Abelian covers, Gale correspondences, and the Hodge conjecture for powers

Proof. The fibers of q are finite, so proper base change gives R^j q_*\Q=0 for j>0. Each fiber is one G-orbit, hence (q_*\Q)^G=\Q. Averaging makes invariants exact, and Leray gives (3.1), also after extending coefficients. Pullback respects mixed Hodge structures. If V is smooth and projective, strictness of the weight filtration and the displayed injection into H^i(V,\Q) give purity.□

Proof. For a projective bundle p:\mathbb P(E)\to S of relative dimension e-1, put \xi=c_1(\mathcal O_{\mathbb P(E)}(1)). The projective-bundle formula is

The powers of \xi restrict to a basis on each fiber, which proves the formula by Leray–Hirsch. Every displayed map is algebraic.

For a blowup b:\widetilde X\to X of a center of codimension c, let j:E_Z\hookrightarrow\widetilde X be the exceptional divisor and p:E_Z=\mathbb P(N_{Z/X})\to Z. With \xi=c_1(\mathcal O_{E_Z}(1)), the blowup formula is

For completeness, the map of relative pairs (\widetilde X,E_Z)\to(X,Z) induces an isomorphism in relative cohomology, since it is an isomorphism off the closed subsets and the quotient spaces obtained by collapsing those subsets agree. On E_Z, the projective-bundle formula and j^*j_*\eta=-\xi\eta show that subtracting the displayed exceptional terms leaves a class whose restriction comes from Z. The long exact sequences of the pairs then show that the remaining class comes from b^*H^q(X). The sum is direct: b_*b^*=\operatorname{id}, b_* kills these exceptional terms, and their restrictions have independent positive powers of \xi on E_Z. All maps in the formula are algebraic. Companion results 2.1(i) prove both supply assertions. The formulas contain only finitely many summands, proving the last assertion.□

Proof. The Fermat hypersurfaces are smooth. The dimension-zero assertion is immediate. The plane curve F_n^1 is connected: distinct positive-degree components in \mathbb P^2 would meet by Bézout’s theorem, making their union singular. For this curve, the coordinate power map is the quotient by G=(\mu_n)^3/\mu_n onto the line z_0+z_1+z_2=0 in \mathbb P^2, branched only at its three coordinate points. The invariant part of first cohomology vanishes by (3.1). For a nontrivial character \chi of G, its summand of the finite pushforward of the constant sheaf is a rank-one local system off the branch points where \chi is nontrivial on inertia, and has zero stalk at those points. At a point where \chi kills inertia it extends across the point, as one sees by passing to the quotient by \ker\chi. If there are r\le3 active points, its Euler characteristic is 2-r. The connected curve has no nontrivial character in degrees zero or two, so

Here a character occurring in first cohomology necessarily has r\ge2. The Hodge group of the Jacobian commutes with the algebraic deck action. Over k its centralizer on first cohomology is a product of multiplicative groups, since every occurring character has multiplicity one. The connected Hodge group is therefore a torus. Thus \operatorname{Jac}(F_n^1) is CM. The Abel map supplies the curve’s first cohomology from this Jacobian, and point correspondences supply degrees zero and two.

We recall the explicit domination associated with Shioda and Katsura [21], including its resolution so that the maps used for supply are algebraic. For a,b\ge1, choose \varepsilon\in\C with \varepsilon^n=-1. There is a rational map

Summing the nth powers of the two blocks gives the target equation. To see dominance, take a target point whose first block has nonzero power sum. Rescale its projective coordinates so that this sum is -1; the second block then has power sum 1. Set x_{a+1}=y_{b+1}=1, use the first block for (x_0,\ldots,x_a) and \varepsilon^{-1} times the second for (y_0,\ldots,y_b). These are preimages on the two Fermat hypersurfaces. Source and target have equal dimension, so the map is generically finite.

The base locus is the smooth codimension-two subvariety

Near each of its points some x_i and some y_j in the displayed blocks are units in projective coordinates. The base ideal of (3.2) is consequently exactly the ideal generated by x_{a+1},y_{b+1}. Blowing up Z removes this ideal and extends the map to a morphism f:\widetilde X\to F_n^{a+b} from a smooth projective variety.

Induct on the target dimension, using a=1, b=d-1 in dimension d\ge2. The product F_n^1\times F_n^{d-1} and the center F_n^0\times F_n^{d-2} are supplied by finitely many correspondence maps from powers of \operatorname{Jac}(F_n^1), by the induction hypothesis and external products. The center may be disconnected; the blowup formula and supply apply to its finitely many smooth components. Lemma 3.2 gives the same supply for \widetilde X. If e>0 is the generic degree of f, the projection formula gives f_*f^*=e\,\operatorname{id} on rational cohomology. Thus f_* is surjective, and composing the supply maps with it supplies the full target cohomology. Every source remains a power of the same Jacobian, and each stage uses finitely many correspondence maps in each degree.□

Proof. The group-algebra character idempotents and all their scalar conjugates preserve the Hodge decomposition, so V_\chi is admissible and its determinant has the meaning fixed in Section 2. We use only the closure assertions of Companion results 2.1(i), not the Hodge-class conclusion.

Let g be the genus of B and let D=t_1+\cdots+t_r be its reduced branch divisor. The images of the branch meridians have product one in \Delta, by the relation in the punctured surface group and commutativity of \Delta. Thus r=0 or r\ge2. The \chi-part of q_*k_C is a rank-one local system on B\setminus D, with zero stalk at every point of D, because \chi is faithful on each nontrivial inertia group. Its Euler characteristic is 2-2g-r. Since C is connected, the nontrivial character does not occur in H^0(C,k) or H^2(C,k). Hence

There is no nontrivial connected unramified cover of a genus-zero base, whose fundamental group is trivial. The numerical cases n=0 are therefore g=0,r=2 and g=1,r=0; in either case the determinant is the unit supplied by a point. We assume n>0 from now on.

The determinant on a finite quotient. Set X=C^n and G=\Delta^n\rtimes\mathfrak S_n. Multiplication of the cyclic coordinates is a homomorphism G\to\Delta; let N be its kernel. The finite quotients are

The symmetric power is smooth: in a small disk about each distinct point of a divisor, elementary symmetric functions are coordinates for its cluster of points.

By (3.1), the pullback of H^n(Z,k)_\chi is the subspace of H^n(X,k) on which (\delta_1,\ldots,\delta_n;\tau) acts by \chi(\delta_1\cdots\delta_n). On each curve factor this nontrivial character excludes degrees zero and two. The only possible Künneth summand is therefore V_\chi^{\otimes n}. A geometric permutation of these degree-one factors includes its Koszul sign, so invariance under \mathfrak S_n means ordinary antisymmetry. Consequently

In particular H^n(Z,k)_\chi is a line.

Put U=(B\setminus D)^{(n)} and \mathcal L=(\pi_*k_Z)_\chi. At a divisor meeting a branch point, the stabilizer of a lifted tuple contains the inertia acting in that coordinate. Its image under the product character is nontrivial. The \chi-part of the permutation representation on that fiber is therefore zero.

Over U, the map \pi is a \Delta-torsor, including along the diagonals. Indeed, suppose that a deck tuple followed by a permutation fixes a lifted tuple. Around each cycle of the permutation, the product of the deck displacements fixes one point of the unramified cover and hence is the identity. Their total product is therefore the identity, so the residual \Delta-action is free. To check local triviality at repeated points, choose disjoint disks about the distinct points of the base divisor and label the unramified sheets over each disk. Quotienting the sheet labels by N retains only their total product, while the coordinate permutations give the symmetric powers of the disks. This yields m local copies of the base neighborhood. Thus \mathcal L is extension by zero of a rank-one local system on U.

Its monodromy can be read from moving divisors. Follow the n points along a loop and multiply the deck displacements of their lifted paths; their order at the endpoint may be permuted. Closing these paths to a fixed base point cancels the auxiliary paths because \Delta is abelian. The product is the original cover monodromy on the sum of the moving paths in H_1(B\setminus D,\mathbf Z), with multiplicities. One may first make this calculation off the diagonals and perturb a loop to avoid them; the local triviality just proved gives the same rule on all of U.

