quasi-étale \pi, so C is klt. In particular it is Cohen–Macaulay and \Q-Gorenstein. The same proposition gives a projective equidimensional map h with connected fibers and smooth canonical-trivial source. Ou’s local reflexivity theorem [46, Theorem 1.3] therefore makes R^jh_*\cO_Z reflexive on C. Both sides of the displayed isomorphism are reflexive and the omitted set has codimension at least two, so it extends uniquely to \phi_{j,C}. At j=0 it is the Stein unit.

Ordinary étale base change preserves both sides. On a normalized main overlap with étale projections, every component meets the inverse image of B_{\mathrm{reg}}, since a main component meets the dense common unbranched open. The two pullbacks there are the same pullback of the fixed global \phi_j. They consequently agree on the entire normal overlap by reflexivity. This also proves compatibility with deck actions and with further étale refinements.□

Proof. Tensor the two maps and their chain homotopies with M. They remain homotopy inverses between C\otimes_RM and D_0\otimes_RM. The maps \gamma_M^i are natural in chain maps, and \gamma_M^i(D_0) is the identity on D_0^i\otimes_RM. Naturality under the two homotopy equivalences proves the assertion for C. This uses no flatness of M, of D_0, or of the terms of C.□

Proof. Suppose first that the characteristic is p>0. Write d=\dim L and \tau=\operatorname{trdeg}_k\kappa(L), where k is the perfect ground field. All residue fields of the A_i are finite extensions of \kappa(L), so their p-degrees are p^\tau. For q=p^a, finite flat Frobenius on a regular local ring, as in Kunz’s theorem [32], gives

Here the module structure on the left is through x\mapsto x^q. The formula follows either from a regular system of parameters and a residue-field q-basis, or by taking the degree of Frobenius on the fraction field. Completion does not change this formula. In particular, the exponent is the same for both algebras, including at a nonclosed point.

The two modules in (5.4) have the same generic rank. Clearing denominators in a generic isomorphism and its inverse gives maps

for one nonzero r\in L. We estimate the size of an invertible block of F_*^a(r) on F_*^aA_i, viewed as a free A_i-module. If \mathfrak n_i is the maximal ideal, its cokernel needs

generators. Since \mathfrak n_i has d generators, \mathfrak n_i^{d(q-1)+1}\subset\mathfrak n_i^{[q]}. The Hilbert–Samuel polynomial of A_i/rA_i, of dimension at most d-1, therefore bounds this number by O(q^{d+\tau-1}). The case d=0 is immediate, since r is a unit. Elementary invertible row and column operations over A_i now exhibit an identity block of size q^{d+\tau}-O(q^{d+\tau-1}) in F_*^a(r).

Apply F_*^a to u,v. The identity block just found, repeated d_2 times, factors through F_*^a(A_2^{\oplus d_1}). Consequently

Finite modules over the complete local ring L have Krull–Schmidt decompositions. For each indecomposable isomorphism class T, let m_i(T) be its multiplicity in A_i. The displayed summand relation, divided by q^{d+\tau} and followed by a\to\infty, gives d_2m_1(T)\le d_1m_2(T). Interchanging the indices gives equality. This proves (5.4).

Here are details of the characteristic-zero passage. Henselize the algebraic base at the chosen point. The idempotents separating the finitely many points of a finite algebra descend over an étale neighborhood. Thus individual factors can be treated in a finite algebraic diagram; after shrinking that diagram they remain regular and dominant over a normal integral base. Let M=A_1^{\oplus d_2} and N=A_2^{\oplus d_1} denote the corresponding finite modules before completion. Choose finite generators \phi_1,\ldots,\phi_s of the ordinary module \Hom(M,N). Its completion is \Hom(\widehat M,\widehat N), because M is finitely presented and completion is flat.

An isomorphism \widehat M\simeq\widehat N exists precisely when some linear combination of the \phi_i is surjective modulo the maximal ideal. For the forward implication, reduce the coefficients of a completed isomorphism. For the converse, Nakayama gives surjectivity, and a surjective map of the same generic rank between torsion-free modules has zero kernel. Existence of the required linear combination is a finite matrix condition: at least one maximal minor of \sum z_i\overline\phi_i is a nonzero polynomial in the z_i.

Spread the finite presentations, the marked point, and this Hom calculation over a finitely generated characteristic-zero domain. Generic freeness applied to the kernels and cokernels of the finite presentation matrices makes the Hom calculation and its reduction at the marked point commute with specialization. Retain normality, regularity, ranks, and the dimensions of the two residual modules. If all the maximal-minor polynomials vanished in characteristic zero, their finitely many coefficients would vanish in every remaining specialization. At any good positive-characteristic specialization, however, the first part supplies an isomorphism of the completed modules. This is a contradiction. A factor that splits further under specialization causes no difficulty: the first part applies to all its regular factors and their sum. Finally, comparisons with one fixed factor and addition prove the assertion for finite products.□

Proof. Set \mathcal K_0=\mathcal P. Relative generation on projective space constructs n+2 exact sequences

where each \mathcal F_r is a finite sum of \mathsf H_Y(-a_{rj}). To obtain such a surjection, choose ordinary coherent generators after twisting and extend their maps \mathsf H_Y-linearly. Since \mathcal K_r is locally projective over \mathsf H_Y, the sequence locally splits over that algebra. Induction shows that every kernel \mathcal K_{r+1} is again locally finite projective over \mathsf H_Y.

The sheaf \mathsf H_Y is flat as a right \mathsf H-module: its tensor functor is \cO_Y\otimes_R-, and Y is R-flat. Therefore all the kernels just constructed are \mathsf H-flat, and every sequence in (5.10) stays exact after tensoring with an arbitrary Q. Choose m_0\ge a_{rj} for all the finitely many twists. For m\ge m_0 each \mathcal F_r(m)\otimes_{\mathsf H}Q is a sum of \cO_Y(m-a_{rj})\otimes_RQ. The coordinate Čech calculation on projective space gives zero higher cohomology for these sheaves for every R-module Q; their sections are the degree m-a_{rj} homogeneous polynomials with coefficients in Q.

For j>0, the long exact sequences consequently give

The n+1 standard affine opens of Y have affine intersections, so quasi-coherent cohomology vanishes in degrees greater than n. Thus the left side is zero for r=0,1,2 and j>0. In particular this proves (5.8). We have made no assertion about the projective dimension or the low-degree cohomology of the final kernel \mathcal K_{n+2}.

The first two sequences, together with the vanishings for \mathcal K_1 and \mathcal K_2, give

For Q=\mathsf H this is a finite presentation of G_m=H^0(Y,\mathcal P(m)) by finite free right \mathsf H-modules. Right exactness of tensoring this same presentation proves (5.9), with its canonical map. Finally, tensoring any short exact sequence of coefficient modules with \mathcal P(m) remains exact by \mathsf H-flatness, and taking sections remains exact by (5.8). Equation (5.9) says that G_m\otimes_{\mathsf H}- is exact. Hence G_m is flat; being finitely presented by (5.12), it is projective.□

Proof. Use the constant polarizable Hodge module on Z in right \mathcal D_Z notation. Its underlying module is \omega_Z, with lowest filtration index -2n. The filtered direct-image construction and projective strictness give

To track the index, use the transfer Spencer resolution. Its term in degree -i is filtered by pieces with source filtration lowered by i and by the nonnegative order of target differential operators. At total index -2n, all i>0 terms vanish; in degree zero only order zero remains. The lowest filtered complex is consequently \mathbf Ra_*\omega_Z. Strictness identifies its degree-zero cohomology with the left side of (5.20). These conventions and strictness are the ones in [49, Sections B.2–B.3 and Theorem 1.4], the latter being [48, Theorem 5.3.1].

The strict-support decomposition is compatible with the filtration. The sheaf a_*\omega_Z is torsion-free on P: multiplication by a nonzero base function is injective on the dominant integral smooth source. Thus its summands on proper supports are zero. After changing from right to left modules by \omega_P^{-1}, the sheaf R\eta lies in the full-support part of \mathcal H^0(a_+\omega_Z), which is generically the n-th relative de Rham cohomology. This part is an intersection extension of semisimple algebraic local systems, hence a semisimple regular holonomic \mathcal D_P-module with no simple constituent on a proper support. Here we use strict support [49, Section 1.2] and algebraic semisimplicity [16, Proposition 4.2.5 and Theorem 4.2.6]; restricting to an arbitrary small analytic open would not justify this semisimplicity assertion.

Inside that semisimple module let U and U_g be the \mathcal D_P-submodules generated by R\eta and gR\eta. At the generic point of P, the finite domain R becomes the field \operatorname{Frac}(R), so g is invertible and these restrictions agree. The quotient U/U_g is supported properly. Semisimplicity makes it a sum of constituents of the full-support module, so it is zero. Consequently \eta belongs locally to \mathcal D_P(gR\eta). Writing this membership using finitely many generators gives (5.18). Quasi-compactness gives a finite principal-open cover and a finite maximum o_g for the orders. On an open D(f), a local section of \mu_*\cO_{\Spec R} has the form r/f^m with r\in R. Absorb multiplication by f^{-m} into the operator acting on gr\eta; its coefficients remain regular on D(f) and its order does not increase. Thus the t_j may be taken in R. Algebraic differential operators on these opens of affine space have the expressions (5.19); their finitely many coefficient denominators are units on the corresponding opens. The restriction of the identity to the generic smooth locus involves only algebraic de Rham classes, connections, and finite-order operations, and thus is finite algebraic data that can be reduced after localization.□

Proof. Put b=b_2(X) and choose a prime \ell>2b. After a finite extension, both the residual representation on a stable lattice of H^2_{\mathrm{et}}(X_{\overline K},\Q_\ell) modulo \ell and \mu_\ell are constant. At a degree-one good place of norm p, geometric Frobenius has an integral trace a_p satisfying

We use integrality and the weight-two Weil bound, and the equality of the étale and crystalline characteristic polynomials [28, Theorem 1], using its formulation and the crystalline applicability stated in [41, Chapter VI, Theorem 3.9 and Section 4].

Discard the finitely many ramified places of the number field and the places where relative de Rham cohomology or its base change has torsion. At a degree-one remaining place the local base is \Z_p. Crystalline–de Rham comparison for the smooth proper lift identifies the reduction of the crystalline trace with the trace of Frobenius on H^2_{\mathrm{dR}}(X_{\F_p}) [7, Corollaries 3.8 and 3.10]. Modulo p, Frobenius on the de Rham complex factors as its projection onto \cO_X followed by the map taking functions to their p-th powers in degree zero. On cohomology write this factorization as \beta\alpha through H^2(\cO_X). The reverse composite \alpha\beta is coherent Frobenius. The identity \tr(\beta\alpha)=\tr(\alpha\beta) shows that zero coherent Frobenius forces p\mid a_p.

In that case a_p/p is an integer in [-b,b] congruent to b modulo \ell. Since \ell>2b, it equals b, so a_p=bp. If coherent Frobenius vanished at all but finitely many degree-one good places, the semisimplification of H^2_{\mathrm{et}}(X_{\overline K},\Q_\ell) would therefore have the same character as \Q_\ell(-1)^{\oplus b}. Indeed degree-one places have density one among primes of a number field; the others have norm at least the square of the underlying rational prime and density zero. Chebotarev for number fields and the characteristic-zero character criterion [51, Theorem 3.2 and Section 5.1.1.3] give the assertion: finite-quotient Frobenius density and continuity identify the two trace functions in the projective limit.

This contradicts Hodge–Tate comparison at a place above \ell: h^2(\cO_X)=1 contributes a weight-zero summand, whereas all Tate summands just found have weight one, in the convention where \Q_\ell(-1) has weight one. Comparison applies to the smooth proper variety over that local field [50, Theorem 1.6 and Corollary 1.8]; de Rham representations are stable under subquotients and their filtered dimensions are additive in exact sequences [11, Theorem 5.2.1(2) and Propositions 6.3.2–6.3.3], so passing to semisimplification has not lost this obstruction. Thus coherent Frobenius is nonzero at infinitely many, and hence arbitrarily large, good residue characteristics.

By (5.21), its n-fold product is nonzero on top coherent cohomology. Serre duality identifies the transpose of this map with Cartier trace on global top forms. The space of global top forms is the frame line and X is geometrically integral and proper, so (5.22) sends 1 to a nonzero constant.□

Proof. Fix b and one of the principal opens in Lemma 5.5 containing \mu(b). In this proof t_j,D_j denote its data, so the t_j are regular at b and the coefficients of the D_j are regular at \mu(b).

First work over K_0=\operatorname{Frac}(R), where the family Z_0=Z\times_RK_0 is smooth and proper. The retained étale locus of \mu identifies du with a basis of \omega_{K_0/k} and gives the relative frame \eta=\sigma_Z/du. Serre duality gives dual lines

Put

Flat field base change identifies these twisted vector spaces with H^n(Z_0^{(p)},\cO_{Z_0^{(p)}}) and H^0(Z_0^{(p)},\omega_{Z_0^{(p)}/K_0}), respectively. Write v^{(p)}=v\otimes1 and \eta^{(p)}=\eta\otimes1. They remain dual under the base-changed Serre pairing. Relative Frobenius F_{\mathrm{rel}}:Z_0\to Z_0^{(p)} and its finite dual trace have the following K_0-linear maps on these lines:

Here the second map is induced by F_{\mathrm{rel}*}\omega_{Z_0/K_0}\to\omega_{Z_0^{(p)}/K_0}. It is the Serre-dual transpose of the first, which proves that their scalars are the same s\in K_0.

Taking p-th powers of a Čech cocycle for v gives a class \lambda in middle relative de Rham cohomology. Its projection to H^n(Z_0,\cO_{Z_0}) is F_{\mathrm{rel}}^*v^{(p)}. Pairing \lambda with the top form \eta, with the order that gives Serre evaluation, therefore gives \langle\lambda,\eta\rangle=s. This distinguishes the coherent Frobenius map in (5.25) from its de Rham lift. The lift is horizontal: its p-power representatives are closed even for absolute differentials over k, so the connecting operation defining the Gauss–Manin derivative annihilates them. The de Rham pairing is compatible with the connection. Pairing the reduced identity (5.18) with this horizontal class gives

The pairing, connection, and (5.18) on the smooth generic family are reductions of their finite characteristic-zero descriptions. The scalar identity (5.26) is obtained in this fiber from the newly constructed horizontal class.

Factor absolute Frobenius as Z_0\xrightarrow{F_{\mathrm{rel}}}Z_0^{(p)} \xrightarrow{w}Z_0, where w is the base change of F_{K_0}. This is the relative Frobenius factorization of [56, Tag 0CC6]. The relative trace in (5.25) sends m\eta to sm\eta^{(p)}. The trace for w, by flat base change and the projection formula on top forms, applies the Cartier map \operatorname{Car}_{K_0}:F_*\omega_{K_0/k}\to\omega_{K_0/k} to its remaining base form. Transitivity of finite trace therefore gives the rational identity below; compatibility of trace with this flat base change is [56, Tag 0E5L].

We suppress F_* in the input notation for \psi_h. The scalar s stays inside the base Cartier operator. At this point it is only a rational element of K_0.