For both cases below, let

be the Abel map, sending an effective divisor to its line bundle. The fiber over L is its complete linear system |L|=\mathbb P H^0(B,L), where projective space parametrizes lines. In both cases \deg K_B(D)=n. Put

Schoen’s construction also uses this distinguished Abel fiber [20, Section 2, pp. 12–13]. We will show that restriction to T detects H^n(Z,k)_\chi and construct a proper surjection onto T from smooth projective varieties with CM-supplied cohomology. Detection maps the determinant into source cohomology. After the two geometric cases, purity and Poincaré duality will turn this detection into supply in the required direction, through a correspondence obtained from the fiber product with C^n\to Z.

Localization in the branched case. Suppose r>0. Then n\ge2g. For L\ne L_0, the evaluations

are surjective. In fact, H^1(B,L)=0 since \deg L=n>2g-2. By the evaluation sequence and Serre duality, the obstruction is dual to H^0(B,K_B\otimes L^{-1}(D)). This is the space of sections of a nontrivial degree-zero line bundle and is zero. Riemann–Roch gives kernel dimension g-1. If an L\ne L_0 exists then g\ge1, since \operatorname{Pic}^n\mathbb P^1 is a point. Coordinates adapted to the independent evaluations give

The restriction of \mathcal L to this complement is nontrivial. Choose a section on one evaluation hyperplane and no other, and take a transverse analytic disk through it in |L|. The chosen evaluation has a simple zero in the disk parameter. Even if the section has a multiple zero at that branch point, Weierstrass preparation says that the product of the nearby root coordinates is that evaluation times a nonvanishing function. On the boundary of the disk the sum of the winding numbers of those roots about the branch point is therefore one. All other roots remain in disks away from the branch set. The moving-divisor rule identifies the monodromy with the nontrivial inertia character there. Collisions away from the branch set can be perturbed apart, by local triviality across the diagonals.

The torus vanishing above now makes the compactly supported cohomology of (3.7) with these coefficients zero. Because \mathcal L is extension by zero and |L| is compact, this is H^*(|L|,\mathcal L|_{|L|}). Proper base change shows that each R^j u_*\mathcal L is supported at L_0. A sheaf supported at one closed point has no higher cohomology. The Leray edge map, followed by proper base change for \pi, therefore gives the restriction isomorphism

When g=0 there is no other Picard fiber, so the same conclusion is immediate. Reduction does not change the underlying cohomology.

Fermat sources over the exceptional fiber. We next construct a proper surjection onto T from smooth projective varieties whose full cohomology is CM-supplied. The residue sequence and the residue theorem give an exact sequence

The surjectivity on the right also follows by Riemann–Roch. Choose a vector-space splitting, so |L_0|=\mathbb P(G_0\oplus R). Its evaluation hyperplane at t_i is w_i=0: a meromorphic differential with at most a simple pole there vanishes at t_i as a section of K_B(D) exactly when its residue is zero. These hyperplanes are distinct for r\ge3; the two hyperplanes coincide for r=2.

The finite reduced map T\to|L_0| is still a quotient by \Delta. Averaging shows that invariants commute with quotienting the ideal defining |L_0|, and taking reductions leaves its reduced quotient unchanged. Every component of T dominates |L_0|. At least one does, since the finitely many closed images of the components cover an irreducible base. Its image is then the whole base, so it meets every fiber. Each fiber is a single \Delta-orbit, and its translates therefore cover T. Every component is one of those translates.

Let F=\C(|L_0|). Over F the cover is a possibly disconnected \mu_m-torsor, after identifying \Delta with \mu_m. Kummer theory writes its algebra as F[z]/(z^m-f) for some f\in F^\times. It is unramified away from the distinct evaluation hyperplanes. Thus the valuation of f at every other prime divisor is divisible by m: a Kummer class extending over that divisor’s discrete valuation ring is represented there by a unit times an mth power.

Write the distinct hyperplanes as H_i=\{\ell_i=0\}, 1\le i\le s, with \ell_i chosen among the residue coordinates. For i\ge2, choose an integer a_i representing the valuation of f at H_i modulo m. The degree of a principal divisor is zero. Since each H_i has degree one and every other valuation is divisible by m, the valuation at H_1 is -\sum_{i\ge2}a_i modulo m. Hence

for an integral divisor E on |L_0|. The Picard group of projective space is torsion-free, so E is principal. It follows that

This includes all prime divisors, not only those in an affine chart. When r=2 we have s=1, so the product is empty and the generic torsor splits.

The coordinate power map

is finite and surjective. Form the smooth projective bundle

The tautological subline of R defines a morphism Y\to\mathbb P(G_0\oplus R)=|L_0|. On each component this map is dominant and generically finite: away from zero R-coordinate its fibers are the finite fibers of s_m. The pullbacks of the hyperplane ratios in (3.9) are (u_i/u_1)^m. The scalar c also has an mth root in \C. The generic torsor algebra therefore splits over the function field of each component of Y. Evaluating at its roots gives rational lifts to every prescribed component of T.

Each rational lift extends to a morphism. Indeed, for a normal integral variety mapping to a base, a rational lift to a scheme finite over that base sends each element of the finite affine algebra to a rational function integral over every local ring of the source. Normality makes it regular; the resulting local lifts glue. The image here is closed by properness and contains the generic point of the prescribed component of T, so the lift surjects onto it. Taking disjoint copies of the required components of Y gives a proper surjection

from smooth projective components of dimension g+r-2. Proposition 3.3 and Lemma 3.2 supply their full cohomology. The vector space G_0 in (3.10) is constant, so this construction uses no cohomology of B. For r=2, F_m^0 is a finite set and Y' is a union of projective spaces, as needed.

Localization and supply in the unramified case. Suppose instead that r=0. Since n>0, (3.3) gives g\ge2 and n=2g-2. The character of the cover factors through H_1(B,\mathbf Z), the period lattice of \operatorname{Jac}(B). It defines a nontrivial rank-one local system \mathcal E on \operatorname{Pic}^n B, after choosing an origin on that torsor. The moving-divisor rule gives \mathcal L=u^*\mathcal E.

Riemann–Roch and Serre duality give

Thus the fibers of u are \mathbb P^{g-2} except for the fiber \mathbb P^{g-1} over K_B. Choose an ample line bundle on B^{(n)}. Its jth Chern power restricts to a nonzero generator in degree 2j on every fiber for 0\le j\le g-2. It defines, by proper base change, a stalkwise isomorphism k\simeq R^{2j}u_*k in each such degree. All odd direct images vanish, and the only remaining one is R^{2g-2}u_*k, a rank-one skyscraper at K_B.

By the projection formula the direct images for u^*\mathcal E are these sheaves tensored with \mathcal E. The cohomology of \mathcal E on the compact real torus underlying \operatorname{Pic}^n B vanishes by the calculation at the start of the subsection. Leray leaves only the skyscraper term. Here L_0=K_B, so the Leray edge map and proper base change give the same restriction isomorphism (3.8). The torsor on |K_B|=\mathbb P^{g-1} splits, since this projective space is simply connected. Hence T itself is a disjoint union of projective spaces of dimension g-1. We may take Y'=T; its full cohomology is supplied by points.

Detection on a smooth source and return to the curve product. In either case we have a proper surjection Y'\to T from smooth projective components whose full cohomology is CM-supplied. We use the following precise consequence of proper cohomological descent: if T is proper and Y'\to T is a proper surjection with Y' smooth, then

This is Deligne’s Proposition 8.2.5 [5, Proposition 8.2.5, p. 39]: its proof injects the weight-n quotient into H^n(Y',\Q). Properness of T bounds its weights above by n, and the proper smooth Y' has pure degree-n cohomology, giving the displayed kernel equality. The proposition requires neither T irreducible nor Y' connected, so applies here.

By (3.1), H^n(Z,\Q) is pure of weight n, since it injects into H^n(X,\Q) and X=C^n is smooth and projective. Restriction to T is a morphism of mixed Hodge structures. Strictness therefore gives

Extend scalars to k and take a nonzero \alpha\in H^n(Z,k)_\chi. Its restriction is nonzero by (3.8). It survives on some component Y_0 of Y' by (3.12). Write p:Y_0\to T\to Z and d=\dim Y_0. Poincaré duality provides

Here (d,2d-n)=(g+r-2,r-2) in the branched case and (d,2d-n)=(g-1,0) in the unramified case. In particular these are valid degrees. The class \beta is CM-supplied because the full cohomology of Y_0 is.

Choose an irreducible component of the finite surjection Y_0\times_Z X\to Y_0 which dominates Y_0, and resolve it projectively. The resulting smooth projective variety \widetilde Y fits into the commutative square

where p factors as Y_0\to T\to Z, h has positive generic degree e, and Ql=ph. The component Y_0 has full CM supply and detects p^*\alpha; the resolving variety \widetilde Y is used only to realize the return correspondence l_*h^*. The algebraic correspondence

induces l_*h^*. Its cohomological degree shift is 2(n-d), so it sends \beta to H^n(X,k). This image is CM-supplied by Companion results 2.1(i). Only the pullback of \beta is used; no supply assertion about the cohomology of \widetilde Y is required. Projection formula gives

Let \rho:G\to k^\times be \rho(\delta_1,\ldots,\delta_n;\tau) =\chi(\delta_1\cdots\delta_n). The inverse-character idempotent is

It is a k-linear combination of algebraic automorphism graphs. The integration pairing is G-invariant, while Q^*\alpha transforms by \rho. Averaging (3.13) therefore gives

The calculation for (3.5) identifies the image of P_{\rho^{-1}} in degree n with D(H^1(C,k)_{\chi^{-1}}); the inverse character has the same rank n by (3.3). This determinant line is thus CM-supplied. Complex conjugation of coefficient scalars sends \chi^{-1} to \chi, since its values are roots of unity. It preserves the rational correspondence span by Companion results 2.1(i), proving the assertion for D(V_\chi).

Finally, in the branched case Proposition 3.3 starts the supply in (3.10) from powers of \operatorname{Jac}(F_m^1). The projective-bundle maps and all subsequent maps merely compose those rational correspondences, and coefficient conjugation preserves their rational image span. The unramified and zero-rank cases use a point. This proves the stated fixed-source refinement.□

Proof. Suppose first that \chi\ne1 and put K=\ker\chi. The quotient C_\chi=C/K is a smooth connected projective curve; locally the quotient of a smooth curve by its finite stabilizer has parameter t^e. The residual group P/K is cyclic and \chi is faithful on it. Its action on C_\chi is effective. Indeed, choose a point of C with trivial P-stabilizer, outside the finitely many fixed points of nonidentity automorphisms. If a coset acted trivially on C/K, it would send this point into its K-orbit, forcing the coset to be the identity.

Equivariantly for the residual action, (3.1) identifies

by quotient pullback. A point of B is branched in C_\chi\to B exactly when its original inertia has nontrivial image in P/K. Lemma 3.4 applies to this connected cyclic quotient and gives its determinant supply; (3.3) gives the dimension formula. The tensor power of the rational graph pullback sends that determinant line to D(H^1(C,k)_\chi), by Companion results 2.1(i). If the character space is zero both lines are the unit. This includes a quotient that becomes unramified although the original cover is branched.

For \chi=1, quotient pullback identifies the character space with H^1(B,k). Choose a point b\in B. The rational algebraic class

is the H^1(B)\otimes H^1(B) component of the diagonal and, up to the cup-pairing convention, is the inverse symplectic tensor. If g=g(B), antisymmetrizing its gth external power gives a nonzero multiple of the top exterior power of this nondegenerate alternating tensor. It therefore generates D(H^1(B,k)) by a rational algebraic class. For g=0 this is the unit. Pullback on the curve slots proves the assertion.

For the last assertion, let \mathcal O(P) be the finite set of orders of nontrivial characters of P, and set

with the empty product interpreted as a point. The faithful lemma uses powers of the indicated factor for a character of order m, and the trivial case uses a point. Each such factor power is supplied from the corresponding power of M_P by its algebraic factor inclusion and projection. The product M_P is CM. This proves the uniform choice of sources.□

Proof. A rational tensor is real. In weight 2a, it has type (a,a) exactly when it belongs to F^a: reality then also places it in \overline{F^a}, whose intersection with F^a is the (a,a) part. Membership in the holomorphic subbundle F^a is the vanishing of the induced section of its quotient bundle, so Z_t is closed analytic. There are countably many rational tensors. A proper analytic subset of the connected complex manifold is nowhere dense, and Baire’s theorem proves the nonemptiness assertion. Odd tensor degrees have no balanced vectors, since the circle element -1 acts on them by -1.

At a point in the displayed complement, the balanced rational tensors are exactly those balanced at every marked parameter. The latter spaces are preserved by \Gamma, because it acts equivariantly on the whole variation. By Companion results 2.1(iii), their pointwise stabilizer in the symplectic group is J_s. Thus every element of \Gamma normalizes J_s.

An individual globally balanced tensor has finite \Gamma-orbit. We use the finite-action argument of [1, Section 4, Lemma 4]. If its denominator is q in the integral tensor lattice, every translate still lies in q^{-1}H_{\mathbb Z}^{\otimes 2a}, and its tensor-polarization norm is unchanged by transport. At a fixed fiber all its translates lie in the real balanced subspace. The tensor polarization, with its fixed weight sign, is definite on that subspace. A fixed norm sphere there is compact and contains only finitely many points of this lattice. The orbit is therefore finite. The connected algebraic group L^\circ fixes every such tensor, so Companion results 2.1(iii) give L^\circ\subset J_s.□

Proof. It is enough to prove the Lie algebra assertion, since the groups in the conclusion are connected. Extending scalars from k to \C preserves irreducibility, semisimplicity, and the rank condition, so we work over \C. In the symplectic alternative the invariant alternating form descends to k: invariant forms are the kernel of linear equations over k, and the determinant is nonzero on a suitable alternating form with coefficients in k. Equality of the connected group images then descends under scalar extension. The center acts by scalars. The Jordan decomposition in an algebraic Lie algebra shows that a nilpotent element has zero central component and nilpotent components in its simple ideals. Write the irreducible representation of the semisimple image as a tensor product \bigotimes_i V_i of nontrivial irreducibles for its simple ideals.

Let N be the given rank-one nilpotent. In particular N^2=0. If two of its simple-ideal components N_i,N_j were nonzero, choose length-two terminal Jordan strings in V_i,V_j and kernel vectors in the other factors. On the tensor product of the top vectors of these strings, N^2 is 2N_i\otimes N_j\ne0. This is impossible. If just N_i is nonzero, its rank on V is \operatorname{rank}(N_i)\prod_{j\ne i}\dim V_j. Rank one therefore forces the semisimple image to be simple.

Put N in the nilradical of a Borel subalgebra of this simple image. A generic one-parameter subgroup of its maximal torus, together with a scalar rescaling, degenerates N to a nonzero root vector x_\alpha. The locus of matrices of rank at most one is closed, so x_\alpha has rank one. Every root vector for a root of the same length has rank one, by Weyl conjugacy. Restriction to the corresponding root \mathfrak{sl}_2 consequently consists of one standard doublet and trivial representations: the rank of the raising operator on an irreducible \mathfrak{sl}_2-module of dimension a is a-1. Thus each reflection in a root of that length exchanges exactly two weights, both of multiplicity one, and fixes every other weight.

Here is the root-system check that finishes the argument. We use the usual classification and weight lattices of reduced irreducible root systems, as in [9, Sections 23–25 and 30]. The roots of a specified length form a full-rank simply laced root system, not necessarily a closed root subsystem of the ambient root system: the whole system in simply laced type; D_\ell or A_1^\ell in types B_\ell and C_\ell; D_4 in type F_4; and A_2 in type G_2. Use the small-rank conventions D_3=A_3 and D_2=A_1\sqcup A_1.

The simple reflections of this root system act as distinct transpositions on the weights. They are distinct because the weights span the dual Cartan space. Orthogonal simple roots give disjoint transpositions, while adjacent simple roots give transpositions sharing one endpoint. A branching vertex of the Dynkin diagram would require three disjoint transpositions each meeting the same two-element set, which is impossible. A connected full-rank path A_\ell therefore acts on precisely \ell+1 moved weights as successive transpositions on a path. All other weights, being fixed by this full-rank root system, are zero. The moved weights form its standard regular simplex: their successive differences are the simple roots with a consistent choice of sign, as follows by pairing with adjacent simple coroots. In full type A_\ell the highest weight is thus the first or last fundamental weight. Irreducibility gives the standard representation or its dual. If this path arose instead from G_2, B_3, or C_3, the full Weyl group contains negation, whereas this simplex, for \ell\ge2, is not stable under negation. Those cases are excluded.