Suppose (5.24) fails, so \psi_h(gR_b)\subset\mathfrak m_{R_b}. Complete the regular base at \mu(b):

The finite completed algebra R\otimes_{k[u]}A is a product of the completed local rings at the points above \mu(b). They are normal complete local domains; keep the factor L at b and put K'=\operatorname{Frac}(L). After tensoring with K, the whole product is finite étale of total dimension e_\mu: the nonzero \Delta_\mu is invertible in K, and its free étale rank was retained. Thus its factors are finite separable fields and

Write T=\operatorname{Tr}_{K'/K}.

We next transport (5.27) to these fields; this step does not require s to be regular in L. Finite Frobenius and the cofinality of the ordinary and Frobenius adic filtrations give F_*R_b\otimes_{R_b}L\simeq F_*L, by (F_*r)\otimes x\mapsto F_*(rx^p). Hence \psi_h extends to L and then to a K'-linear map F_*K'\to K'. The monomials v^\alpha, 0\le\alpha_i<p, are a p-basis of K: decompose power series according to their exponents modulo p and use perfection of k. They remain a p-basis of the finite separable extension K'. They are also a p-basis of the embedded K_0, since K_0 is finite separable over k(u_1,\ldots,u_n) and translation by constants of k changes u_i to v_i. The Cartier maps computed in these bases agree on K_0. The extended \psi_h and m\mapsto\operatorname{Car}_{K'}(F_*(sm\,du))/du agree on the p-basis monomials by (5.27); they are both K'-linear maps on F_*K', so they agree everywhere. The same uniqueness of separable extension shows that the extended derivations in (5.26) agree on K_0, so that identity also holds in K' with these operators.

The failed containment extends to \psi_h(gL)\subset\mathfrak m_L. For every base monomial a=v^\beta and each t_j it follows that

Indeed T(L)\subset A by integrality and normality. An element of \mathfrak m_L has all its conjugates in the maximal ideal above \mathfrak m: the integral closure in a finite normal closure is finite over the complete regular ring A and is local by henselianity. Their sum consequently lies in \mathfrak m.

The common p-basis also proves that Cartier commutes with this rational trace. If x=\sum v^\alpha c_\alpha^p in K', then T(x)=\sum v^\alpha T(c_\alpha)^p, since T(c^p)=T(c)^p. Cartier for du=dv_1\wedge\cdots\wedge dv_n extracts the coefficient with \alpha=(p-1)\mathbf1 in either field. Hence

Derivations likewise commute with T, by uniqueness of their separable extensions. Combining this identity with (5.27) on K' and (5.29), and setting z_j=T(gt_js)\in K, gives

Write uniquely

Multiplication by v^{(p-1)\mathbf1-\alpha} followed by Cartier extracts exactly c_{j,\alpha}. For another exponent \beta the exponent vector lies componentwise between 0 and 2p-2 and has Cartier residue (p-1)\mathbf1 only when \beta=\alpha. Consequently every c_{j,\alpha} belongs to \mathfrak m. This proves the required regularity of the traced products, namely

No regular extension of s or \eta across b has entered this argument.

The reduced expressions (5.19) have coefficients in A at \mu(b) and ordinary differential order at most o_g<p. They preserve \mathfrak m^{[p]}, since the ordinary partial derivatives annihilate every v_i^p. Their coefficients lie on the regular base P, so they pass through the K-linear trace T. Taking T in (5.26) and using (5.30) gives T(s)\in\mathfrak m^{[p]}. Cartier consequently sends T(s)du into \mathfrak m\,du. But (5.27) with m=1 gives

where (5.28) and \delta\in k^* give the final inequality. This contradiction proves the lemma.□

Proof. The preceding argument makes C_\nu(\mathcal V) homotopy equivalent to a zero-differential complex near every closed point. Its coefficient identifications satisfy (5.2) and (5.3). Lemma 5.1 therefore gives the canonical map (5.35) for every module on each such neighborhood. Since the base is affine, the cohomology modules correspond to the quasi-coherent direct-image sheaves; the map is multiplication by the pulled-back coefficient.

The local homotopy splitting extends to a neighborhood of each closed point: its maps involve finite modules, the local function t, and the inverse of the unit u. These neighborhoods are chosen before any coefficient module M. They cover the finite-type chart, since any nonempty closed complement would have a closed point. Consequently the local statement holds as a sheaf statement for every module. The maps are compatible with the étale overlaps and deck actions because they are canonical multiplication maps. Proposition 4.5 therefore descends them to every supported thickening. The ordinary supported pullback on X is exactly the one constructed there. Taking the Euler characteristic of Leray and using the spread chartwise direct-image isomorphisms of Proposition 4.7 gives (5.36). Here N_i\otimes M has the ordinary bounded-support meaning fixed in Section 4; all supports are proper.

The only choices required before reduction were the finite diagrams, resolutions, one cleared homotopy per chart, and bounded differential identities. The proof of Lemma 5.4 works for all coefficient modules in each retained fiber, with the old bound for C_{\nu_0} retained by the split sequences. The exponent \nu, the comparison at p\nu, and the local functions t were constructed anew in that fiber. No spreading of infinitely many Frobenius identities or of the individual modules M occurs. A finite chart cover and Lemma 5.6 give the asserted simultaneous choice of reductions.□

Proof. Ambient vector fields are tangent to the orbit and restrict onto its tangent bundle. Their restriction kernel is the trace kernel: a vector field vanishing at an orbit point pairs to zero there with every reflexive one-form, whereas differentials of ordinary functions detect every nonzero tangent vector. The perfect barred pairing between V and V^\vee therefore identifies the two multiplicity bundles with \Omega^1_{\cS} and T_{\cS}. Evaluation and contraction give mutually inverse inclusion and projection maps for the trivial-isotypic part. They are intrinsic, so agree on overlaps.

Exterior multiplication applied to V=\mathcal U\otimes\mathbf1\oplus W in this quotient category gives \Lambda^iV=\bigoplus_{a+b=i}\Lambda^a\mathcal U\otimes\Lambda^bW. Writing this identity in local dual frames gives matrices of barred Hom maps and their inverses. Applying these matrices to a left module proves (6.2); an unbarred algebra splitting is not needed.

At each point, Corollary 3.15 supplies a simple constituent of \Lambda^\bullet W which occurs once and in exactly one degree j_0. The fixed transverse descriptions transport this fiberwise selection, and its degree is locally constant, hence constant on the connected \cS. In \Lambda^{j_0}V its multiplicity is still one: a positive exterior degree of \mathcal U would require its occurrence in a smaller degree of W. Its isotypic projector is u, and Schur’s lemma gives (6.3) fiberwise.

The selection in the corollary is intrinsic to the independent standard factors and is unchanged by their permutations. The centers of the semisimple block algebras are finite étale over \cS; in a product presentation their primitive components are constant. The invariant selection therefore defines an algebraic idempotent on the whole component. Finally every automorphism acts on an endomorphism algebra by conjugation and fixes the identity of the selected simple object. This proves the assertion about the character of its identity line, including for ineffective stabilizers.□

Proof. First let (R,\mathfrak m) be a complete transverse local ring and M the corresponding finite direct sum of transverse modules. Put

For every fixed j, the images of E_l\to E_j, l\ge j, form a descending sequence of subspaces of a finite-dimensional vector space. Write D_j for its stable value. The maps D_{j+1}\to D_j are surjective. Indeed, for a\in D_j, the images in E_{j+1} of its lifts to all sufficiently large E_l are nonempty descending affine subspaces; they stabilize and give a lift in D_{j+1}. Recursively choose compatible lifts. Completeness of M identifies \varprojlim_jE_j with E, so D_1 is exactly the image of E\to E_1. This is an all-level argument; stabilization at level one alone would not justify the lifting assertion.

Choose k for which \im(E_k\to E_1)=D_1. A map in the kernel lowers M/\mathfrak m^kM into its \mathfrak m-multiple, so the k-th power of this kernel is zero. Also \ker(E\to D_1)\subset\rad(E): if a lowers M into \mathfrak mM, then 1-ba is invertible for every b\in E, by the convergent \mathfrak m-adic geometric series. The trace-radical calculation consequently identifies

The composite E_k\to D_1\to D_1/\rad(D_1) is onto and has nilpotent kernel. Indeed its kernel modulo the lowering kernel is the radical of a finite-dimensional algebra and is nilpotent. Notice that no claim has been made that every element of E_k lifts to E; only its residual action must lift.

On a product presentation, finite quotients and their Hom modules are these fixed transverse objects extended by the smooth parameter ring. Flat extension commutes with their finite presentations and kernels. Thus the chosen exponent works throughout that presentation. Take the maximum of the exponents for a finite cover of \cS. Assertions about the quotient, its kernel, and a fixed nilpotence exponent descend from faithfully flat completions.

To check canonicity, let \mathcal A_{\mathrm{res}} be the common image of the full and ambient endomorphism algebras in \End(T/IT). The original trace quotient \cE\to\overline{\cE} kills the kernel of \cE\to\mathcal A_{\mathrm{res}}, by the transverse calculation, and hence factors uniquely through \mathcal A_{\mathrm{res}}. Composing with \End(T)\to\mathcal A_{\mathrm{res}} defines (6.4), including on maps without ambient lifts. Its kernel is the transverse nilpotent ideal extended along the smooth parameters; it need not be the ordinary Jacobson radical of the full algebra over the parameter ring. The quotient is locally a product of matrix algebras over a reduced smooth base and has no nonzero nilpotent two-sided ideal. Thus this kernel is intrinsically the largest nilpotent two-sided ideal. Transition isomorphisms preserve it and the factorization through the residual-action image, so the map glues on the stack independently of the product coordinates.

After setting the smooth parameters to zero, let C be the image of ambient endomorphisms in the full finite algebra \End(T_o). Product structure identifies C with the transverse ambient image. It maps onto the barred algebra because the original E does. Its kernel is its intersection with the nilpotent kernel of (6.4). Changing the smooth parameter lifts while keeping the transverse coordinates is a formal automorphism preserving I; differential tensors transform functorially under it. This conjugates the calculation and does not change any exponent or create an additional lifting obstruction.□

Proof. We first prove, in complex cohomology with supports, that for one positive rational number \rho

The supported class [S_0] exists because B is an oriented rational homology manifold by Proposition 3.2. Equidimensionality implies that every component of f^{-1}(S_0) of dimension 2n-c=n+d dominates S_0: one whose image has smaller dimension has dimension at most n+d-1.

Choose one maximal chart and one lifted component S' of its stratum. For each top-dimensional component of f^{-1}(S_0), restrict it to the chosen base neighborhood and choose a component of its inverse image under the source torsor of Proposition 4.5. Its base image is a stratum lift. The deck group is transitive on such lifts over a general coarse point, so translating the chosen source component makes its base image S'. This translation leaves its image on X unchanged. Thus one chosen lift S' sees every top component, without requiring transitivity on components of the fibers of f. Use the finite projection of Section 5 on this source chart,

Each chosen lifted component has dimension n+d and is finite over S'\times\PP^n. It therefore dominates that entire irreducible support. We can now choose one common general closed point y\in S'\times\PP^n at which all its reduced preimage support branches are smooth and étale over the support. In analytic germs each of these branches maps isomorphically to the support germ.

Let R=\widehat{\cO}_{Y,y} and let A_\alpha=\widehat{\cO}_{Z,z_\alpha} run through the regular local factors above this same point. Write r_\alpha for their generic ranks over R, and J\subset R for the support ideal. The quotient R/J is regular. If m_\alpha is the scheme multiplicity of the ordinary reference pullback on this branch, the preceding degree-one assertion gives

Proposition 5.2, in characteristic zero and over this single target ring, gives

Quotienting by J and taking ranks yields

Thus m_\alpha/r_\alpha is one common scalar.

We compare the cohomological multiplicities by the finite local-degree rule. For a finite local branch, the projection formula with supports, capped with the complex orientation, multiplies the target support class by the generic degree of the branch after pushforward. Dividing by the degree of its reduced support map gives the coefficient of the pulled class. Applied first to the chosen base-chart branch, this gives \lambda[S'] as the pullback of [S_0], with one positive factor \lambda on S'. The intervening étale base refinement does not change that factor. Applied next to a branch of \pi, whose reduced support map has degree one at the common point, the same rule gives coefficient r_\alpha. Hence the coefficient of f^*[S_0] on that branch is \lambda r_\alpha. Equation (6.7) makes the scheme-to-cohomology ratio m_\alpha/(\lambda r_\alpha) independent of the branch. The source charts are étale, so both kinds of multiplicities descend unchanged. Top support classes are determined at these generic points; this proves (6.6) globally.

By Propositions 3.2 and 2.1, ordinary H^{2c}(B,\Q) is one-dimensional, and [S_0] is a multiple of c_1(H)^c. Its pullback is therefore a multiple of e^c. For every integer t, the IHS Riemann–Roch polynomial gives

since q(e)=0; see [23, §1.11]. The coefficient of t^c in ordinary Hirzebruch–Riemann–Roch is \frac1{c!}\int_Xe^c\td_{2n-c}(X). It vanishes for c>0. Together with (6.6) this proves (6.5).

For completeness, specialization of this number does not use rational smoothness of a characteristic-p base. The ordinary coherent sheaf F_0 on the smooth projective X has a finite locally free resolution. Include its finitely many terms, maps, kernels, the descended quotient ideal defining F_0, and the identification with the ordinary pullback in a finite coherent model. Generic flatness and coherent base change on a nonempty principal open preserve the resolution of that quotient on fibers and support codimension at least c. Thus its restriction represents the actual reference pullback on every retained fiber. Its Chern character below codimension c vanishes and its component in codimension c is its actual support cycle. The degree

is constant: it is the degree of a fixed relative Chern polynomial in the vector bundles of that resolution and the relative tangent bundle on a smooth proper family. This gives the last assertion. The argument neither keeps track of individual cohomology classes on the reduced base nor requires a constant number of support components.□

Proof. There are finitely many connected components. The functional and its representation dependence are componentwise; if the estimate holds on them with constants C_j, summing gives the stated bound with C_{\mathcal D}=\sum_j C_j under our rank convention. We may therefore assume \mathcal D connected.

Let A be the pullback of an ample line bundle from the coarse space and set

The coarse pushforward is exact by tameness [1, Definition 3.1 and Theorem 3.2]. The usual Euler polynomial on its projective coarse space shows that \Theta_m(G) is a polynomial in m of degree at most d.

For m sufficiently large, global sections of T_{\mathcal D}\otimes A^m evaluate onto the invariant tangent vectors at every residual gerbe. To see the uniform assertion, push this bundle to the coarse space, use an ample twist to generate that coherent sheaf, and use exactness of invariants to restrict to a residual gerbe. At a point x with stabilizer J_x, the possible values are exactly (T_x\mathcal D)^{J_x}. The locus where this dimension is at most r has dimension at most r. On a finite quotient chart, its stabilizer strata lie in fixed loci of subgroups; these are smooth for tame actions and their tangent spaces are the invariant tangent spaces. There are finitely many relevant subgroup strata on a finite atlas.

For a stratum of dimension s with invariant tangent dimension r, the incidence of sections vanishing at a point has codimension r in the space of sections over that point. Since s\le r, a general section has a zero scheme Z_m of dimension at most zero. At every zero its d components generate an ideal of height d in a regular local ring. They are a regular sequence, and their Koszul complex gives

This remains valid when Z_m is empty.