It remains to consider the orthogonal-axis root system A_1^\ell. Its transpositions are disjoint. A moved weight for one axis is fixed by every other axis reflection, so lies on that axis; the doublet condition makes its two weights \pm\alpha/2. For the long roots 2e_i in C_\ell, this gives the weights \pm e_i, and the highest weight is that of the symplectic standard representation. For the short roots e_i in B_\ell, the vectors \pm e_i/2 are not weights when \ell\ge2. In the rank-two D_2 root system the half-roots (\pm e_1\pm e_2)/2 are not weights of C_2; in B_2 they give its spin representation, which is the standard representation under B_2=C_2. Rank one gives the ordinary \mathfrak{sl}_2 standard. These are exactly the alternatives in the statement.□

Proof. The eigenspace on the unbranched cover is the cohomology of the corresponding rank-one local system on the punctured base, by averaging over the finite deck group. Filling an inactive puncture gives ordinary cohomology on the filled base. At an active puncture the local complex is k\xrightarrow{t_i-1}k, which is acyclic, so no local term changes the character cohomology. The cellular cochain model, or the standard crossed-homomorphism model of degree-one group cohomology, gives Equation (4.2). Euler characteristic gives Equation (4.1); in the unramified case both degree zero and degree two vanish for a nontrivial rank-one system.

For the twist, conjugate the two meridians by their product \delta=g_ig_j and fix the exterior generators. The identity

gives Equation (4.3) by direct subtraction. In particular \ell_\alpha(u_\alpha)=t_it_j-1. All such disk twists preserve \phi: they act trivially on the abelianization of the punctured surface group. For other mapping classes preserving t, a positive power preserves \phi, which suffices for all Lie-algebra statements.

Set s_i=z(g_i)/(1-t_i). These are the fixed points of the affine transformations represented by the meridians. The fixed-hyperplane condition is s_i=s_j after transport along \alpha. More explicitly, if \alpha is represented by p_i^{-1}xp_j in terms of based access paths and a based loop x, it is

This description is unchanged by endpoint windings. For r\ge3 the image vector in Equation (4.3) is nonzero modulo coboundaries, since another active meridian has zero coordinate in it. When r=2, if that vector is a coboundary, all exterior multipliers are one; the surface relation then also makes \ell_\alpha zero.

The polarization pairs opposite character spaces perfectly and gives a nondegenerate Hermitian form on W_{\chi,\C}, preserved by the monodromy. If T=1+u\ell is a nontrivial isometry, every fixed vector is orthogonal to (T-1)W=ku. Both spaces have codimension one, proving the fixed-hyperplane assertion. A common zero intersection of fixed hyperplanes makes the directions span by nondegeneracy. A nonzero invariant subspace contains a vector moved by one transformation, hence contains its image direction. Nonorthogonality then propagates this inclusion along the graph and gives the whole space.

Finally, if a simple closed curve has trivial t-monodromy, lift its annulus to the cyclic cover and twist simultaneously about its lifts. The ordinary intersection formula restricts on a character space to rank at most one: the projected lift classes are scalar multiples of each other. If evaluation of the cocycle on one closed lift is nonzero, the intersection-dual projected class is nonzero as well, by the perfect opposite-character pairing. The resulting transformation is a nontrivial rank-one unipotent. Its logarithm, or a nonzero integral multiple obtained by taking a power, belongs to the Lie algebra of connected monodromy.□

Proof. The unipotent radical of the identity component is normal in the full closure. Its common fixed subspace is nonzero and is invariant under the full group, so it is all of V; the radical is trivial in the given faithful image. The identity component is therefore reductive. Its isotypic summands are permuted transitively by the full group. A rank-one nilpotent can act nontrivially on only one isotypic summand, and there with multiplicity one; that summand has dimension at least two. Transitivity gives the same two properties for every summand. A scalar multiple of a rank-one change cannot permute two summands of dimension at least two: its difference from that scalar identity would have rank at least two. Thus all the generators preserve every summand. Irreducibility forces a single summand, with multiplicity one, as required.□

Proof. Take the based handle bouquet on the handle side of a small vertex disk, and fan the access paths p_i to the punctures out on its other side. Choose the first two access paths to avoid the interior of the standard 2,3 arc when that arc is present. Concatenate p_1^{-1}xp_2. Outside the vertex disk its pieces are embedded and disjoint. In the vertex disk there are two germs from the puncture side and two from the handle side. Choose the orientation of x so that the incoming and outgoing germs can be joined in noncrossing pairs. Replacing these joins by small smooth arcs removes the vertex and gives the required embedded arc. Its regular neighborhood with the two endpoint disks is a disk; with the 2,3 arc added it is a regular neighborhood of an embedded three-vertex tree, also a disk. This explains why the consecutive-meridian formulas apply to the winding and standard arcs together. Auxiliary marks can be avoided by small perturbations.

Use the standard separated handles for the last assertion. The winding arc and its endpoint disk can be placed in the last-handle and puncture region. The other handle loops lie outside. Carrying all these choices by the homeomorphism used in Equation (4.5) preserves the assertion. A change of their based access paths only conjugates those loops; because their t-multipliers are one, zero cocycle displacement on them remains zero after that change.□

Proof. We exhibit an irreducible group of twists and a rank-one nilpotent in the Lie algebra of its connected closure. There are three cases.

No active punctures. Here \eta\ne1, since t is nontrivial. The surface relation is (\eta-1)z(b_h)=0. Thus z(b_h)=0, and a coboundary removes z(a_h). The remaining coordinates are precisely the ordinary first cohomology coordinates of the first h-1 handles. The twist character is trivial on this subsurface. Twists along a symplectic handle basis and along connecting band sums give rank-one transvections whose directions span with connected nonorthogonality graph. Their positive powers preserving \phi have the same logarithm directions. They act irreducibly already on the identity component. When h=1 the character space is zero, as asserted.

At least three active punctures. Start with consecutive arcs inside the ordered puncture disk. Their common fixed equations make all s_i equal. Adjacent directions pair nontrivially: in a common three-puncture disk this follows by substituting the vector of one pair twist into the functional of its neighbor in Equation (4.3); the factors that occur are nonzero multiples of t_j, 1-t_i, and 1-t_k.

For each handle generator add the winding arc of Lemma 4.6 from 1 to 2. Its direction is linked to the standard 2,3 direction by the same calculation in their three-puncture neighborhood. In the common fixed space, use a coboundary to make all s_i=0. The winding equations then force z(x)=0 for every handle generator. Thus the common fixed space is zero and the direction graph is connected. Lemma 4.4 gives irreducibility.

The twist about b_h supplies a rank-one nilpotent. In compatible generators it replaces a_h by a_hb_h and fixes the other generators, so its cocycle displacement is supported on a_h and is \eta z(b_h). The coordinate z(b_h) can be chosen freely and the single surface relation compensated at an active meridian. The image vector is not a coboundary, since it vanishes on all active meridians. This is a nonzero square-zero rank-one operator. A positive power of the twist preserves \phi; a further power removes any finite deck scalar. Adjoin this twist to the irreducible disk-and-winding subgroup. The enlarged group is still generated, up to scalars, by rank-one changes of the identity. Lemma 4.5 now gives connected irreducibility for its closure.

Two active punctures. Write t_1=\lambda, t_2=\lambda^{-1}. The cocycle relation is

If \eta=1, remove the common value s_1=s_2 by a coboundary. All the handle coordinates are then ordinary symplectic coordinates, and their twists and band sums prove the assertion exactly as in the first case.

Suppose \eta\ne1 and impose the gauge s_2=0. The handle coordinates are arbitrary and Equation (4.6) determines s_1. Take the standard two-puncture arc and the winding arcs corresponding to a_h and to a_j,b_j for j<h. The standard fixed equation is s_1=0; the others are s_1=z(x), after reversing x if necessary. Reversing a trivial-multiplier generator merely changes its sign, and for a_h it multiplies its coordinate by the nonzero scalar -\eta^{-1}; these changes do not affect the argument. The common kernel is zero: first s_1=z(b_h)=0, then z(a_h)=0 and all remaining handle coordinates vanish.