The right side is a finite sum of invariant dimensions on zero-dimensional quotient charts. Filtering their Artin rings by powers of the maximal ideal expresses each summand as an integral linear functional of the fiber representation G_x. Its absolute value is bounded by a fixed constant times \dim G_x. In particular, it depends only on these fiber representations, not on the transition functions of G. Choose d+1 distinct sufficiently large integers m. Polynomial interpolation expresses \Theta_0 as a fixed rational linear combination of the corresponding \Theta_m. This proves both assertions for bundles, with one constant independent of G. Additivity extends the representation dependence to virtual classes, and the triangle inequality gives the displayed presentation bound. The construction uses the full finite groups at zeroes and applies without an effectiveness assumption.□

Proof. If I^bM=0, choose generators x_1,\ldots,x_v for the maximal ideal of the transverse ring. Its Frobenius pullback is killed by (\mathfrak m^b)^{[q]}, which contains (x_1^{bq},\ldots,x_v^{bq}) and hence \mathfrak m^{vbq}. A fixed number of generators for M also generates its ordinary Frobenius pullback. Hilbert–Samuel growth in transverse dimension c therefore bounds its length by O(q^c). Tensoring with each of the fixed N_i changes this by only a fixed generator bound. The total rank of the graded pieces of L(F^{a*}M), and hence the sum of the ranks in its expression (6.8), is O(q^c) on a finite atlas.

The annihilating thickness just obtained grows with q. The all-modules, all-orders conclusion of Proposition 5.8 therefore matters here: it applies to each new coefficient F^{a*}M in this same reduction. The Frobenius square commutes with ordinary pullback on the ambient charts and with their étale descent, so f^*F^{a*}M=F_X^{a*}f^*M. Applying (6.9) and Lemma 6.5 proves the bound with the original test and prime fixed.□

Proof. If M has support of dimension less than d, its ordinary pullback has codimension greater than c by equidimensionality, and its codimension-c character is zero. Otherwise repeat the common-chart and lifted-component choice from Lemma 6.4 in this fixed reduction, using the retained labelled source torsor. Translating a lift of each top component to the same S' again makes every component visible there. Now take the generic point \eta of S'\times\PP^n. Complete its target local ring and call it R; its residue field is K=\kappa(\eta). Let A_\alpha be all the regular complete local factors over it. Write

for a finite-length R-module Q. Here Q is the module induced by M; the reference module is K. The finite-regular-module identity and length over R give, for every pair of factors,

The second lengths are positive. Division cancels both generic ranks and residue degrees and shows

Proposition 5.2 applies at these nonclosed points with their residue-field Frobenius degrees; no perfection of K is being assumed. All top components are visible over this one generic point, because every chosen lift dominates the support under the finite projection. If factors split in reduction, include every new factor in (6.12). Descent through the étale source charts shows that

as support cycles, for a scalar \rho_M depending possibly on M and on the good prime. Lemma 6.4 proves (6.10).

It remains to explain why this one coefficient suffices. Frobenius is flat on the smooth X, and Chern characters of ordinary Frobenius pullbacks satisfy \ch_i(F_X^{a*}F)=q^i\ch_i(F); one can check this on a finite vector-bundle resolution by the splitting principle. Hirzebruch–Riemann–Roch, applied additively to that finite resolution [5, §0.1], therefore gives the polynomial

The growth bound of Lemma 6.6 kills its coefficients for i>c. Equation (6.10), proved independently, kills the coefficient for i=c. Thus the polynomial is identically zero; evaluating at q=1 and using (6.9) proves (6.11). The scalar \rho_M is fixed before the Frobenius exponent varies. Uniformity of that scalar in M, in the prime, or in Frobenius pullbacks is unnecessary.□

Proof. In a product chart the restricted ordinary cotangent is \Omega^1_{S_U}\oplus (\mathfrak m_{t_0}/\mathfrak m_{t_0}^2)\otimes\cO_{S_U}, hence is locally free. This is the ordinary cotangent, which records all embedding directions, not its reflexive hull. A finite-order automorphism of a complete local ring that is trivial on its cotangent space and has order invertible in the residue field is trivial: it is the identity on the associated graded, and at the first nonzero order of its difference from the identity, iteration multiplies that difference by the order, a contradiction. Equivalently, average lifts of a cotangent basis and apply complete Nakayama. A deck stabilizer acts faithfully on the ambient germ, so its cotangent representation is faithful.

The surjection from the ambient maximal ideal to the cotangent space of the smooth stratum admits an H-equivariant linear section by averaging. Its finite-dimensional image gives the desired lifted functions. Their restrictions are coordinates by the formal inverse function theorem. They can be chosen as algebraic local functions before completion, since a finite group average of such functions is again algebraic locally. Lemma 4.6 permits this change of parameters. The arguments use only finite group averaging and invertibility of the parameter Jacobian, and hence apply in the indicated reductions as well.□

Proof. Choose a minimal A-generating list q_1,\ldots,q_m of Q_i. Lift these constant elements through the given surjection. Their residue classes in C^{r_i}/\mathfrak m_CC^{r_i} are independent, since their images form a basis of Q_i/\mathfrak m_AQ_i. Complete them to a basis of the free module. The first m columns are then exactly the q_j; subtract suitable combinations of these columns from the others to make the latter zero. These invertible column operations on C^{r_i} induce invertible operations on (E_i\otimes_A C)\otimes_C C^{r_i} even if E_i is not A-flat. Evaluation respects the same column operations, so the resulting map has constant coefficients in the asserted sense. Flatness of A\to A[[t]] preserves its kernel, image and cokernel. A finite A-module extended in this way is finite free over K[[t]] as a K[[t]]-module, which proves exact specialization.□

Proof. By Lemma 6.8, the ordinary ambient cotangent restricted to \cS is a vector bundle faithful on every full stabilizer. Since q_0\ge k, ambient Frobenius factors through a map \Phi_{q_0}:\cS_k\to\cS. Pulling back that bundle gives a bundle on \cS_k whose stabilizer representations remain faithful: raising roots of unity of order prime to p to the q_0-th power is a permutation. A finite sum of bounded tensor powers contains every irreducible representation of each stabilizer. One elementary justification is that matrix coefficients of a faithful representation separate elements of a finite group, hence generate its function algebra; their products are matrix coefficients of tensor powers. There is one bound for a finite quotient atlas and its finitely many subgroup types. Dual powers may equally be included.

Let G be such a generating bundle. If \mathcal A is any coherent sheaf on \cS_k, exact coarse pushforward and sufficiently ample coarse twists generate the invariant Hom sheaf from G to \mathcal A. The evaluation maps then surject on every residual gerbe, since G contains all its simple types. Nakayama makes them surjective on stalks. Taking suitable finite sums of twists of G gives the P_i. This is the generating-sheaf criterion of [43, Definition 5.1 and Theorem 5.2], here realized by the faithful ambient cotangent bundle.

For the product assertion, complete at a point and forget the action temporarily. Put

Every T_i is M_i\otimes_KD for a finite A-module M_i. In these coordinates u is a constant primitive central idempotent in the N_{j_0} block: its center is the fixed transverse center tensored with the local ring D, and its idempotents are constant. Thus the finite-dimensional transverse Hom calculation and (6.4) give \mathcal A_i=A_i\otimes_KD, where A_i\subset\Hom_A(M_i,M_{j_0}) is the kernel of the transverse map a\mapsto u\bar a in (6.15). Trivialize P_i, so its surjection is C^{r_i}\to A_i\otimes_KD. Apply Lemma 6.9 with E_i=M_i, F=M_{j_0} and Q_i=A_i. It makes the evaluation a fixed transverse map extended by D and proves that its kernel, image and cokernel are product modules, flat over D, whose sequences specialize exactly modulo (t_1,\ldots,t_d).

These basis changes establish formal product form only; they need not be equivariant. The actual bundles, maps and kernel in (6.17) keep their original equivariant structures. For character calculations, choose equivariant lifts of the smooth stratum parameters by tame averaging, as in Lemma 6.8. Their ideal is stable under the stabilizer. The formal automorphism fixing the transverse ring in Lemma 4.6 transports the tensor, full Hom, trace and ideal data to these new product coordinates. Reapplying Lemma 6.9 there proves flat specialization for this stable ideal. Completion detects the coherent local properties claimed here; it does not assert an algebraic equivariant product splitting.□

Proof. Let the stabilizer be J, choose the equivariant smooth parameters from Lemma 6.10, and denote specialization modulo them by a subscript o. Let A be the resulting transverse Artin ring over the algebraically closed residue field K, and let \delta=\det(\Omega^1_{\cS,o}) be the one-dimensional cotangent determinant representation of J.

The completed finite map to the smooth parameter disk computes the canonical module \omega_C as finite Hom to the disk’s canonical module [56, Tag 0AX0]. The ring is finite free over the parameter ring, so this dualizing complex is concentrated in its canonical degree. The chosen parameter span is stabilizer-stable, with determinant \delta, and the parameters form a regular sequence on C=A[[t_1,\ldots,t_d]]. Equivariant Koszul adjunction gives

Thus the specialized dualizing module is A^*\otimes_K\delta with its actual stabilizer action. Product form justifies specialization of Hom, and finite duality gives \Hom_A(T'_o,A^*)\simeq(T'_o)^* equivariantly, where (-)^*=\Hom_K(-,K). The other line in (6.18) has character \delta^{-q_0}. Consequently

The last isomorphism is an assertion about the stabilizer representation: (6.16) makes \delta^{1-q_0} trivial. It does not assert that the corresponding global line bundle is trivial.

For each j, consider the image C_j of the ambient N_j endomorphisms in \End_A(T_{j,o}). By Lemma 6.3, it surjects onto the barred corner with nilpotent kernel. This is a J-equivariant surjection. Exactness of tame invariants makes C_j^J surject onto the invariant barred corner, whose idempotent p_j we want to lift. The invariant kernel is still nilpotent, so idempotent lifting gives

An explicit lifting proof starts with a lift a and corrects a^2-a successively modulo powers of the nilpotent ideal; 2a-1 is invertible modulo that ideal because its square is 1 there. The corrections take place inside C_j^J. A global unbarred projector is not asserted.

Being in the ambient image has an additional consequence. Choose an ambient endomorphism inducing p'_j after specialization. It preserves every \cJ^s(N\otimes M_{T'}) because \cJ is a two-sided ideal. On the specialized associated graded its action is p_{j,o}, since the residual specialization of this representative is p_{j,o}; no assertion about its residual action before specialization is needed. Product form makes the filtration specialize exactly. The idempotent p'_j therefore splits the specialized filtered module with graded image p_{j,o}L(M_{T'})_o. Thus in the representation ring,

The second equality is ordinary finite-dimensional tensor–Hom duality: dualizing its left-hand tensor gives \Hom_A(p'_jT_{j,o},T'_o).

Put Q_j=p'_jT_{j,o}. Lemma 6.10 identifies Z_{0,o} with the actual kernel of the specialized evaluation \Phi_o:Y_{0,o}\to T_{j_0,o}. Apply (6.14) with Q=Q_j and this specialized evaluation. We compute the image of its Hom map explicitly. Each evaluation column belongs to the right ideal (6.15), and precomposing it by any map from Q_j remains in that ideal. Conversely, a right-ideal map Q_j\to T_{j_0,o} extends by zero on (1-p'_j)T_{j,o} to an element of \mathcal A_{j,o}. Express that element in the generating columns of P_{j,o} and restrict to Q_j. These expressions lift it through Y_{0,o}. This verifies equality of the image with the indicated ideal, using all A-linear maps and with their genuine J-actions.

The full quotient (6.4) now identifies that cokernel with u\overline{\cE}_op_j. Surjectivity on this corner follows by lifting any barred map to a full truncated map and precomposing with p'_j; its kernel is exactly the inverse-image ideal just described. Dualizing the signed identity and using (6.21) leaves [(u\overline{\cE}_op_j)^*]. By (6.3) this is zero unless j=j_0, when it is the invariant identity line. Summing with the signs (-1)^j in (6.8) proves (6.19).□

Proof of Proposition 6.1. Equation (6.11) vanishes for the three modules (6.18). Take their signed sum. By Lemma 6.11 its coefficient has at every point the representation (-1)^{j_0}\mathbf1. Lemma 6.5 therefore gives

This coherent Euler characteristic is retained from characteristic zero in the chosen model. Indeed the finitely many exterior bundles, proper coarse pushforward and coherent cohomology can be spread with base change; tame coarse pushforward is exact. Thus (6.22) holds for the original complex stack.

Finally on smooth finite quotient charts the holomorphic de Rham complex resolves the constant sheaf, and taking invariants is exact in characteristic zero. Hypercohomology Euler characteristics, and proper GAGA [47, Theorem 7.1], identify

The rational or complex cohomology of a finite stabilizer has no positive degrees; hence passage to the coarse space in this identity does not insert stabilizer weights. This remains true for stabilizers acting trivially on a chart. The left side is (6.22), proving the claimed ordinary Euler vanishing.□

Proof. The smooth fibers have one common cohomology class, and intersecting n general hyperplanes of the base gives e^n=\deg_H(B)[F]. Polarizing the Fujiki relation, and using q(e)=0, gives a positive constant a such that

Indeed, in every nonzero pairing of the 2n entries e,\ldots,e,\xi_1,\ldots,\xi_n, each copy of e must be paired with a different \xi_j. Positivity of a follows by taking all \xi_j to be an ample class h; in particular q(e,h)>0.

The restrictions of \sigma_0 and \bar\sigma_0 vanish. If \xi\in H^{1,1}(X_0,\R) and q(e,\xi)=0, Equation (7.2) gives

when n\geq2. The Hodge index theorem on F makes the second quadratic form negative definite on the primitive real (1,1)-classes, so i^*\xi=0. For n=1, the first integral already proves this, since H^2(F,\R) is one-dimensional. Thus e^\perp is contained in the kernel. Conversely, the first equality in Equation (7.2), with one entry \xi and all other entries h, shows that every class whose restriction vanishes is orthogonal to e. The Hodge decomposition and complexification give Equation (7.1).□

Proof. Choose m>0 so that H^m is very ample, and put L_0=f_0^*H^m. The local universal deformation of the pair (X_0,L_0) is the restriction of the Kuranishi family to

and carries an extension \mathcal L of L_0; this is [39, Theorem 1.1]. Thus its restriction to a small polydisc is exactly the family \mathcal X\to D chosen above. Theorem 1.2 and Corollary 1.3 of the same source give a locally free sheaf \mathcal E=\pi_*\mathcal L and a surjective evaluation \pi^*\mathcal E\to\mathcal L. Their hypotheses are the original Lagrangian fibration over the projective B and L_0=f_0^*H^m with H^m very ample; the nearby fibers may be nonprojective compact Kähler manifolds. The resulting proper morphism is

Here \PP_D(\mathcal E) parametrizes one-dimensional quotients. The cohomology-and-base-change argument in Theorem 1.2 identifies \mathcal E\otimes k(0) with H^0(X_0,L_0). Since f_{0*}\cO_{X_0}=\cO_B, projection formula identifies this last space with H^0(B,H^m). Hence \Phi_0 is f_0 followed by the very ample embedding of B defined by the complete system |H^m|.