We verify connectedness of the direction graph. Before the gauge, the standard arc direction has zero handle coordinates and s_1=s_2=-\lambda/(1-\lambda)\ne0. Adding the coboundary with parameter \lambda/(1-\lambda) puts it in the gauge s_2=0 and gives

with every other handle coordinate zero. The winding a_h functional is therefore nonzero on this direction, with either orientation of a_h. The fixed-hyperplane description and Hermitian symmetry give the reciprocal nonzero pairing, so the a_h-winding direction has s_1\ne0. That direction has zero coordinates on all a_j,b_j for j<h, by the last assertion of Lemma 4.6; subtracting its gauge coboundary does not alter these coordinates, whose multipliers are one. Hence every remaining winding functional is nonzero on this direction. The graph is connected. All these pair twists have exceptional eigenvalue \lambda\lambda^{-1}=1, so their nontrivial changes are rank-one nilpotents. Lemmas 4.4 and 4.5 finish this case.

In all cases the connected twist closure is irreducible and its Lie algebra contains a rank-one nilpotent. It lies in the projected Hodge group at every s\in\mathcal T_\phi^{\mathrm{Hg}} by Lemma 4.1. The latter is reductive and irreducible, so Lemma 4.3 applied to it gives the stated standard special linear or symplectic image.

The same constructions apply after every scalar conjugation: the active set is unchanged, a multiplier equal to one remains one, and every displayed nonzero factor remains nonzero. Thus every conjugate space has a nontrivial unipotent preserving its polarization Hermitian form. A definite Hermitian space admits no such unipotent isometry. Polarization is definite on a pure character space, so both Hodge types occur on every conjugate.

For the trivial character use the ordinary handle transvections on H^1(B), again taking powers in the finite-index stabilizer. They give the full symplectic group by the same spanning and connectedness argument and Lemma 4.3; the polarization rules out the larger special linear alternative in dimension greater than two.□

Proof. Choose a cyclotomic number field E containing all character values. The projector for \chi and the monodromy representation on its image are defined over E. The determinant of an integral monodromy operator on this image is an algebraic integer: its eigenvalues are among those of the operator on integral first cohomology. Applying the same observation to the inverse shows that the determinant is an algebraic unit. At every embedding of E the operator preserves the nondegenerate Hermitian form obtained from the polarization, so the absolute value of its determinant is one. An algebraic integer all of whose conjugates have absolute value one is a root of unity. For example, the monic polynomials whose roots are the conjugates of its positive powers have bounded integral coefficients; their finite number forces two powers of a conjugate, and hence of the original integer, to coincide. The roots of unity in the fixed field E form a finite group. This finite root set is Zariski closed, so the determinant of the closure M_\chi has the same finite-image property. This proves the first assertion.

The matrices at any other embedding are obtained by applying that embedding to the matrices over E. A finite set of matrices therefore remains finite after every embedding. Their direct sum is the scalar extension of the rational orbit representation, proving the second assertion. For the last assertion, change the sign of each negative definite Hermitian form and take their direct sum. It gives an invariant positive definite real form on the rational orbit representation, after restricting from its complexification. This representation also has an invariant lattice, its intersection with integral first cohomology. The image of each lattice basis vector has fixed length and hence finitely many possibilities. The monodromy image is finite.□

Proof. Let V_\Q be the rational character-orbit summand. The orbit contains the inverse character, so the polarization restricts nondegenerately to V_\Q. It is a polarized subvariation of first cohomology. By Lemma 4.8 its monodromy is finite. We pass to an algebraic quotient of the full marked family. Since r\ge4, the ordered moduli space of the marked points on \mathbb P^1 is a smooth quasi-projective configuration space modulo projective coordinates; one may fix three labels. Include the labelled base point and the chosen point above it. The finite-index stabilizer of \phi gives a finite topological cover of this configuration space, which is algebraic and finite étale by the Riemann existence theorem. The base-point section splits the fundamental-group sequence of the universal punctured curve: in the homotopy sequence of this surface bundle, the section kills the boundary map into the fiber fundamental group and splits the projection on fundamental groups. Since the base subgroup preserves \phi, sending its loops to the identity and the fiber loops by \phi defines a homomorphism from the total fundamental group to P. Its finite topological cover is algebraic by the same theorem. Normalize the universal compact curve in this cover. Analytically near a branch section, a small simply connected base neighborhood trivializes the topological data, and the normalization is a disjoint union of charts (u,s)\mapsto(u^e,s) with the fixed inertia order. The normalization is finite over the universal compact curve, so these charts make it a smooth proper family of compact covers. Thus this is a quotient of the full marked variation, not a restricted family. Indeed its monodromy on first cohomology is the action of the chosen lifts in \Gamma: the lifted base-point section fixes the sheet during transport. On pulling back to \mathcal T_\phi, it is the original marked family, since the topological covering epimorphism and the chosen sheet agree and the complex structure is uniquely obtained by lifting the base charts and filling the same power charts. In particular the pulled-back orbit variation is exactly V_\Q above.

A finite-index kernel in \Gamma kills the finite monodromy of V_\Q. The corresponding finite topological cover of the same configuration quotient is algebraic and finite étale by Riemann existence; the smooth proper family is simply pulled back to it. The period map of V_\Q, now with a constant marking, is a single-valued holomorphic map from this cover to a Siegel period domain. A Cayley transform realizes the domain as a bounded domain in a finite-dimensional complex vector space.

Each coordinate of the period map is consequently a bounded holomorphic function. Take a smooth projective compactification with normal-crossing boundary. The removable-singularity theorem, in local polydisks along the boundary, extends each coordinate holomorphically over the compactification. Such a function is constant on a connected projective variety. The period map of V_\Q is therefore constant.

We show that a mixed character would contradict this constancy. Pass to the cyclic quotient

and let D_\chi consist of its r active branch points. The \chi and \chi^{-1} spaces in first cohomology are unchanged by this quotient. The map forgetting inactive and auxiliary markings is locally a submersion onto the configuration space of D_\chi: the active points can be moved independently while the additional points are kept away from them. It thus suffices to show that the character period on the cyclic quotient has nonzero differential whenever it is mixed.

Here is a deformation calculation which also covers genus one for Y. If a branch chart is x=u^e, an invariant vector field on Y has the form u b(u^e)\partial/\partial u; its pushforward is e x b(x)\partial/\partial x. Consequently

The second equality uses finiteness of \pi and exactness of invariants in characteristic zero. The tangent map for varying the branch points is this identification. Indeed, on a Čech cover, coordinate changes preserving the branch sections lift to the cyclic cover; differentiation gives precisely the inverse of the displayed pushforward on invariant vector fields. This describes the full tangent space of ordered branch configurations modulo projective coordinate changes. Since r\ge4, there are no infinitesimal automorphisms of the pointed base. This argument does not require H^0(Y,T_Y)=0.

If the \chi-space is mixed, there are nonzero forms

Their product \alpha\beta is a nonzero invariant quadratic differential. It pairs nontrivially with some element of H^1(Y,T_Y)^{P/\ker\chi} by Serre duality and averaging over the finite group. Equivalently, invariant quadratic differentials descend to H^0(\mathbb P^1,K_{\mathbb P^1}^{\otimes2}(D_\chi)): in the chart x=u^e, invariance permits at most a simple pole at x=0, and its pullback is holomorphic. This is the dual of (4.8).

For clarity, the period differential here follows directly from the same local deformation. If the transition functions of the deformed curve differ to first order by the vector-field cocycle \theta_{ij}, Cartan’s formula for the closed holomorphic form \alpha shows that its first-order changes differ on overlaps by \mathcal L_{\theta_{ij}}\alpha =d(\theta_{ij}\mathbin{\lrcorner}\alpha). Its variation modulo holomorphic forms is therefore the class of \theta_{ij}\mathbin{\lrcorner}\alpha in H^1(Y,\mathcal O_Y), up to an immaterial common sign. Serre duality pairs this class with \beta by evaluating \theta on \alpha\beta. The chosen tangent direction makes that value nonzero. This calculation also applies in genus one. Thus the period differential is nonzero. The contradiction proves purity, separately at every scalar translate.

Finally write F=\Q(\chi(P)). The rational orbit summand is an F-vector space whose scalar embedding spaces are exactly these character spaces. Since each is pure, the Hodge circle acts by scalars on each of them. It therefore centralizes the entire rational algebra \operatorname{End}_F(V_\Q). The definition of the rational Hodge group at each fiber forces it to centralize the same algebra. Its action on V_\Q lies in the scalar torus \operatorname{Res}_{F/\Q}\mathbb G_m, as claimed.□

Proof. We first check that the arc family acts irreducibly. When an actual rank-one unipotent is available, its logarithm yields the nilpotent needed to prove connected irreducibility; the remaining pseudoreflection case requires a separate proof of connected irreducibility. In that case an adjoint-eigenspace projection then produces a rank-one nilpotent in the Lie algebra.

Consecutive two-puncture arcs give an irreducible family on W_\chi. To check the hypotheses of Lemma 4.4, use the cocycle coordinates z_i=z(g_i) and put s_i=z_i/(1-t_i). Their common fixed hyperplanes say that all s_i are equal; subtracting the resulting coboundary kills the cocycle. Consecutive arc directions have a nonzero mutual pairing by the formula in that lemma. Thus the directions span and their nonorthogonality graph is connected. The full group M_\chi acts irreducibly.

Its identity component is reductive in this representation. Indeed, the unipotent radical is characteristic in the identity component and hence normal in M_\chi. Its nonzero fixed-vector space is then M_\chi-stable, and irreducibility makes that space all of W_\chi. The radical acts trivially. The isotypical summands for M_\chi^\circ are permuted transitively by M_\chi.

First suppose that the Lie algebra already contains a rank-one nilpotent N. On an isotypical summand S\otimes A it acts as N_S\otimes1_A. A summand on which it is nonzero must therefore have \dim A=1 and \dim S\ge2. Transitivity gives these same properties on every isotypical summand. An operator differing from a scalar by rank one cannot permute two summands of dimension at least two: its displacement on either of the summands would have rank at least two. Every arc pseudoreflection therefore fixes each summand. The irreducibility of the arc family forces there to be just one, proving connected irreducibility in this case.

There are two immediate sources of such an N. If t_it_j=1 for some active pair, its twist is a nontrivial rank-one unipotent by Lemma 4.4. Its logarithm belongs to \operatorname{Lie}(M_\chi^\circ). If all active pair products are -1, then all the t_i are equal to the same primitive fourth root of unity. Equation (4.7) implies 4\mid r; since r\ge5, we have r\ge8. Choose a disk containing four active points. If g is the product of their consecutive meridians, then t(g)=1. The boundary twist conjugates each inside meridian by g, and its cocycle displacement is

This is a rank-one unipotent. Its image is nonzero modulo coboundaries because there are active points outside the disk. Its functional is nonzero: one can prescribe a nonzero value of z(g) on the inside and satisfy the global cocycle relation using an outside entry. Its square-zero displacement follows also from t(g)=1.

It remains to treat the case with no pair product equal to 1 and with some pair product \lambda\ne\pm1. All pair twists are then semisimple pseudoreflections. Suppose first that the isotypical summands for M_\chi^\circ have dimension at least two. The rank argument just given makes every pair twist fix every summand, so there is just one summand S\otimes A. If \dim S=1, the connected group acts by scalars; its finite determinant then makes the connected group trivial, contrary to infinitude. Thus \dim S>1.

An operator normalizing this isotypical action has the form B\otimes D, with B\in\operatorname{GL}(S) and D\in\operatorname{GL}(A). To see this, choose an intertwiner B for the induced automorphism of the irreducible representation on S; what remains centralizes that representation and is 1\otimes D by Schur’s lemma. A nontrivial semisimple pseudoreflection cannot have this form when both factors have dimension at least two. Indeed the factors must themselves be semisimple (a nontrivial unipotent Jordan part would survive on the tensor product). Their eigenvalue products would then form a rectangular grid with just one exceptional entry. A two-by-two subrectangle containing that entry contradicts the equality of the products of its two diagonal pairs. Hence \dim A=1, and the connected action is irreducible.

The only remaining reducible case has distinct one-dimensional isotypical summands. The Hermitian form pairs each such line nondegenerately with exactly one other line: for characters a,b of the connected group, a nonzero pairing requires a\overline b=1 on the finite-index monodromy subgroup dense in that group. Distinct characters preclude two choices of b. Hermitian symmetry makes this a pairing involution on the set of lines. Each pair twist preserves the involution and permutes the lines by either the identity or one transposition. Indeed, a permutation cycle of length \ell contributes rank at least \ell-1 to the difference from a scalar identity, and disjoint cycles contribute independently; here that rank is one. The permutation group generated by the twists is transitive, by irreducibility. If the pairing involution had no fixed point, a transposition commuting with it would have an invariant two-element support, so could only exchange the members of one of its pairs. Such transpositions cannot act transitively on at least three lines. Thus there is a fixed point, and transitivity makes every line self-paired. Distinct lines are orthogonal; transitivity also makes their signs equal. The Hermitian form is definite.

The decomposition into distinct one-dimensional connected isotypicals and the pseudoreflection action are algebraic properties over k. They persist at every scalar translate. The same argument therefore makes the Hermitian form definite on every translated character space. By Lemma 4.8 the rational orbit monodromy is finite, a contradiction. This proves connected irreducibility in the remaining case.

Use now a pair twist with eigenvalue ratio \lambda\ne\pm1. After a scalar normalization it is R=\operatorname{diag}(\lambda,1,\ldots,1) for a decomposition W_\chi=L\oplus U, \dim L=1. Conjugation by R preserves \mathfrak m=\operatorname{Lie}(M_\chi^\circ) and has the three distinct eigenvalues 1,\lambda,\lambda^{-1} on

respectively. Polynomial projection onto these eigenspaces preserves \mathfrak m. Connected irreducibility implies that some off-diagonal component is nonzero, since otherwise L and U would be invariant. A nonzero element of either off-diagonal space has rank one and square zero. It supplies the asserted nilpotent.□

Proof. For n\ge3, Lemmas 4.10 and 4.3 give either the full standard special linear or the full standard symplectic derived algebra of connected monodromy. Its determinant is finite, so its connected scalar center is trivial. In the symplectic case all of M_\chi normalizes the connected symplectic group and hence preserves its unique invariant alternating form up to scalar.

In dimension at least four a semisimple pseudoreflection with ratio different from 1 cannot be such a similitude. Normalize its fixed hyperplane U to have eigenvalue 1. The alternating form does not vanish identically on U, since \dim U=n-1>n/2; thus the similitude multiplier is 1. Its exceptional eigenline must then be orthogonal to U and to itself, and hence be in the radical, an impossibility. It follows that every active pair product is 1. With at least three active points this forces all t_i=-1. The sphere meridians generate the deck group, so \chi has order two. Conversely an order-two character space has a nondegenerate alternating polarization and monodromy contained in its symplectic group. Its pair twists are nontrivial unipotents, so the symplectic alternative occurs. This proves the stated connected monodromy groups.

The algebraic image of L_k on W_\chi is closed and is the Zariski closure of the character action, hence is M_\chi. The image of L_k^\circ is M_\chi^\circ, by the identity-component property of algebraic group images. By Lemma 4.1, L^\circ\subset J, so connected character monodromy is contained in the image of J_k on W_\chi. In the special linear case its action already contains the full traceless algebra. In the order-two case the Hodge group also preserves the restricted alternating polarization. Hence the derived Hodge projections are exactly the groups asserted in (i). Finite monodromy gives the toric conclusion by Lemma 4.9.