Let \mathcal Y be the image of \Phi, with its reduced analytic structure, and let

be its relative Stein factorization. Write p:\mathcal B\to D. The space \mathcal B is normal because \mathcal X is normal, and p is proper. For each t, take the normal Stein factorization of the induced map to its image,

Here Z_t is normal and finite over a projective variety, and g_t has connected fibers. Corollary 1.3 gives a general Lagrangian fiber. The symplectic form on the compact Kähler manifold X_t is d-closed. Thus the proper map g_t satisfies [36, Theorem 1], which makes every irreducible component of every fiber an n-dimensional Lagrangian subvariety. The underlying set of each fiber of F is a connected component of a fiber of \Phi_t, and hence is the underlying set of a fiber of g_t. In particular F is equidimensional of relative dimension n. This determines its fiber dimensions before any identification of a fiber of \mathcal B with a fiberwise Stein target.

We prove that p is locally an analytic product before making any base-change assertion for its central fiber. Fix b\in p^{-1}(0) and x\in F^{-1}(b). A general codimension-n smooth slice through x in X_0 meets F^{-1}(b) only at x as a germ. Extend its local defining equations to \mathcal X. The resulting slice Y is smooth over D, and its map to the germ (\mathcal B,b) is finite and dominant. This follows from the isolated-fiber criterion for a finite map of analytic germs and equality of dimensions. Let R\subset A be the corresponding finite extension of local rings and let r be its generic degree. Here R is normal, A is regular, and the parameter functions t_1,\ldots,t_s are part of a smooth coordinate system on Y.

Choose derivations \delta_i of A satisfying \delta_i(t_j)=\delta_{ij}. Define

Trace takes A into R: its values are integral over R and belong to its fraction field, and R is normal. Trace is R-linear, so the Leibniz rule shows that \partial_i is a derivation of R, with \partial_i(t_j)=\delta_{ij}. Holomorphic derivations of an analytic space have local flows. For example, embed the germ in a polydisc, lift the derivation to a vector field preserving its defining ideal, and restrict the ambient flow. Successive flows of the \partial_i give an isomorphism of germs

over D. The lifts need not commute: inverse flows in the reverse order give the inverse product map.

Normality of \mathcal B now implies normality of p^{-1}(0). The central factorization has connected first fibers and a finite second map, and its composite is \Phi_0, the complete system |H^m| pulled back by f_0. Uniqueness of Stein factorization therefore identifies p^{-1}(0) with the original B.

It remains to globalize Equation (7.4). It gives an exact sequence

The relative tangent sheaf is flat over D and restricts to T_B, as is seen in the local products. By Corollary 2.2, H^1(B,T_B)=0. Proper coherent semicontinuity consequently gives R^1p_*T_{\mathcal B/D}=0 after shrinking. Also p_*\cO_{\mathcal B}=\cO_D, since each fiber of p is the image of the connected X_t and is therefore connected. Pushing forward the tangent sequence and using that D is Stein therefore gives global holomorphic lifts of the coordinate vector fields of D. Properness supplies one neighborhood on which their successive flows are defined along the whole central fiber. These flows give \mathcal B\simeq B\times D over D. The resulting \mathfrak f has the asserted fiber properties. The line bundle \mathfrak f^*\operatorname{pr}_B^*H on \mathcal X restricts centrally to f_0^*H, so its fiber Chern classes are the flat class e under the chosen marking.□

Proof. First take a smooth slice T through y finite over (V,g(y)). For a vector v\in T_yT, choose a local derivation \delta of T with that value. The normalized trace construction in Equation (7.3) gives a derivation \partial of V. Because the chosen local finite branch has only y over g(y), the trace of a function specializes to its value at y times the local degree. Thus \partial(g(y))=dg_y(v). Evaluations of all local derivations of the chart are precisely the tangent space of its orbit, by Proposition 4.3. This proves Equation (7.5) for vectors tangent to the slice.

The tangent spaces of finite smooth slices span T_yY. Indeed, the condition that a complementary linear slice miss the projectivized fiber tangent cone is a nonempty open condition in the appropriate Grassmannian; its tangent planes cannot all lie in a proper linear subspace. Taking these slices proves the claim for every vector. The argument is analytic and applies unchanged to the compact Kähler deformations. The normalized source covers used here are smooth: their maps to \mathcal X are finite and unramified outside a set of codimension at least two, by equidimensionality, and hence are étale by purity.□

Proof. On a maximal chart, lift a form on the smooth orbit to an ordinary holomorphic one-form \widetilde\theta on the ambient base chart. On its smooth source define the relative Hamiltonian vector field v by \iota_v\sigma_t=g^*\widetilde\theta. It is vertical on the dense locus where the fibration is smooth, because those fibers are Lagrangian, and hence it is vertical everywhere. Equivalently, Hamiltonians of two functions pulled back from the base have identically zero Poisson bracket. Thus the local flow preserves the reduced inverse image of the stratum.

If two extensions of \theta are chosen, their difference belongs to I\Omega^1_V+dI, where I is the ideal of the orbit. Its pullback as a covector on the smooth source is zero at every point over the orbit, by Lemma 7.4. Nondegeneracy of the ambient symplectic form then says that the difference of the two vector fields is zero on this reduced inverse image. Consequently they induce the same derivation there.

Functoriality for the smooth joint flow map and for the product with its parameter disc lifts the relative flow on the reduced total inverse image to the total analytic resolution over D^\circ. The lifted flow fixes the deformation parameter, so differentiation gives relative vector fields there. Additivity, \cO-linearity, and compatibility on overlapping charts hold on the dense isomorphism locus of the total resolution. Their differences are holomorphic relative vector fields, so these identities extend over that total resolution. This gives a relative bundle map. The contraction and verticality identities also hold on the same dense locus and extend as holomorphic relative tensor identities over the total resolution. Restricting the map and identities to Q_t gives Equation (7.6). In particular the image vectors are pairwise isotropic and annihilate every form pulled back from \cS.□

Proof. Suppose a stratum \cS as above exists. We prove

If d=0, its coarse space is a point, already contradicting Proposition 6.1; we may assume d>0.

Fix \alpha\in H^{p,q}(\cS) with p>q, and set k=d-p+1. On every Q_t,

To verify it, first note that the pullback of \sigma_t by every smooth space mapping into a fiber of f_t is zero. On a resolution of each irreducible fiber component this follows from the Lagrangian assumption and extension of holomorphic forms. For a map whose image is contained in a proper subvariety of that component, resolve a component of its proper base change to that resolution which dominates the source. Pullback of forms by a dominant map is injective, giving the same conclusion. Thus, at a general point where h_t has constant rank, \omega_t vanishes on two vertical vectors. Each of its terms has at least one horizontal one-form factor. Wedge multiplication by \omega_t^k consequently kills the pullback of every holomorphic p-form on \cS, since k+p>d. This equality of holomorphic bundle maps extends from the dense open set and implies Equation (7.8) on Dolbeault cohomology.

Work over a simply connected small open subset of D^\circ. Smooth proper transport makes the maps j_t^* and h_t^* constant on cohomology. Accordingly

is one fixed homogeneous polynomial with values in a fixed cohomology group. It vanishes on all period vectors over that open set, by Equation (7.8). The quadratic form on e^\perp_\C has radical \C e and rank b_2-2\geq3. Its null cone is therefore irreducible. Local Torelli and Proposition 7.2 imply that the periods just used contain an analytic open subset of its projectivization. Such an open set is Zariski dense. Hence the polynomial vanishes on the whole cone, including the conjugate period of our chosen fiber. Suppressing t, we have obtained

We give the contraction step with its coefficients retained. This avoids replacing a sheaf-valued class by its exterior product before the contractions have been applied. If \alpha\ne0, Poincaré duality supplies \beta\in H^{d-p,d-q}(\cS) with \alpha\beta\ne0. Pass to the conjugate complex structures and put

Write \rho:\mathcal A\to\Omega^1_{\bar Q} for differential pullback. In this notation

The second class is kept in the indicated coefficient bundle; it has not yet been wedge-mapped to \Omega^r_{\bar Q}. Since p>q, we have r\geq k. Apply the exterior coproduct

to its coefficients. Cup Equation (7.9) with this class, contract the first k coefficient factors by \overline{\mathcal H} from Lemma 7.5, and wedge the remaining r-k factors as basic forms. All these are holomorphic sheaf maps on \bar Q, so they induce well-defined operations on its sheaf cohomology. Their output belongs to H^d(\bar Q,\Omega^d_{\bar Q}).

Here is the exterior-algebra computation of that output. If a_1,\ldots,a_k are local sections of \mathcal A and \eta is a basic q-form, the identities

give

For a decomposable r-form the signed shuffle coproduct followed by wedge multiplication is \binom rk times the original form. It follows, with only an overall sign depending on conventions, that the preceding sheaf cup operation is

Its input was zero, so Equation (7.10) forces h^*(\alpha\beta)=0.

That is impossible. Choose a component Q' dominating S_0, a Kähler form \kappa on Q', and a Kähler orbifold form \tau on \cS. If N=\dim Q', then

the integrand is nonnegative and positive on the nonempty open set where h has rank d. Thus the top class of \cS pulls back nontrivially. As \alpha\beta is a nonzero top class, it cannot pull back to zero. This proves the vanishing for p>q; complex conjugation proves Equation (7.7).

The Euler number is now

because the coarse stratum is nonempty and connected. This contradicts Proposition 6.1. There can therefore be no nonquotient germ.□

Proof. Since h^{2,0}(X)=h^{0,2}(X)=1, we have h^{1,1}(X)=2. An ample class h and the nonzero isotropic class e are linearly independent, so they span \NS(X)_{\Q}. Their Beauville form is nondegenerate, and its orthogonal complement is the stated rational plane.

Let F be any divisor dominating D. Its restriction to a general smooth fiber is zero, whereas h restricts nontrivially and e restricts to zero. Therefore [F]=a e with a\in\Q. Intersecting with an ample class to the power 2n-1 shows a>0. Choose positive integers k,l with k[F]=l e. After multiplying once more if necessary, this is equality of integral classes. Since H^1(X,\cO_X)=0, the exponential sequence gives

The section defining kF is pulled back from a section of H^{\otimes l}, because f_*\cO_X=\cO_B. Its zero set consequently contains every component over the generic point of D. Its support is F, so there is only one such component. A local uniformizer of D now has divisor m_DF_D.□

Proof. Choose an ample line bundle on X, and use its restriction to polarize the smooth fibers. Constancy of the polarized moduli map means that the geometric generic fiber, after choosing an origin, is a fixed polarized abelian variety P. The origin and a polarized isomorphism are defined over a finite extension of \C(B): each is specified by finitely many algebraic data over its algebraic closure. Alternatively, a sufficiently high level structure gives this isomorphism on a dense open by the fine moduli space of polarized abelian varieties.

We can also make the actual line bundle constant. Once its numerical polarization has been matched with a fixed ample bundle L_P on P, their difference lies in \Pic^0(P). The polarization isogeny

allows us to remove that difference by a translation after a further finite extension. Take a Galois closure and normalize B in it. The resulting T is projective and normal. The descent automorphisms of P preserve L_P. The group \Aut(P,L_P) is finite: its translation subgroup is the finite kernel of the polarization isogeny, and its origin-preserving subgroup is a discrete subgroup of the compact unitary group of the polarization. Thus each descent automorphism, initially over the function field of T, is a constant affine automorphism over \C. This proves the first assertion.

Take a G-equivariant projective resolution \widetilde T\to T. The rational map P\times\widetilde T\dashrightarrow X pulls \sigma back to a holomorphic two-form \widetilde\sigma on P\times\widetilde T. To see regularity, resolve the graph; its pulled-back form descends across the birational map to the smooth product, since holomorphic differential forms are unchanged by a proper birational modification of smooth varieties. Choose a basis \alpha_1,\ldots,\alpha_n of H^0(P,\Omega_P^1). The Künneth decomposition and the Lagrangian condition give

There is no two-form term from P. At a general point the pulled-back form is nondegenerate and P is Lagrangian; hence the \gamma_i are pointwise independent there.

We spell out the rationality needed for an abelian quotient. Graph pullback and projection onto the Künneth summand give a morphism of rational Hodge structures

Contract its image with H^1(P,\Q)^\vee, and denote the resulting rational subspace of H^1(\widetilde T,\Q) by W_{\Q}. By Lemma 8.2 and (8.1),

It is a rational weight-one Hodge substructure of dimension 2n. To form the corresponding Albanese quotient, take the annihilator of W_{\Q} in H_1(\operatorname{Alb}(\widetilde T),\Q). It is a rational Hodge subspace, so its real span and its lattice define a connected complex subtorus. Restriction of an ample polarization makes this an abelian subvariety. The quotient is an abelian variety A of dimension n; its one-forms pull back to the span of the \gamma_i. The induced morphism \widetilde a:\widetilde T\to A is generically finite. The subspace W_{\Q} is G-stable; the Albanese functor consequently gives an affine action on A making \widetilde a equivariant.

We next descend this morphism to T. Let C' be a curve contracted by \widetilde T\to T. The rational map from P\times C' to X, interpreted on a resolved graph, has image in one fiber of f. The pullback of \sigma on that graph is zero. Here the Lagrangian condition also applies if the image lies in a smaller subvariety of a fiber component: resolve the component, on which the pulled-back form vanishes, and dominate the smaller subvariety by proper base change. Restriction of (8.1) and contraction in the P directions now show that each \gamma_i vanishes on the normalization of C'. It follows that \widetilde a is constant on every contracted curve. It is constant on each connected resolution fiber as well: a positive-dimensional projective image of such a fiber in an abelian variety would contain the image of a noncontracted curve. The fibers of \widetilde T\to T are connected and T is normal, so this proves descent to a morphism a:T\to A.

Equivariance gives a generically finite morphism \bar a:B=T/G\to A/G. Pull back an ample line bundle from the projective quotient A/G. The result is a nef and big line bundle on B; because \rho(B)=1, it is numerically a positive multiple of an ample class and hence is ample. Thus \bar a contracts no curve and is finite. The commutative quotient square then shows that a is finite too.

Finally the cross pairing in (8.1) is perfect between the tangent spaces of the two abelian factors. Its invariance shows that an element has trivial linear action on P if and only if it has trivial linear action on A. Such an element acts on A by a translation. Quotienting its finite translation image gives A_l, and equivariance gives the stated finite map from T/K. The derivative representation is unchanged by an isogeny, so the linear action of G_l on A_l is faithful.□

Proof. The maps and divisors being compared fit into the following diagram. The upper square commutes as rational maps, and the lower square is the quotient square from Lemma 8.3.

Here t is a local uniformizer of D, and \phi is induced by the birational identification over T. The integers M and L are local indices along E, not the degrees of the finite maps. They compare to the multiplicity m_D on the original source X, which may be greater than one.

All assertions concern generic points of divisors. We may therefore discard smaller subsets, split the unramified tangential extension, and work with smooth transverse polydiscs. Choose a transverse parameter z for E, absorbing a unit so that p^*t=z^M. Cyclic inertia fixes E pointwise. Since a is finite and equivariant, it fixes the divisor D_A pointwise. A finite affine automorphism with a fixed point is conjugate by a translation to its linear part; fixing a divisor makes its linear part either the identity or a pseudoreflection. The perfect cross pairing gives the same assertion on P.

Let \omega=\sigma^n/n!, with an irrelevant nonzero scalar absorbed into the choice of a translation-invariant volume form on P. In the top exterior power of (8.1), every term must contain all n P-directions. Thus the term \beta makes no contribution, and on the smooth product model

Its order along P\times E is L-1. More precisely, near the generic point of E, it is u z^{L-1}\operatorname{vol}_{\mathrm{prod}}, where \operatorname{vol}_{\mathrm{prod}} =\operatorname{vol}_P\wedge\operatorname{vol}_T for a nowhere vanishing local top form \operatorname{vol}_T on T, and u is pulled back from a unit at the generic point of E.