A connected reductive subgroup of \operatorname{GL}_2 either acts through a torus or, when its derived image is nontrivial, has the full standard \operatorname{SL}_2 as its derived image. Indeed a reducible two-dimensional representation of a reductive group is a sum of characters, while the only nonzero semisimple subalgebra of \mathfrak{sl}_2 is \mathfrak{sl}_2 itself. Dimension one is toric, and dimension zero is omitted. This proves (ii). Finally each displayed derived image is simple, and a surjective map from a semisimple Lie algebra to a simple one is supported by exactly one simple ideal.□

Proof. Pass to the cyclic quotient on which the character is faithful, of order m, and write v_i=a_i/m, 0<a_i<m. Choose a coordinate with infinity away from the branch points. The quotient has an equation

The eigendifferentials for the inverse of the character of y are f(x)\,dx/y. At a branch point with d_i=\gcd(m,a_i), a uniformizer has x-b_i=u^{m/d_i} and y equal to a unit times u^{a_i/d_i}. The differential dx/y is holomorphic there, whereas a pole of f would destroy holomorphicity. Thus f is a polynomial. At infinity holomorphicity requires \deg f\le S-2, giving dimension S-1. Replacing y by \prod_i(x-b_i)/y computes the opposite eigenspace with exponents m-a_i and dimension 3-S. The choice of whether t labels this character or its inverse exchanges the two dimensions everywhere and does not affect the assertion.□

Proof. Write \mathfrak f for the shared ideal. The standard representations of a simple special linear or symplectic algebra determine their dimension and are preserved, up to duality, by isomorphisms of these algebras. The coincidence \mathfrak{sl}_2=\mathfrak{sp}_2 has the same standard two-dimensional representation. Invert one character when necessary, so that an \mathfrak f-linear isomorphism

exists. At this stage F need not respect the Hodge circle.

The projective actions of the entire group \Gamma are intertwined by F. Indeed, \Gamma preserves the character spaces and normalizes J by Lemma 4.1. The kernel of either character projection on \operatorname{Lie}(J^{\mathrm{der}})_k is preserved, because the character space is preserved. Its unique complementary supporting ideal \mathfrak f is therefore preserved as well. Each lift induces the same automorphism of this ideal on both spaces. The two implementations of this automorphism differ, after conjugation by F, by a scalar, by irreducibility and Schur’s lemma.

A two-puncture twist is projectively trivial if at least one endpoint is inactive: fill that inactive puncture in the rank-one local system, after which the twist is supported in a disk with at most one puncture. It is projectively nontrivial when both endpoints are active. For the latter assertion use r=n+2\ge4 and the nonzero rank-one formula of Lemma 4.4; the global cocycle relation can be satisfied using an active point outside the disk, so its functional is nonzero as well as its direction. The graph of nontrivial pair twists is therefore the complete graph on the active set. Projective equivalence shows that the active sets of \chi and \chi' coincide. Label them identically and write t_i,t_i' for their multipliers.

Suppose first that n\ge3. The pair twist has one exceptional eigenvalue and a fixed hyperplane. Its ratio is distinguished projectively by the multiplicities, so

If the ratio is 1, the operator is nontrivial unipotent and the same conclusion holds. These twists also intrinsically identify their image directions: for a semisimple twist this is its exceptional eigenline, and for a unipotent twist it is the image of its normalized displacement.

Choose three active points in an embedded disk, with consecutive meridians g_i,g_j,g_\ell. Cocycles supported on these entries and with value zero on their product form a two-dimensional subspace U\subset W_\chi. There are active points outside the disk, so passage to the cocycle quotient is injective on this space. The two consecutive pair directions are independent and span U. Thus F identifies this space with its primed counterpart. The twist enclosing the three points acts on U by the scalar t_it_jt_\ell and on W_\chi/U by 1. Indeed its displacement on a cocycle is supported in the disk and has value zero on the product; on a cocycle with that product value zero, conjugation by the product multiplies every inside value by its multiplier. The quotient has dimension n-2>0. Its identity action fixes the projective scalar in the comparison, even when the full operator is a nontrivial unipotent extension of the identity. We obtain

Equation (4.9) makes every ratio t_i'/t_i equal to a common sign; the triple equality makes that sign 1. All local multipliers are equal. Since sphere meridians generate the deck group, the characters coincide after the permitted inversion. This proves the first part of (i). The symplectic assertion follows from Proposition 4.11: its character has order two, so complex conjugation acts internally and exchanges its two Hodge parts. Both parts have dimension n/2 at every translate.

Now suppose n=2, so the common active set has four points. The eigenvalue ratio of a projective two-by-two operator is defined up to inversion. Hence

Put A=t_1t_2, B=t_1t_3, and D=t_1t_4. The Klein four permutations on the four indices give the following triples of pair products:

Uniformly inverting the four entries supplies the other four inversion patterns. Choose \pi in this Klein four group and \epsilon\in\{1,-1\} realizing the three choices in (4.10) involving index 1. Since t_1^2=ABD, recovery of the four entries from these three products leaves only a common sign. Thus

The choices \delta,\epsilon,\pi are made once and are unchanged at every scalar translate of this identity.

This classifies only the projective twist data. It remains to use the shared ideal to align pure and mixed behavior and to make one fixed choice between the original model and its polarized dual for every scalar translate.

Mixedness agrees between the two character spaces at each translate. To justify this without any assumption about the pure orientations, split the infinitesimal Hodge-circle action into its central scalar part and its component in the shared simple ideal. On a standard two-dimensional module, the ideal component is zero precisely when the circle is scalar, that is, precisely when the character space is pure. The same ideal component acts on both equivalent modules, and this observation remains valid after every scalar translation.

We compare the two pure orientations. Permuting the four multipliers has no effect on their signatures; uniformly inverting them reverses the two pure signatures by Lemma 4.12. Consider therefore the additional common sign \delta=-1. At a fixed translate write the four entries before this sign change as \exp(2\pi i v_i), 0<v_i<1, and put S=\sum_i v_i. No v_i equals 1/2, since otherwise the sign change would produce an inactive multiplier. If a entries exceed 1/2, the sum after the change is

For S=1 one has a\in\{0,1\}; agreement of purity excludes S'=2, so a=0 and S'=3. For S=3 one has a\in\{3,4\}; the same argument gives a=4 and S'=1. For S=2, agreement of mixedness forces a=2 and S'=2. Thus the common sign reverses pure orientation wherever the spaces are pure. Combining it with \epsilon, the relative orientation of the two pure spaces is the same fixed sign \delta\epsilon at every pure scalar translate.

The standard \mathfrak{sl}_2 module is self-dual. According to this fixed sign choose either the original model or its polarized dual, whose circle weights are the negatives of the original weights. Its pure type now agrees with that of W_{\chi'} at every translate. Choose an intertwiner over k of the resulting standard ideal modules. At a mixed translate the two circle weights are +1,-1; their sum is zero, so the central scalar shift is zero on both modules. The intertwiner respects the common ideal action and hence the whole circle. At a pure translate both circles are the same scalar, so it respects them there as well. The same reasoning applies to every scalar conjugate of the chosen intertwiner. This proves (ii). Notice that polarized duality reverses the weights at every translate; no commutation between an arbitrary scalar automorphism and complex conjugation is being assumed.□

Proof. The intertwiner line is one-dimensional by Schur’s lemma and is stable under the Hodge group, since its derived group is normal. If the Hodge circle acts on this line by z^a, the intertwiner shifts the spectrum of one space by a. Both spectra are \{-1,1\}, so \{-1+a,1+a\}=\{-1,1\} forces a=0. Scalar conjugation gives the identical argument for the conjugated intertwiner and spaces. It is therefore an admissible Hodge isomorphism, and hence algebraic in the sense of Companion results 2.1(ii).

An invariant nondegenerate alternating form also spans a Hodge-stable line. Its scalar weight must be zero: nondegeneracy pairs the spectrum \{-1,1\} with its reflection about half that weight, so the midpoint is zero. The inverse tensor consequently has type (1,1), and so do all its conjugates. It is the required pair tensor.□

Proof of Proposition 4.2 and Theorem 1.1. Fix s\in\mathcal T_\phi^{\mathrm{Hg}} and put J=J_s. Set A=\operatorname{Jac}(C_s) and choose an Abel map a:C_s\to A. Its pullback a^*:H^1(A,\Q)\to H_\Q is an isomorphism of rational Hodge structures. Its inverse is an algebraic degree-one operation by Companion results 2.1(ii). Transport the admissible character projectors e_\chi through this isomorphism. Their images are then actual summands of H^1(A,k), as required by the tensor criterion; we use the same notation for the identified curve and Jacobian spaces.