Let v=\operatorname{ord}_{F_D} be the normalized valuation of the divisor in Lemma 8.2. The product field is the compositum \C(P\times T)=\C(X)\C(T). Geometric integrality of the generic fiber makes this a Galois extension of \C(X) with group G. Extend v to a normalized divisorial valuation w of the product field \C(P\times T), choosing the extension above E. Indeed, conjugating any extension by G preserves its restriction v and moves its base center through all divisors of T over D. Write r for the ramification index of w/v. Since v(t)=m_D,

The center of w on the proper product projects dominantly to E. In particular the product is smooth at the generic point of this center, and the unit u above is a w-unit. Denote its log discrepancy by A_{\mathrm{prod}}(w). The volume form on the smooth X has order zero at v. The order of the same pulled-back top form can therefore be computed in the finite extension of valuation rings and on the product model:

The first line uses the different exponent r-1 of a tame extension of discrete valuation rings; the residue extension is separable in characteristic zero. The second line uses \operatorname{ord}_w(\operatorname{vol}_{\mathrm{prod}}) =A_{\mathrm{prod}}(w)-1. On the smooth product, the pair defined by the smooth divisor z=0 is log canonical, so A_{\mathrm{prod}}(w)\ge w(z). Consequently (8.5) gives

This last inequality can also be read in local coordinates: the pullback of a form containing dz has order at least w(z)-1. Combining (8.4) and (8.6) proves (8.2).

Let k be the inertia order of T\to T_l along this divisor. Its image on A is a group of translations fixing a(z=0), so that image is trivial. Quotienting A by the translation group is an isogeny and is everywhere unramified. The indices in the two compositions consequently give

In particular L_lm_D\le M_l, which implies (8.3).□

Proof. Set m=m_D, and make the base change t=s^m. Near the generic point of D, the normalized source X' of this base change is smooth and its map to X is étale. Indeed f^*t has the single divisor mF_D. In the regular local rings of X one can write f^*t=u h^m, with u a unit; adjoining s and normalizing adjoins an m-th root of the unit. This is étale. Equivalently, the extension is unramified in codimension one and purity applies. The divisor s=0 on X' is reduced. These statements hold on a neighborhood of the whole fiber over the generic point of D, which is the neighborhood used below.

Put \delta=\gcd(M,m). On a component of the normalized compositum of the two base covers, use a parameter z' so that

Its cyclic inertia over the rooted base has order

The order of the pulled-back volume on the new product is L'-1, where

This follows by substituting z=(z')^{m/\delta} in z^{L-1}dz. In particular 0<L'\le M', by (8.2).

Write T' for the chosen normalized cover, E' for its divisor z'=0, and \phi':P\times T'\dashrightarrow X' for the induced rational map. Let \omega' be the pullback of \omega to X'; it is nowhere vanishing on the smooth neighborhood just described. We first show that the full affine inertia action has a common fixed point. Take an actual divisor of X' over s=0, with normalized valuation v', and extend it to a normalized divisorial valuation w' of the product field above E'. Geometric integrality of the generic fiber makes this field extension Galois with the same group as T' over the rooted base. Let r'=e(w'/v'); in this tame Galois extension it is also the inertia order. Since v'(s)=1, reducedness gives

The center Z of w' on the proper product projects dominantly to E'. Thus

Every inertia element fixes \kappa(w'); its restriction to the base therefore fixes \kappa(E') and belongs to the cyclic base inertia of order M'. Hence r'\le M', and therefore

Thus the whole cyclic base inertia fixes \kappa(w'), and hence \kappa(Z). The generic point of Z supplies a point of P over \kappa(Z) fixed by this full affine group. Its common fixed locus is a constant closed subscheme of P, so nonemptiness after that field extension implies nonemptiness over \C. Translation by a complex fixed point conjugates the affine action to the linear action. In particular their orders agree.

Suppose now that the linear action has order two. It is a pseudoreflection, so its determinant on the top fiber form is -1. A generator rotates z' by a primitive M'-th root \zeta. The leading coefficient of the volume is independent of the P-variable, and inertia fixes the tangential base residue field. Invariance of the volume therefore gives

Thus M' is even and L'\equiv M'/2\pmod {M'}. The bound 0<L'\le M' makes this

Consider instead the product divisor P\times E' itself, with its normalized valuation w_0. The cyclic base inertia acts on \kappa(w_0) through its affine action on P, of order two. If v_0 is the normalized restriction to the field of X', every element of the inertia of w_0/v_0 fixes \kappa(E') and therefore lies in this base inertia. It is exactly the kernel of the displayed residue action, so

Properness over the rooted base supplies a center for v_0 on X'. It lies in the smooth neighborhood of the whole fiber over the generic point of s=0. Thus, even before knowing whether v_0 is a divisor on X', the nowhere vanishing form \omega' satisfies \operatorname{ord}_{v_0}(\omega')=A_{X'}(v_0)-1. Computing the top-form order and the order of s in the finite extension gives

Since L'=e_0=M'/2, these equalities say

A normalized divisorial valuation over a smooth variety with log discrepancy one is the valuation of a divisor on that variety: a center of codimension at least two has log discrepancy at least two, as is seen by taking regular parameters at its generic point. Hence v_0 is an actual divisor of the smooth source X'. Its displayed multiplicity two contradicts the reduced divisor defined by s. This proves the lemma.□

Proof. Choose a connected dense open B^\circ on which f is smooth, the base cover is unramified, and the product description in Lemma 8.3 holds. Write X^\circ=f^{-1}(B^\circ). The smooth proper map X^\circ\to B^\circ has connected fibers, so its map of fundamental groups is surjective. Since X is simply connected, \pi_1(X^\circ) is normally generated by small meridians around the divisorial boundary of X^\circ. One obtains this statement by filling loops and putting the filling discs in general position; the subset of complex codimension at least two does not contribute. Equidimensionality ensures that the inverse image of a subset of base codimension at least two also has codimension at least two. By Lemma 8.2, a meridian around F_D projects to the m_D-th power of a meridian around D.

The unramified restriction of T\to B is connected, and hence its monodromy has full image G. Translation parts act trivially on H^1(P); the linear monodromy is therefore exactly G_l. The preceding normal generation says that it is generated by conjugates of the linear parts of g_D^{m_D}. Lemma 8.4 makes each nontrivial such part a pseudoreflection; Lemma 8.5 excludes order two. Notice that generation uses the powered meridians. Replacing them by unpowered inertia would lose this conclusion.

The global invariant cycle theorem, applied to the smooth projective family X^\circ\to B^\circ and its smooth projective compactification X, identifies

The hypotheses here are the algebraic smooth family over the connected open and the proper smooth compactification; no assertion about invariants on an arbitrary analytic disc is being used [16, Théorème 4.1.1(ii)]. The classes e,[\sigma],[\overline\sigma] restrict to zero, and the ample class restricts nontrivially. These four classes span H^2(X,\C), so the displayed space has dimension one.

If \operatorname{Lie}(P) had a nontrivial invariant splitting, average a positive Hermitian form over the finite group and take an orthogonal invariant splitting. The imaginary parts of its restrictions to the two summands give two independent invariant real alternating forms on the underlying real tangent space. Their constant forms represent independent classes in H^2(P,\C)^{G_l}, a contradiction. Thus the representation is irreducible.

We use the following precise consequence of the reflection classification. An irreducible finite complex reflection group generated by reflections of order greater than two has dimension at most four. Indeed, in the imprimitive case the imprimitivity spaces are lines and every reflection is either diagonal or has a two-coordinate block

The latter has order two; diagonal reflections alone cannot generate an irreducible representation of dimension greater than one. These are exactly the alternatives in [53, §2, pp. 276–277]. In the primitive case, [14, Proposition (5.9)(iii), p. 430] states that dimension at least five permits no reflection of order greater than two. Applying this consequence proves n\le4.□

Proof. First describe the branch divisors of a_l. If R\subset T_l has ramification index L_l>1, then (8.3) implies M_l>1 for p_l. Its nontrivial cyclic inertia fixes R pointwise and therefore fixes the divisor a_l(R) pointwise. Since the linear action of G_l is faithful, a generator acts on A_l by an affine pseudoreflection. Consequently a_l(R) is a component of its fixed locus: it is a translate of an abelian subvariety of codimension one. We call such a translate an abelian hyperplane.

After shrinking about z_0, every ramification divisor that meets the neighborhood contains z_0. Its inertia is contained in J, since an automorphism fixing the divisor pointwise fixes z_0. The tangent hyperplane of its image therefore contains V_J.

Let v\in V_J. The finite separable map a_l gives a unique meromorphic lift \widetilde v of the constant field v. It is regular in codimension one. At an unramified generic point this is immediate. At a general point of a ramification divisor choose coordinates in which a_l is

Since v is tangent to the abelian branch hyperplane, its normal component is zero and the lift has no pole. Normality then extends the derivation everywhere: for any regular function h, the meromorphic function \widetilde v(h) belongs to every local height-one ring and thus to the normal local ring itself. Uniqueness of the lift and J-invariance of v show that \widetilde v is J-invariant. The germ of p_l is the quotient by J, so the field descends to that germ of B.

These descended fields have independent evaluations. To verify this without assuming either germ smooth, center exponential coordinates of A_l at a_l(z_0), so the finite affine J-action becomes linear. Choose J-invariant linear coordinate functions \ell_1,\ldots,\ell_r, dual to a basis v_1,\ldots,v_r of V_J. The invariant functions a_l^*\ell_i descend to functions u_i on the base germ, and the descended fields \xi_j satisfy

They therefore split r smooth directions by their local flows. The same directions persist on the maximal quasi-étale chart: the fields lift uniquely across a quasi-étale finite map (check regularity in codimension one and use normality), and the pulled-back functions retain these identities. On S_0 the evaluation rank on that chart is d=1, by Proposition 4.3. Thus r\le1.□

Proof. The coefficients are uniquely defined meromorphic functions, since a_l is generically étale. It is enough to check regularity at every prime divisor of T_l. There is nothing to prove where a_l is unramified. At a general ramification point write M=M_l, L=L_l. Use linear coordinates (x,y_2,\ldots,y_n) on A_l adapted to the affine fixed hyperplane x=0. The inertia fixes the y_i and acts on x by a root of unity. The restrictions of a_l^*y_i are generically independent along the divisor. They can be used as tangential coordinates on T_l, and, after absorbing a unit in the normal parameter,

Choose z so that the cyclic inertia rotates it by a primitive M-th root. The local invariant ring of this cyclic quotient has coordinates z^M,y_2,\ldots,y_n. Thus p_l^*u=F(z^M,y) for a holomorphic function F, and

All coefficients are holomorphic because L\le M. The displayed coordinate covectors are constant linear combinations of the \alpha_i, so this proves the required codimension-one regularity. Normality extends every h_i.

Expression (8.9) is unique on the function field. Its left side is J-invariant and its right side transforms by the linear representation on the \alpha_i. Evaluation at the J-fixed point z_0 therefore gives an invariant covector, as asserted.□

Proof. All components C are projective curves mapping finitely onto S_0. Let J_0 be their generic pointwise stabilizer. More explicitly, discard the proper fixed loci of those finitely many group elements that do not fix C identically. On the remaining open the stabilizer is the subgroup J_0 fixing C pointwise. Equivariance implies

Every component of this fixed locus is a translate of an abelian subvariety with tangent space V_{J_0}. Lemma 8.7 bounds its dimension by one, whereas finiteness of a_l makes the image of C one-dimensional. It follows that the image is exactly an elliptic translate E, and V_{J_0}=\operatorname{Lie}(E).

Write \nu:\widehat C\to C for the normalization and let

be the two finite morphisms. At any q\in\widehat C, choose a parameter u on the smooth curve S_0 near b(q), and extend it to a local function on B. Such an extension exists because S_0 has its reduced closed-subspace structure. Restricting (8.9) gives

near q, where \theta is a nonzero translation-invariant one-form on E and \eta is holomorphic. There is no assumption here that T_l is smooth along C. To justify restriction, first pull (8.9) to a resolution of T_l. Both sides are regular ordinary differentials there and agree generically, hence everywhere. A component of the proper base change dominating \widehat C then gives the stated equality on \widehat C, since pullback of differentials along a dominant map to a smooth curve in characteristic zero is injective.

Equation (8.10) implies pointwise

Both targets have genus one. Riemann–Hurwitz therefore gives the same total ramification degree, 2g(\widehat C)-2, for both maps. The pointwise inequalities must all be equalities. In particular \eta(q)\ne0.

Put z=\nu(q) and J=\operatorname{Stab}_{G_l}(z). The covector in Lemma 8.8 is J-invariant, and the nonvanishing of \eta(q) says that its restriction to \operatorname{Lie}(E) is nonzero. Since J contains J_0, we have

A finite-dimensional representation of a finite group in characteristic zero has equally dimensional invariant vector and covector spaces, by the averaging projector. Thus V_J\ne0, and the displayed inclusion is equality. This holds at every point and on every normalization branch.

Fix a nonzero v_0\in\operatorname{Lie}(E). Its unique meromorphic lift through a_l is holomorphic near each point of C, by Lemma 8.7, and the lifts agree on overlaps by uniqueness. Call the resulting field \widetilde v_0. It has nonzero evaluation everywhere on C, since da_l(\widetilde v_0)=v_0. Locally it descends through p_l to a field on B. Its flow preserves the intrinsically defined minimum stratum S_0, and hence its lifted flow preserves the reduced inverse image and each of its irreducible curve components. Thus \widetilde v_0 is tangent to C.

A reduced curve carrying such a nonvanishing ambient holomorphic field is smooth: choose a function whose derivative under the field is nonzero, and use the local flow to split off its disc direction. An invariant reduced curve in this product is the disc times a reduced zero-dimensional germ. It follows that C=\widehat C is smooth. Since da_l(\widetilde v_0)=v_0\ne0, the map C\to E is unramified. It is consequently an étale cover of an elliptic curve, so g(C)=1. The equality of ramification divisors already proved also makes C\to S_0 unramified.□

Proof of Proposition 8.1. If a nonquotient germ remained, (8.8) and Lemma 8.9 would supply C and a nonzero translation direction v_0 as above. Choose a positive integer r such that \mathcal L=\cO_B(rK_B) is a line bundle, and put

We construct a holomorphic connection on \mathcal M|_C; in particular we must check both its extension and its gluing.

Near z\in C, the field \widetilde v_0 is invariant under the point stabilizer and descends to a holomorphic field \xi_z on the corresponding base germ. On B_{\mathrm{reg}}, Lie differentiation by \xi_z acts on r-fold canonical tensors. It extends to an operator

Indeed \mathcal L is the reflexive extension of its restriction to B_{\mathrm{reg}}, so a differentiated section extends uniquely across the subset of codimension at least two. The operator obeys

Pull it to the chosen neighborhood in T_l using the lifted symbol \widetilde v_0: in a local pulled-back frame its rule is

The Leibniz identity and \widetilde v_0(p_l^*u)=p_l^*(\xi_z u) make this definition independent of the frame.

On an overlap, both operators \widetilde{\mathcal D}_z have the same symbol. They agree on the dense open where p_l is étale and both varieties are smooth: there the differential of p_l identifies the pulled-back canonical tensors with canonical tensors upstairs, and both operators are Lie differentiation by the same field \widetilde v_0. Their difference is an \cO_{T_l}-linear endomorphism of the line bundle \mathcal M; its coefficient is holomorphic and zero on a dense open, hence is zero. Thus the local operators glue on a neighborhood of C. This argument compares the operators upstairs and does not require a single descended vector field on a neighborhood of all of S_0.

Tangency of \widetilde v_0 to the reduced curve C implies that the glued operator preserves \mathcal I_C\mathcal M, by (8.11). It therefore induces a first-order operator

with symbol the nowhere vanishing field \widetilde v_0|_C. Let \tau be the holomorphic one-form on the elliptic curve C satisfying \tau(\widetilde v_0)=1. Then

is a holomorphic connection on \mathcal M|_C.

A line bundle with a holomorphic connection on a smooth projective curve has degree zero, a classical connection obstruction [3]. One elementary proof takes a nonzero meromorphic section s: the meromorphic one-form \nabla s/s has at each point the residue \operatorname{ord}_q(s), and the residue theorem gives \deg\operatorname{div}(s)=0. Consequently

On the other hand, -K_B is ample as a \Q-Cartier divisor, and p_l|_C is finite onto the positive-dimensional curve S_0. Hence

This contradiction excludes the nonquotient germ and proves Proposition 8.1.□

Proof. Given a realized pair (P,\kappa), nearby planes T' contain oriented period planes P' close to P and orthogonal positive vectors \kappa' close to \kappa. Local Torelli realizes P' by a small deformation of that manifold. Openness of the relative Kähler cone makes \kappa' Kähler, proving openness.

Recall the two parts of the MBM theorem used here: the negative rational wall classes are invariant under deformation that keeps them of type (1,1), and the Kähler cone is a component of the positive cone after removing their orthogonal hyperplanes [2, Theorems 1.17 and 1.19]. No bound on their squares is used. A realized plane cannot be orthogonal to such a class: that class would be of type (1,1) and orthogonal to a Kähler class. Conversely, suppose T has no such orthogonality. Choose a generic oriented plane P\subset T, so that every rational class orthogonal to P is already orthogonal to T. Indeed, each rational vector not orthogonal to T excludes a proper closed subset of the sphere of planes in T, and there are only countably many such vectors. Period surjectivity gives a marked manifold with period P. Its rational (1,1) classes include no MBM class. The cone theorem therefore identifies its Kähler cone with the whole positive component. The orthogonal ray to P in T, with sign fixed by the orientation, is in that component. This realizes T.

For rank four one can avoid even the MBM formulation in this step. If the negative line T^\perp is irrational, the same generic choice of P has zero Picard lattice. Its Kähler cone is the whole positive component, so T is realized. Thus a nonrealized plane has rational negative normal line.

Finally, an intersection between the Kähler cones of two marked models with the same period gives a Hodge parallel-transport isometry carrying a Kähler class to a Kähler class. Torelli makes it the isometry of an isomorphism. The cones then agree.□

Proof. Choose b\in U_{\Q} with q(a,b)=1 and q(b)=0; subtracting q(b_0)a/2 from any b_0 paired to 1 with a gives such a vector. Let V=\langle a,b\rangle^\perp\cap U, a positive rational plane. For x\in V_{\Q} the null rotation is

Direct pairing verifies that it preserves q, and substitution gives T_{a,x}T_{a,y}=T_{a,x+y}. Thus T_{a,x}^m=T_{a,mx}. Its matrix entries in an integral basis are polynomials in m of degree at most two with rational coefficients and constant term the identity. For sufficiently divisible m, both T_{a,mx} and its inverse are integral. This accounts also for the finite gluing between the sublattices in U and U^\perp.

For x\ne0, choose a rational positive vector \ell\in V\cap x^\perp. The rotation fixes \ell. Replace it by a primitive integral vector on the same line. A generic period in \ell^\perp has Picard lattice \Z\ell. Period surjectivity and the cone theorem give an ample marked deformation with polarization \ell or -\ell; choose the ample sign. This sign does not alter the stabilizer of \ell.

We now use the prescribed-polarization conclusion of [44, Lemma 4.2]. It applies to this primitive positive class \ell, ample on a manifold in the chosen marked component. Write D_\ell for the component of its type-IV period domain containing that ample period, and put

The cited lemma supplies a smooth projective family of IHS manifolds over a smooth connected quasi-projective variety S, with a relative polarization of class k\ell for some k>0 and a reference marking obtained by parallel transport from X_0. It also supplies a torsion-free finite-index subgroup \Gamma\subset G_\ell(\Z) containing the family monodromy and an algebraic period map

Here Q_\ell is smooth and quasi-projective, the map D_\ell\to Q_\ell is a covering with free \Gamma-action, and a local marked lift of \Phi is submersive. The type-IV domain D_\ell is contractible, including in rank four, so this covering identifies \pi_1(Q_\ell) with \Gamma.

The remaining finite-index assertion concerns the image of this actual family monodromy, and we prove it here. The submersive local lift makes the algebraic map \Phi dominant.

Choose an irreducible algebraic slice S'\subset S of dimension \dim Q_\ell dominating Q_\ell. For example, intersect with general hyperplanes through a smooth submersive point, transverse to its period fiber. After normalization and restriction, this slice is smooth and generically finite over Q_\ell. There is a nonempty Zariski-open Q^\circ\subset Q_\ell over which its restriction is a nonempty finite étale cover: generic finiteness gives a finite open after removing the boundary in a proper compactification, and characteristic zero allows removal of the branch locus. Therefore its fundamental-group image has finite index in \pi_1(Q^\circ). The map \pi_1(Q^\circ)\to\pi_1(Q_\ell) is surjective, since a loop in a complex manifold can be moved off a proper algebraic subset. It follows that

The period map on the universal cover of S, based at this reference marking, is equivariant for the actual second-cohomology monodromy. Because D_\ell\to Q_\ell is a covering with free action, the deck transformation induced by each loop is exactly that cohomology operator. Hence the subgroup in (9.6) consists of actual parallel transports, with the original marking retained.

Our rotation fixes \ell; after taking an integral power, a further power preserves D_\ell and therefore lies in G_\ell(\Z). Since \Gamma has finite index in G_\ell(\Z), a further power lies in \Gamma; by (9.6), one more positive power lies in the family monodromy. This uses only that every element has a positive power in any finite-index subgroup; normality of the subgroup is unnecessary. Transport to the original reference fiber finishes the proof. The zero parameter gives the identity.□

Proof. Choose a rational null partner b of e on the line of w_* with q(e,b)=1, and a rational orthogonal basis v_1,v_2 of their positive perpendicular plane. Write q(v_1)=A>0, q(v_2)=B>0. For a rational number t\ne0, set x=v_1, y=v_1+t v_2. Choose t so that

This is possible: the expression lies in (-2,2) and assumes each of the three possible integral values at only finitely many real t. By Lemma 9.3, for some positive integers m_1,m_2 the opposite rotations T_{e,m_1x} and T_{b,m_2y} are integral monodromies.

For completeness, the spectral calculation can be performed in a real orthonormal transverse basis without any rationality assumption on \sqrt A or \sqrt B. Identify a e+b' b+c v_1+d v_2 with the Hermitian matrix

Its determinant is the negative of q. The action H\mapsto MHM^* of SL_2(\C) induces SO^+(3,1). Formula (9.5) corresponds, for the first rotation, to an upper triangular unipotent with parameter -m_1\sqrt A/\sqrt2; the opposite rotation corresponds to a lower triangular unipotent with parameter m_2(\sqrt A-i t\sqrt B)/\sqrt2. Put

Its spin trace is

Let \lambda_k,\lambda_k^{-1} be its spin eigenvalues, choosing |\lambda_k|>1. For large k, \lambda_k/\tau_k\to1. The four eigenvalues on U_{\R} are

The sum of the last pair converges to 2\cos(2\arg\zeta)=2(A-Bt^2)/(A+Bt^2). By (9.7), it is nonintegral for all sufficiently large k. In particular, this conjugate pair is nonreal and neither eigenvalue is 1 or -1.

The characteristic polynomial on U is monic with rational coefficients, and its roots are algebraic integers because \gamma_k preserves \Lambda. It therefore belongs to \Z[z]. It is reciprocal, has constant term 1, and has no rational root: a rational root would be 1 or -1. If it factored into two quadratics over \Q, real coefficients would force the nonreal pair to belong to one quadratic. That monic factor has integral coefficients, so the sum of the pair would be an integer, a contradiction. Thus it is irreducible. Fix a sufficiently large k.

Finally, a nontrivial null rotation fixing [w_*] has powers converging to [w_*] on every other point of the null boundary. In boundary coordinates with [w_*] at infinity these powers are translations z\mapsto z+jc, with c\ne0. Such a rotation has an integral monodromy power by Lemma 9.3. Neither eigenline of \gamma_k is rational, by irreducibility, so neither is [w_*]. Conjugating \gamma_k by a sufficiently high power of this rotation brings both eigenlines as close to [w_*] as desired and preserves all the asserted properties. Call the conjugate \gamma.□

Proof. Choose future eigenvectors for a_+,a_- with eigenvalues \rho,\rho^{-1}, respectively. The remaining real invariant plane is positive definite and \gamma acts on it by a rotation. Since e is a future null vector different from both eigenlines, its coefficients along both future eigenvectors are positive. Indeed, pairing its eigenspace decomposition with either eigenvector shows the coefficient of the other to be positive. It follows that

where v_+,v_- are positive representatives of a_+,a_-.

If the assertion failed, choose N\to\infty and z_N=[s_Ne+\gamma^Ne]\in J, with s_N>0. We first claim

To prove the first limit, normalize \gamma^Ne by c_N=-q(e,\gamma^Ne); by (9.12), c_N/\rho^N tends to a positive constant, and [\gamma^Ne]\to a_+\in\mathcal V. The tail exclusion (9.11) bounds s_N/c_N above. If a subsequence of s_N/\rho^N had a positive limit, z_N would converge to an interior point of (e,a_+). Closedness of J would put that point in J, contradicting (9.9). This proves the first limit. Apply the same argument to

The uniform tail at e now uses a_-\in\mathcal V and gives s_N^{-1}\rho^{-N}\to0, which is the second limit.

Choose integers k_N such that

The two limits in (9.13) imply k_N\to\infty and N-k_N\to\infty. Passing to a subsequence, suppose s_N\rho^{2k_N-N}\to r\in[1,\rho^2]. Apply \gamma^{-k_N} to z_N and divide its representative by \rho^{N-k_N}. Equation (9.12) gives

The limit has negative square, so its line is an interior point of the axis (a_-,a_+). Invariance and closedness of J put that line in J, contradicting (9.9).□

Proof. Nonzero variation means that, at some regular fiber, the derivative in a base direction of its weight-one period map is nonzero. This condition persists under small deformations of the fibration: on a small smooth product neighborhood in the family supplied by Proposition 7.2, the relative Hodge bundle and that derivative vary holomorphically, so a chosen nonzero matrix entry stays nonzero.

The fiber polarization can also be transported. Its class is the restriction of an integral ample class on X_0. By Lemma 7.1 and topological transport, the restriction image on a nearby smooth fiber is a rational one-dimensional space with kernel e^\perp. The restriction of a Kähler class spans its real form, since it pairs positively with e, so the transported integral class remains of type (1,1). Positivity on the smooth torus is open. This gives a polarized family near the chosen fiber, and the same argument applies at the conjugate fibration.

In rank four the preliminary point was X_*=X_0. Complex conjugation of its hyperkähler complex structure reverses the period-plane orientation and the Kähler sign. The resulting marked manifold \overline X_* has period \overline P_*, ample class -h, and conjugate fibration \overline f_*:\overline X_*\to\overline B with fiber class -e. It lies in the same marked deformation component: a half-circle of the twistor sphere joins the complex structure to its negative. Nonzero variation of \overline f_* is equivalent to that of f_*.

We choose the endpoint neighborhood in Lemma 9.4 small enough for the following comparison with this conjugate fibration. Under the actual parallel transport \gamma^{-N}, the underlying plane of P_N becomes

It approaches an unoriented plane near P_*, since [\gamma^{-N}e]\to a_- and a_- can be chosen arbitrarily close to [w_*]. The transported positive cone is

By (9.4), this cone contains a vector close to -h. Preservation of the orientation of maximal positive three-spaces therefore makes the transported oriented period close to \overline P_*.

Choose the endpoint neighborhood sufficiently small, and then N sufficiently large, that this period lies in a neighborhood where the (-e)-deformation of \overline f_* exists and has nonzero variation. Let Y_N be that actual small deformation of \overline X_* with period \gamma^{-N}P_N. It has a Kähler class near -h, in the component (9.18). In rank four, H^{1,1}(X_N,\R)=\langle e,u_N\rangle_{\R}, so (9.15) says that the entire positive component of X_N is Kähler. The Hodge parallel-transport isometry between Y_N and X_N induced by \gamma^N consequently carries a Kähler class to a Kähler class. Torelli identifies these manifolds by an isomorphism inducing that isometry. The class -e on Y_N becomes u_N on X_N.

Line bundles on an IHS manifold are determined by their Chern classes. Thus the transported fibration and the fibration obtained from u_N have the same connected fibers: positive generated powers define the same Stein factorization. It follows that g has nonzero polarized variation. The isometry in this comparison is the actual parallel transport constructed in Lemma 9.3.□

Proof. We first check the independence needed to define (10.1). On a maximal chart, lift a form on the smooth orbit to an ordinary ambient differential, using the local product presentation of Proposition 4.5. Every tangent vector to the smooth finite cover of this chart maps, at a point above the orbit, into the orbit tangent space. Here is the trace argument in this finite-map situation. Work with a pointed finite local branch R\subset E of generic degree m, with E regular. Extend a tangent vector upstairs to a local derivation \delta of E. The rule

is a derivation of R: its values are regular by normality, and the Leibniz rule follows from R-linearity of trace. Its value at the marked point is the differential image of the chosen tangent vector, since all conjugates in the pointed branch have the same residue and trace on the residue is multiplication by m. Evaluations of ambient derivations are precisely the orbit tangent directions, by Proposition 4.3.

The normalized pullback of a maximal chart to A_t is smooth and étale over A_t. Indeed the quasi-étale exceptional locus has codimension at least two, as does its inverse image under the finite map A_t\to B, and purity applies. Thus the preceding tangent calculation applies there. The difference of two lifts of an orbit form vanishes on the orbit tangent, so its pullback vanishes as an ambient cotangent vector at every point above the orbit. Restricting to the reduced inverse image and then pulling to its resolution kills this difference. The resulting maps are compatible on the chart overlaps and with the finite group actions; hence they descend to (10.1).

The map of the entire family to the fixed stack \cS makes h^*\alpha a global flat section. It is nonzero on every fiber. To see this, choose \beta\in H^{d-a,d-b}(\cS) with \alpha\smile\beta\ne0, using Poincaré duality and the Hodge decomposition on \cS. Pullback on H^{2d}(\cS,\C) is nonzero: the dth power of a coarse ample class has positive integral on a dominating component of Q_t. This also holds for d=0, where the class is the unit. Since \cS is connected, its top cohomology is one-dimensional; consequently h_t^*(\alpha\smile\beta)\ne0 and h_t^*\alpha\ne0.

Finally, take the ath exterior power of (10.1), and use

On H^b the resulting sheaf maps factor Dolbeault pullback through

Under the Dolbeault identification the last arrow is cup product with the pulled-back holomorphic forms. This is exactly (10.3).□

Proof. First we verify the stated fixed-part condition from the geometric hypotheses. Write U=U^b. A pure tensor word has the form

The polarizations give V^\vee\simeq V(1) and U^\vee\simeq U(b), so

The untwisted word is a flat Hodge direct summand of R^{r+r'+b(s+s')}\varpi_*\Q for the corresponding fiber product \varpi:\mathcal Y\to T of copies of \pi and \rho. The canonical Künneth inclusion and projection commute with monodromy and Hodge decomposition; no algebraic Künneth correspondence is required. The product remains smooth and projective when fibers of \rho are disconnected. Deligne’s Theorem 4.1.1(ii) and Corollary 4.1.2 of [16] supply invariant lifts from a nonsingular compactification of this product and make their Hodge components globally flat. Applying the Künneth projection and then the Tate twist proves the fixed-part condition for \mathbb M. It also proves it for flat Hodge direct summands such as exterior and symmetric powers. The summand projectors decompose every tensor construction on V\oplus U into these pure words. Since T is connected, global flat sections are exactly full-monodromy invariants, and taking the Zariski closure does not change them.

The same geometric hypotheses give semisimplicity of V_\Q\oplus U^b_\Q by Deligne’s Proposition 4.2.5 and Theorem 4.2.6 of [16]. Both cohomology systems have rational structures, integral lattices modulo torsion, and polarizations on the smooth algebraic base T. Semisimplicity and faithfulness imply that the unipotent radical of G^\circ is trivial. The component group G/G^\circ is finite, so averaging complements in characteristic zero makes the full, possibly disconnected group G linearly reductive.

A linearly reductive subgroup of a general linear group is the pointwise stabilizer of its invariant mixed tensors. One way to check the pointwise assertion is to realize the subgroup as the stabilizer of a line in a rational representation, realize that representation in mixed tensors, and choose an invariant complementary subspace to the line by linear reductivity. Its projection is an invariant tensor; fixing that tensor preserves the line and hence belongs to the subgroup. Applying this to the faithful representation V\oplus U^b, with its invariant summand projectors included, gives

For a flat invariant tensor of a single Hodge type, the three parts of its connection have different Hodge degrees. Its lowering part therefore vanishes separately. Applying this to every flat Hodge component just described shows that the simultaneous pair in (10.6) annihilates every tensor in (10.9). This proves (10.6); it is an assertion about the Lie algebra of the pointwise stabilizer. Also, the invariant tensor spaces are Hodge graded, so their common infinitesimal annihilator is Hodge graded. It is flat by conjugation transport, and therefore is a subvariation of the indicated weight-zero endomorphism variation.

The grading-preserving part of its connection is the restriction of the ambient Chern connection. Projection to \End(V) is a morphism of variations; its image is a subvariation, and its degree (-1,1) part is E. This proves constant rank and Chern parallelism. The weight-one polarization identifies \overline H_1 with H_1^* and makes these lowering matrices symmetric.

For completeness, the bracket calculation takes place in the polarization-preserving Lie algebra. In a unitary Hodge frame a lowering operator has block matrix

The conjugate of a lift of L_B in \mathfrak g^{-1,1} is in \mathfrak g^{1,-1} and has upper block B^*, up to the same irrelevant polarization sign for all B. The double bracket of lifts of L_A, this upper operator, and L_C has lower block AB^*C+CB^*A, up to sign. Its Hodge degree is (-1,1); projection to the first factor proves (10.8).□

Proof. Let r=\rk A; the assertion is immediate if r=0. By unitary symmetric diagonalization, take coordinates in which A=\operatorname{diag}(\lambda_1,\ldots,\lambda_r,0), with \lambda_i>0. The triple operation gives all odd powers \operatorname{diag}(\lambda_i^{2m+1},0). Interpolation on the finitely many nonzero singular values therefore gives

The linear map Z\mapsto PP^*Z+ZP^*P preserves E and has eigenvalues 2,1,0 on the FF, off-diagonal, and KK blocks. Its polynomial spectral projections thus separate these blocks inside E. A nonzero KK block would, when added to P, have rank larger than r. Hence every member of E has zero KK block. Write

Here D is symmetric and B:K\to F is a rectangular matrix. Triple closure, first with (P,D,P) and then with (D,P,\left(\begin{smallmatrix}0&B\\B^{\mathsf t}&0\end{smallmatrix}\right)), gives

For B\in\mathcal B, the Schur complement computes

Maximality of r implies B^{\mathsf t}B=0. Polarizing in B shows that, for a fixed k\in K, the space

is totally isotropic for the complex symmetric form (u,v)\mapsto u^{\mathsf t}v. In detail, B^{\mathsf t}C+C^{\mathsf t}B=0, and evaluation on (k,k) gives (Bk)^{\mathsf t}(Ck)=0. Equation (10.12) shows that W_k is stable under every D\in\mathcal D and every D^*.

Put S=c(k,-,-). Every cubic contraction has zero KK block; by symmetry, for f\in F and k'\in K,

Thus S has only an FF block, also denoted by S, in \mathcal D. For f\in F, let B_f be the off-diagonal block of c(f,-,-). Cubic symmetry gives Sf=B_fk, so

Since S is symmetric and W_k is isotropic, for w\in W_k and f\in F we have

It follows that S annihilates W_k. But S^* preserves W_k by (10.12); together with (10.13) this proves

The singular values of SS^*S are the cubes of those of S. Hence S=0. This is the Hermitian positivity step; no positivity of the complex symmetric form was asserted.□

Proof. Let r be the maximum rank in E_t. It is independent of t, because Chern parallel transport is unitary and carries E_t onto E_{t'} by congruence. Define intrinsically

Unitary Chern transport preserves the entire collection of these kernels, and therefore their span. Thus (10.15) is a global Chern-parallel subbundle of constant rank. No matrix P, chosen kernel, or finite cover enters its definition.

By Lemma 10.4, the cubic vanishes when any argument lies in K^{\mathrm{all}}. In particular all lowering Higgs operators annihilate this subbundle. Write the real flat connection on V as

with D the Chern connection. On K^{\mathrm{all}}\subset V^{1,0}, \theta vanishes by the cubic and \overline\theta vanishes by Hodge degree. Complex conjugation in the real variation shows that \overline\theta vanishes on \overline{K^{\mathrm{all}}}\subset V^{0,1}, while \theta vanishes there by degree. Consequently

is the complexification of a real flat subbundle L\subset V_\R; its flat connection is unitary. Positivity of the Hodge metric shows that the alternating polarization is nondegenerate on L. Its symplectic orthogonal L^\perp is also a global flat subbundle. These statements concern the full flat connection on T, and hence full monodromy around every loop.

We now compute the relevant full-monodromy invariant space. The use of the one-dimensional fiber-restriction image to prove irreducibility over \R also appears in Bakker’s proof of Matsushita’s variation conjecture [4, Lemma 5], building on Voisin’s argument. Here it rules out the specific flat summand extracted from the maximal-rank kernels. By the global invariant-cycle theorem, in the form of [16, Theorem 4.1.1(ii)] applied to g^{-1}(C^\circ)\to C^\circ with smooth projective compactification X, the image of

is precisely the invariant subspace. Its dimension is one: the independent classes \sigma, \overline\sigma, and g^*c_1(H_C) restrict to zero, whereas an ample class restricts nontrivially; here H_C is ample and b_2(X)=4. The inclusion T\hookrightarrow C^\circ induces a surjection of fundamental groups, because loops in a smooth complex variety can be moved off a closed subset of positive complex codimension. Since V|_T is the restriction of the same local system, its monodromy image is unchanged. Therefore

Invariants commute with extension from \Q to \R: they are the common kernels of rational monodromy-minus-identity maps, and a finite subcollection suffices in finite dimension.

If L were nonzero and proper, the polarization composed with the two flat projections onto L and L^\perp would give two independent real invariant alternating forms. The polarization identifies the dual alternating-square representation with \Lambda^2 V_\R, so this contradicts (10.16). Rationality of L is not needed. If L=V_\R, the Higgs field vanishes identically, contradicting the chosen nonzero variation of g (which remains nonzero on every dense open). Thus L=0. If r<n, every maximum-rank matrix would have a nonzero kernel, making K^{\mathrm{all}} nonzero. Hence r=n.□

Proof of Proposition 10.1. Suppose 0\ne\alpha\in H^{a,b}(\cS) with a>b. The flat section z in (10.4) is fixed by the joint algebraic monodromy G, and its component (10.5) is nonzero. At a point t\in T, choose an invertible matrix A\in E_t and a lift \nu\in\mathfrak g_t^{-1,1}. Let N_V be its induced action on \Lambda^a V_t and N_U its action on U^b_t, acting on their respective tensor factors. Invariance gives

Since U^b_t has Hodge types (p,b-p) with 0\le p\le b, the operator N_U satisfies N_U^{b+1}=0. The two tensor actions commute, so induction gives

because a>b.

On V_t, the operator \nu maps H_{1,t} isomorphically onto \overline H_{1,t}, and kills \overline H_{1,t}. Consequently N_V^a kills every summand of \Lambda^a V_t having fewer than a factors in H_{1,t}. On the whole sector with all a exterior factors holomorphic, including its U^b_t factor, it is

The projection of z_t to this sector is nonzero, because its further projection to (U^b_t)^{0,b} is the nonzero component (10.5). The displayed map is injective and every other exterior sector is killed, so N_V^a z_t\ne0, contradicting (10.17). Thus no H^{a,b}(\cS) with a>b is nonzero; complex conjugation excludes a<b as well.

It follows that all odd cohomology of \cS vanishes and

since h^{0,0}(\cS)=1. Rational cohomology of the smooth stack is that of its coarse space, including when its stabilizers are ineffective. This proves the stated positive Euler characteristic.□

Proof. Take a local Galois chart U\to U/G as above. Its non-étale locus on the coarse space has codimension at least two. The inverse image of that locus has codimension at least two in X, by equidimensionality, and in A, by finiteness. Consequently, the full normalizations of the components of the pullbacks of U\to U/G that dominate these smooth sources are finite, by excellence, and étale over the sources, by purity of the branch locus [56, Tag 0BMB]. Their deck actions are torsor actions: the torsor identities hold on the dense étale locus and extend uniquely to the finite étale covers. The torsors and their equivariant maps to U are precisely the data defining the two lifts. Uniqueness on normalized overlaps makes these data compatible and gives the stated representable lifts. The normalized-source package of Proposition 3.3 gives the asserted properties of \widetilde f on each chart: its source is smooth and symplectic, its map to U is projective with Lagrangian equidimensional fibers, and Stein factorization gives connected fibers and direct image \cO_U.

The pullback of \widetilde p to U is finite, since it is finite over the ordinary finite base change A\times_B U. It is a morphism between smooth n-dimensional varieties and is flat by miracle flatness [56, Tag 00R4]. Thus \widetilde p is finite flat, and it is surjective because its coarse morphism is surjective.□

Proof. Use the global isomorphisms \phi_j:R^j f_*\cO_X\xrightarrow{\sim}\Omega_B^{[j]} chosen once on the original base in Section 2, with \phi_0 the unit isomorphism. We apply the chart comparison of Proposition 4.7. Choose a connected affine quotient chart U\to U/G with U/G\to B an integral separated étale neighborhood, and view U\to\mathfrak B as its atlas map. Write f_U:X_U\to U for the normalized lift, put U^0=U\times_B B_{\rm reg}, and let \pi:U^0\to B_{\rm reg} be the induced étale morphism. The complement of U^0 has codimension at least two. Over this open the lift is the ordinary base change of the original fibration. Ordinary flat base change for \pi, followed by the étale cotangent identification, gives

This open includes the discriminant over the regular part of B; the flat map used here is \pi, so no smoothness of the fibers is required for the base-change step [56, Tag 02KH].

For j>0, Ou’s local reflexivity theorem applies to f_U: Proposition 3.3 supplies its projective, equidimensional map with connected fibers and smooth canonical-trivial source, and U is smooth [46, Theorem 1.3]. Both R^j(f_U)_*\cO_{X_U} and \Omega_U^j are therefore their reflexive extensions from U^0. The displayed isomorphism extends uniquely across the complement. For j=0 the same conclusion is the unit isomorphism from (f_U)_*\cO_{X_U}=\cO_U. On a chart overlap the extended maps agree over the inverse image of B_{\rm reg}, by transitivity of flat base change from the same \phi_j. Every component of the overlap meets this open, since it maps étale to a chart and cannot be supported over its codimension-at-least-two complement. Uniqueness of reflexive extension makes the maps agree everywhere. Thus they glue to degreewise isomorphisms

For j>n, the restriction to U^0 is zero by Proposition 2.1, and reflexivity gives zero on U. Thus no higher direct-image terms occur.

These isomorphisms first give (11.3) in coherent K-theory. In this setting coherent and vector-bundle K-theory agree. Indeed, T_{\mathfrak B} is faithful on every stabilizer by linearization of finite group actions. A finite direct sum \mathcal G of bounded tensor powers of this bundle and its dual contains every irreducible stabilizer representation occurring on a finite system of quotient charts. Exactness of finite-group invariants makes the evaluation

surjective for every coherent \mathcal F. For sufficiently large k, the coherent sheaf c_*(\mathcal G^\vee\otimes\mathcal F)\otimes H^k is globally generated on B. Its generators yield a surjection from a finite sum of \mathcal G\otimes c^*H^{-k} onto \mathcal F. Iterating and using regularity of the smooth stack gives a finite locally free resolution. This proves the claimed comparison and allows the identity to be pulled back in vector-bundle K^0, even along an inertia morphism which is not flat.

We spell out the nilpotence needed for (11.4). If V is a vector bundle of rank r on a smooth projective variety Y, then [V]-r is nilpotent in K^0(Y). Indeed, pull back to the full flag bundle of V. This pullback is injective on K^0, since pushforward of its structure sheaf is \cO_Y, with no higher direct images. The pulled-back class is a sum of classes [L]-1 for line bundles. For a globally generated line bundle L, a finite set of generating sections and its exact Koszul complex show that (1-[L^{-1}])^N=0 for some N. An arbitrary line bundle is a quotient of two sufficiently positive globally generated line bundles. Therefore its class minus one is also nilpotent: it is an invertible multiple of a difference of two commuting nilpotent elements. A finite sum of commuting nilpotents is nilpotent. This proves the assertion.

Apply it on the original projective X to \widetilde f^*T_{\mathfrak B}. The projection formula and (11.3) give (11.4), with one exponent m before any inertia component is chosen.□

Proof. An inertia component is a smooth proper stack: on a quotient chart its presentation is a fixed locus U^h modulo the appropriate centralizer. The fixed eigensubbundle of j^*T_{\mathfrak B} is T_S. Also, W is a scheme, since it is finite over A. The morphism W\to S is finite flat, being a base change of \widetilde p. Thus W is projective, pure-dimensional of dimension d, and Cohen–Macaulay. These assertions hold even if W is nonreduced.

Restrict (11.4) to S and take the character of the tautological automorphism. This is the inertia eigenbundle character used in Toën’s Riemann–Roch framework [57, Section 3.3 and Lemma 4.6]; only its explicitly described ring homomorphism and moving-factor cancellation are needed here. More explicitly, for a bundle V on \mathfrak B, decompose j^*V=\bigoplus_{\zeta} V_\zeta into the h-eigenbundles and use the ring homomorphism

Pull this identity to W and take its Chern character in H^{\rm even}(W,\C). The degree-zero term coming from [T_{\mathfrak B}]-n is \operatorname{tr}(h|T_{\mathfrak B})-n, which is nonzero. Indeed, the tangent action is faithful and all eigenvalues have absolute value one; their sum can be n only if every eigenvalue is one. This would make h the identity by finite-order linearization. The character of each moving cotangent factor in the alternating exterior sum has nonzero degree-zero term (1-\zeta)^{\operatorname{rk}V_\zeta}, for \zeta\ne1. The eigenbundle ranks are constant on connected S, so each of these degree-zero terms is the same nonzero scalar on every connected component of W. Each full class is therefore a unit: every positive-degree element of the cohomology of the finite-dimensional projective scheme W is nilpotent. Cancelling them gives

The term of degree 2d on the left is c_d^{\rm top}(E), since for Chern roots x_i of E the expression is \prod_i(1-e^{-x_i}). Hence

In particular, a nonempty nonidentity inertia component of dimension zero is impossible, as in that case this says 1=0.

Consider the bundle map

used in (11.5). For every reduced irreducible subvariety Y\subset W of dimension s, this map has rank at least s at its generic point. In fact, Y\to S is finite onto its image and is generically separable. On a smooth dense open of Y, and after an étale chart on S, its differential consequently has rank s. The differential identity for the fiber square says

because the latter differential has values in the fixed tangent subbundle. This proves the rank assertion.

It follows that the locus on W_{\rm red} where \operatorname{rk}\alpha\le r has dimension at most r: apply the preceding assertion to each of its irreducible components. Over the stratum of rank r, the incidence variety

has dimension at most r+(n-r)=n. There are finitely many rank strata. Thus a general u gives a section whose zero scheme has at most zero-dimensional support. It is a finite scheme because W is proper.

At every such zero, the d local entries of the section are a system of parameters in a d-dimensional Cohen–Macaulay local ring, hence a regular sequence [56, Tag 02JN]. Let [W]_d=\sum_i\operatorname{length}(\cO_{W,\eta_i})[W_i] be the scheme-theoretic fundamental cycle, where the W_i are the irreducible components with generic points \eta_i. The algebraic zero-section formula gives

[56, Tag 0FA9]. The complex Borel–Moore cycle class commutes with vector-bundle Chern classes and proper pushforward [19, Chapter 19]. With the same generic multiplicities in \operatorname{cl}_{\rm BM}([W]_d), this gives

where the last equality is (11.6). Each nonempty isolated zero has strictly positive length. Thus there are no zeros. The inertia has only finitely many connected components, so the intersection of the corresponding nonempty open sets of choices of u supplies the single field v in the statement.□

Proof. Let Z be the normalization of such a component, let z_0\in Z map to the given arrow, and suppose that \delta_Z:Z\to A is dominant. Take its Stein factorization

Here T is normal, \gamma is finite and dominant, and \beta has connected fibers. The connected fiber through z_0 lies entirely over nonidentity arrows: its image lies in \delta^{-1}(0), where the identity and nonidentity parts are open and closed.

Put t_0=\beta(z_0), and take the field v of Lemma 11.4. A finite dominant map to the smooth n-fold A is locally open. For completeness, a sufficiently small local branch of T at t_0 is finite over a neighborhood of 0; its analytic image is closed and has dimension n, so contains a neighborhood of 0. We can therefore lift a sequence \exp(\varepsilon_\nu v), with 0\ne\varepsilon_\nu\to0, to points of T tending to t_0. Lift these points farther to Z. Properness gives a subsequence tending to a point z_\infty\in\beta^{-1}(t_0). Its arrow is nonidentity, and the arrows in the sequence have coordinates

Choose a local quotient chart at \widetilde p(x_\infty), a lift \psi of \widetilde p on a small analytic neighborhood of x_\infty, and finite-group linearizing coordinates at \psi(x_\infty). After passing to a subsequence the labels are one fixed group element h\ne1, fixing \psi(x_\infty), and

Projecting to the h-fixed coordinate space yields

Division by \varepsilon_\nu and passage to the limit give \operatorname{pr}_{h=1}d\psi_{x_\infty}(v)=0. This is a zero of the section (11.5) on the inertia component containing h, contrary to Lemma 11.4.□

Proof. Let U_Z\subset D_Z be a nonempty open over which every irreducible fiber component of \delta_Z has dimension n-r, let a\in U_Z, and let F be any one of those components, with its reduced structure. On F the second coordinate map is t_a composed with the first. The equality of the two pulled-back polarizations therefore says

Let H_F\subset A be the connected abelian subvariety generated by differences of points of s(F). Since s is finite,

Resolve F and choose a base point on its resolution. Its map to A factors through a surjective homomorphism from its Albanese variety onto H_F, followed by a translation. The dual homomorphism \widehat H_F\to\operatorname{Pic}^0(\widetilde F) has finite kernel. Equation (11.7) thus implies that the restriction of \lambda_P(a) to H_F is torsion.

For a fixed abelian subvariety H\subset A, the set of points with this property is a countable union of torsion translates of H^\perp. To check the assertion including disconnected kernels, the homomorphism

is surjective and has H^\perp as the identity component of its kernel; the induced map A/H^\perp\to\widehat H is an isogeny. The inverse image of any torsion point is consequently a finite union of cosets of H^\perp represented by torsion points of A. Here torsion points lift through a quotient of abelian varieties; this follows, for example, by choosing an isogeny complement to the kernel.

There are only countably many abelian subvarieties of A, since their rational first-homology subspaces determine them. There are also only countably many torsion points. The open U_Z is therefore covered by countably many closed torsion cosets, each of dimension at most r. A complex irreducible variety cannot be covered by countably many proper closed subvarieties: apply the Baire theorem in a small analytic ball in its smooth locus, and then use irreducibility. Hence one such coset contains D_Z. The dimension bound forces equality, proving D_Z=\tau+D with \dim D=r.

This argument also determines the subgroup generated by every component over a dense set of very general points. Let a range over U_Z minus every coset in the preceding countable collection which does not contain D_Z. This complement is dense by the same Baire argument. For each fiber component F over each such a, its character constraint supplies a coset of H_F^\perp of dimension at most r containing a. That coset must contain D_Z and hence equal it. Thus

The image s(F) lies in one K-coset and has dimension n-r=\dim K, so it is the whole coset. Consequently the relation over every such very general a is stable under diagonal K-translation. These fibers are dense in Z: every nonempty Zariski open in Z has image containing a nonempty Zariski open in D_Z, which meets this set. Taking closures therefore proves this invariance on the full image \Gamma\subset A\times A.

The finite map from the original relation component to \Gamma is generically one-to-one: over B_{\rm reg} there is a unique isomorphism label for a pair of points. Each component dominates A, so this open is dense on it. Thus Z is also the normalization of \Gamma. The morphism K\times Z\to\Gamma induced by simultaneous translation is dominant and has normal source. It therefore lifts uniquely to the normalization Z. The action law, equivariance of the coordinate maps, and invariance of the difference follow by uniqueness of this lift, or by checking on the dense normal isomorphism locus and extending.□

Proof. Since \widetilde p is generically étale, the determinant of d\widetilde p defines an effective Cartier ramification divisor \operatorname{Ram}(\widetilde p) on A, and

There is no divisorial stabilizer on \mathfrak B. A sufficiently divisible power of its canonical bundle is therefore the pullback of \cO_B(mK_B). Thus

The right side is ample because p is finite and -K_B is ample. Hence the line bundle of the total ramification divisor is ample.

Since D\ne A, the group K is positive-dimensional. If every prime component of \operatorname{Ram}(\widetilde p) had proper image in A/K, a general K-coset would miss the entire divisor. Its defining section would trivialize the restricted ample line bundle on that positive-dimensional projective coset, a contradiction. Thus one component L dominates A/K; properness makes this map surjective. This uses ampleness of the total ramification divisor and does not assert that each of its components is ample.

The generic point of L lies over B_{\rm reg}, because finiteness preserves the codimension-at-least-two bound for the inverse image of B_{\rm sing}. Near a general point of L, the map between smooth spaces has the transverse form

Its self fiber product has, besides the identity branch, branches v_1=\zeta u_1,\ v_j=u_j for \zeta^e=1,\ \zeta\ne1. They meet the identity along L. There are only finitely many global irreducible components of \mathcal R. Their closed intersections with i(L), excluding the identity component, contain a dense open of i(L). Irreducibility therefore gives a single residual component \mathcal R'\ne i(A) containing all of i(L), including its special points.

Suppose, for a contradiction, that \delta(\mathcal R')\subset D. Lemma 11.6 says that this difference image is a torsion coset. It contains zero, since i(L)\subset\mathcal R', so it is an abelian subvariety D'\subset D. Its polarized complement K'=(D')^\perp contains K. The same lemma gives a K-action on the normalization Z' of \mathcal R'.

Consider the closed set

It is proper: otherwise \mathcal R' would contain i(A), and the two irreducible n-dimensional components would be equal. For any x\in F, choose an individual preimage of i(x) in Z'. Its connected K-orbit remains over difference zero and remains in the inverse image of the identity part of \delta^{-1}(0), since that part is open and closed. Its image at k is therefore precisely i(x+k). Hence F is K-invariant. This reasoning does not require a global lift of i(L) to Z'.

The divisor L\subset F is an irreducible component of F, because F is proper in the n-fold A. A connected group preserves each irreducible component of a closed invariant set with finitely many components. Thus K+L=L. But L\to A/K is surjective, which gives K+L=A. This contradiction proves the required assertion about \delta(\mathcal R').□

Proof. The action of K supplied by Lemma 11.6 preserves difference zero. Connectedness and the open-and-closed identity partition preserve the nonidentity condition. Since L\to A/K is surjective, a translate of the first coordinate of z_0 belongs to L. Choose such a k, and put e_0=k\cdot z_0 and x_0=s(e_0)=t(e_0). This is the translated normalization point retained below.

The divisor L is Cartier because A is smooth. Its local equation pulls back to a nonzero nonunit in the integral local ring \cO_{Z,e_0}. A minimal prime over that equation has height one, by the principal ideal theorem. Taking its closure with reduced structure gives a prime divisor E through e_0 and contained in s^{-1}(L). The finite map s:Z\to A preserves dimension, so its closed image s(E) has dimension n-1 and equals the irreducible divisor L.

The image of the proper action morphism K\times E\to Z is closed and irreducible. Its first-coordinate image is K+L=A. Since Z\to A is finite, this image has dimension n and hence is all of Z. As \delta_Z is K-invariant,

These are equalities of actual images; projectivity ensures that no special point is lost by taking closures.□

Completion of the proof of Proposition 11.1. Apply Lemmas 11.7 and 11.8 to the fixed D,K,Z,z_*. They provide a divisor L\subset A, a residual component \mathcal R'\subset\mathcal R containing all of i(L), and a reduced prime divisor E\subset Z through the translated point e_0, with

Write x_0=s(e_0)=t(e_0)\in L and \alpha_0=\nu(e_0). The arrow \alpha_0 is nonidentity and has difference zero. It is this translated arrow that composition will retain. Form the space of composable arrows

The inclusion of all of i(L) in the actual, possibly nonnormal component \mathcal R' gives a global morphism i|_L:L\to\mathcal R': both L and \mathcal R' carry their reduced structures. Hence the map

is a closed embedding, the graph of this morphism over the closed subscheme E\subset Z. Since E is irreducible, one irreducible component \mathcal T_0 contains its entire image. Give \mathcal T_0 its reduced structure. Since E is reduced, \iota factors as a closed embedding into \mathcal T_0; in particular, this component contains the retained marked point. Figure 2 records this component choice and the subsequent composition.

This same component dominates both factors. Indeed, \mathcal R'\times Z has pure dimension 2n. The equations for \mathcal T are the pullback of the diagonal in A\times A, which is locally cut out by n equations. The dimension inequality for an ideal with n generators implies \dim\mathcal T_0\ge n. Both projections from \mathcal T to its factors are finite, so \dim\mathcal T_0\le n. Its images in both irreducible n-dimensional factors are closed of dimension n, and thus are the whole factors. The dimension estimate uses smoothness of A through the equations of its diagonal.

Composition of the two labelled arrows gives \mu:\mathcal T_0\to\mathcal R. Both \mathcal T_0 and \mathcal R are finite over the outer source coordinate in A, so \mu is finite. Its closed irreducible image has dimension n, since that source coordinate is surjective. Consequently its image is an irreducible component \mathcal R'' of \mathcal R. On the embedded E, composition with the identity is exactly \nu(e), with its label. Thus \mathcal R'' contains the marked nonidentity arrow \alpha_0=\nu(e_0) over difference zero, and its difference image contains all of \delta_Z(E)=D.

It also contains a point outside D. In fact, dominance of \mathcal T_0\to\mathcal R' allows us to choose composable arrows

with x-z\notin D, whereas y-x\in D, because the second arrow comes from Z. Since D is a subgroup, y-z=(y-x)+(x-z)\notin D. The closed irreducible difference image of \mathcal R'' therefore contains D properly, and has dimension greater than r. It still meets nonidentity inertia over zero, contradicting the choice of r.

Thus every stabilizer is trivial. A Deligne–Mumford stack with trivial inertia is an algebraic space, and its coarse morphism is then an isomorphism. Therefore \mathfrak B=B, and B is smooth.□

Proof of Theorem 1.1. The case n=1 was settled in Section 2; assume n>1. By Lemma 2.3, b_2(X)\geq4. If b_2(X)\geq5, Proposition 7.6 gives quotient singularities for B. If b_2(X)=4 and the polarized family of smooth fibers of the original f is isotrivial, Proposition 8.1 gives the same conclusion. In the remaining case, Proposition 9.6 chooses the two systems of Proposition 9.1 on a projective IHS deformation X', still with b_2(X')=4: projective Lagrangian fibrations f':X'\to B and g:X'\to C, with C normal and projective, such that g has nonzero polarized variation and f' restricts to a finite surjection on every smooth fiber of g. If B had a nonquotient germ, a maximal chart would be singular. Propositions 4.3 and 4.5 would supply the fixed stratum (S_0,\cS) and its product presentations. Proposition 6.1, applied to the original fibration, would give \chi_{\mathrm{top}}(S_0)=0. For the two fibrations on X', Proposition 10.1 instead gives \chi_{\mathrm{top}}(S_0)>0, a contradiction. Thus every germ of B is a quotient singularity in this case as well. The three cases exhaust the possibilities.

Using the fixed-base family, Proposition 9.1 supplies in every case a finite cover of the unchanged B by an n-dimensional abelian variety: restrict its first fibration to a smooth fiber of the second. This cover, the quotient singularities just proved, and the original projective fibration f:X\to B meet the hypotheses of Proposition 11.1. Hence B is smooth. The original X is projective, f is surjective with connected positive-dimensional fibers, and B is now smooth and projective. Hwang’s theorem applies with precisely these hypotheses [26, Theorem 1.2, in the setting of Theorem 1.1], giving B\simeq\PP^n_{\C}.□