Collect the rational Galois orbits of character spaces on which J^{\mathrm{der}} acts trivially into a rational Hodge summand H_0. Its Hodge group is a torus: its projections on the orbits are tori, and the group on their direct sum is a subgroup of their product. Companion results 2.1(iv) realize H_0 as the first cohomology of a CM isogeny factor. For h\ge1, every remaining nonzero character space has standard special linear or symplectic derived projection by Proposition 4.7. For h=0 the same assertion, with the toric alternatives accounted for, is Proposition 4.11.

Each remaining character space is supported by exactly one simple ideal of \operatorname{Lie}(J^{\mathrm{der}})_k. Indeed its image is a simple standard special linear or symplectic algebra. In a semisimple algebra mapping onto a simple one, exactly one simple ideal has nonzero image; the commuting images of two nonzero ideals could not both be that simple algebra. The supporting ideal maps isomorphically. If two character spaces share it, their standard representations are equivalent or dual. The only coincidence between the two group types is \mathfrak{sl}_2=\mathfrak{sp}_2, with the same standard module.

In positive genus, Lemma 4.14 makes these comparisons admissible Hodge isomorphisms, using the opposite character for a dual comparison. On the sphere, Proposition 4.13 gives the same conclusion, including the simultaneous alignment of pure scalar translates in dimension two. Group the spaces with the same supporting ideal into one block. Treat a two-dimensional block as linear. A higher-dimensional symplectic block has its alternating pair tensor: it comes from Lemma 4.14 in positive genus and from the polarization on an order-two character on the sphere. A linear block uses the polarization pairing of a character and its opposite. Since finite character values are roots of unity, the opposite space is its coefficient-conjugate space \overline W.

The simple-ideal decomposition gives simultaneous independent standard actions on these blocks. More explicitly, each ideal acts on precisely the copies in its own block, with the chosen standard or dual identifications, and acts trivially on the other blocks and on H_0. The connected image of J^{\mathrm{der}}_k consequently has the Lie algebra of the product of these standard groups. The connected groups have the same image, so this is the product required in part (c) of Companion results 2.2. No independence of the central torus is needed.

A representative of every block is the transported image of an actual character space on C_s. Corollary 3.5 supplies its determinant singleton: a nontrivial character uses the faithful cyclic quotient C_s/\ker\chi, and the invariant character uses the polarization. The curve–Jacobian degree-one maps and all the block identifications are algebraic operations by Companion results 2.1(ii). They transfer these determinant lines to the chosen Jacobian slots. Thus every linear determinant singleton is CM-supplied; there are no orthogonal blocks. The data are saturated.

Companion results 2.2 now prove the assertion for every power after adjoining any CM M. This is a pointwise application after saturation, so M does not enter the definition of \mathcal T_\phi^{\mathrm{Hg}}. Finally, the algebraic curve–Jacobian correspondences identify their first cohomology, and a point and the fundamental class supply the other two curve degrees. External products of these maps and Companion results 2.3 transfer the tensor conclusion to every C_s^a\times M^b.□

Proof. The ring map \C[t_1,\ldots,t_m]\to\C[x_1,\ldots,x_m], t_i\mapsto x_i^n, is free with basis the monomials \prod_i x_i^{e_i}, 0\leq e_i<n. It is flat, so the independent linear equations for \ker A remain a regular sequence after pullback. The projective coordinate-power map is finite and surjective. Its inverse image of \PP(\ker A) therefore has dimension m-c-1, proving the first assertion.

At a point let I=\{i:x_i\ne0\}. The Jacobian has column n a_i x_i^{n-1} for i\in I and zero otherwise. If every c-minor is nonzero, every at most c columns are independent. The equation \sum_{i\in I}a_i x_i^n=0 forces |I|>c, and the Jacobian has rank c. Conversely, a vanishing c-minor gives a nonzero dependence on those columns. Taking nth roots of its nonzero coefficients and setting the other coordinates to zero gives a point with Jacobian rank below c. Here n\geq2 makes the zero coordinates give zero Jacobian columns. The Jacobian criterion proves the equivalence.

Projection \ker A\to\C^S, for |S|=m-c, is an isomorphism exactly when no nonzero vector of \ker A is supported on the complementary c coordinates. This is the nonvanishing of that complementary minor, proving the coordinate-generality assertion.□

Proof. We first choose a smooth partial-intersection flag. Let Q be the row space of A, of dimension c. For every j\leq c and every j-element coordinate set I, the projection Q\to\C^I is surjective: a dependent set of j columns could be extended to a dependent set of c columns, contrary to Proposition 5.2. In the complete flag variety of Q, the condition that this projection restricts to an isomorphism on the j-plane is a nonempty open condition. It is nonempty by choosing a complement of the projection kernel and extending it to a flag. The flag variety is irreducible, so the intersection of these finitely many nonempty opens is nonempty. Choose such a flag 0=Q_0\subset Q_1\subset\cdots\subset Q_c=Q.

Let Y_j be the diagonal zero locus of Q_j, with Y_0=\PP^{m-1}. Every maximal minor of Q_j is nonzero, so Proposition 5.2 makes Y_j smooth of dimension m-j-1. Choose an equation basis adapted to the flag. Under the degree-n Veronese embedding of \PP^{m-1}, the next degree-n equation is a hyperplane, so Y_j is a hyperplane section of Y_{j-1} in that embedding.

The Lefschetz hyperplane theorem in [15, Corollary 7.3, p. 41] says that for a projective variety V of dimension k and a hyperplane containing its singular locus, the inclusion of the section induces homology isomorphisms in degrees below k-1. Here Y_{j-1} is smooth, so its singular locus is empty. Applying the theorem successively gives

Universal coefficients over \Q give the corresponding cohomology restriction isomorphisms. This gives connectedness at i=0 and the formula below d.

Poincaré duality gives the same dimensions above d. In even degree the ambient class is nonzero: the complete intersection has degree n^c, so \int_{X_A}h^d=n^c>0; vanishing of any h^r, 0\leq r\leq d, would force h^d=0. Thus it spans the one-dimensional even cohomology above the middle as well.□

Proof. Any at most d+1 coordinate restrictions to P are independent. Thus at most d coordinates vanish at a point of \PP(P). If a vanish, their ratios to a nonvanishing coordinate have independent differentials and extend to relative local coordinates t_1,\ldots,t_d. After an étale base change taking roots of the remaining units, the cover has equations

and free coordinates t_{a+1},\ldots,t_d. It is smooth, with normal-crossing boundary z_1\cdots z_a=0. The same calculation on the universal projective bundle proves the relative assertions. Proposition 5.2 gives the complete-intersection description.

Fibers of the coordinate-power map are exactly phase orbits, giving the finite quotient. In dimension zero the base point has no zero coordinate, so its inverse image is a free transitive phase set. For d\geq1, Lemma 5.3 gives connectedness and ambient cohomology outside the middle degree.

For distinct a_1,\ldots,a_m\in\C, the subspace \{(f(a_1),\ldots,f(a_m)):\deg f\leq d\} has nonzero maximal minors by the Vandermonde determinant. Thus \U_{d,m} is a nonempty open in an irreducible Grassmannian, hence connected as a complex manifold. The universal incidence equations are projective over it; the local calculation proves smoothness and respects phases.□

Proof. Each curve point has at most one zero coordinate. Since j<m, the products are never all zero. A product of evaluations of linear polynomials is the evaluation of their product, so the formula gives a morphism to X_{P_j}. Its defining sections give

which is ample. A positive-dimensional projective fiber would contain a curve on which this line bundle has degree zero. Thus the proper morphism is finite.

Homogenization f(z)\mapsto v^j f(u/v) identifies polynomials of degree at most j with binary forms of degree j. On the bases the map multiplies j binary linear forms. Every degree-j form factors into such forms, including the factor v representing a zero at infinity. Choose roots of the factors’ evaluation vectors; a phase change on one factor then realizes any prescribed root vector above the product. This proves surjectivity. Where the factors are distinct and no evaluation vanishes, the only identifications are their permutations and phase tuples of total product one. The indicated finite quotient therefore maps finitely and birationally to the smooth target. Normality of the target makes this map an isomorphism.

By Lemma 3.1, target cohomology is the invariant cohomology of the product. In residual character b, product kernel invariance forces character b in every factor. Connectedness of C kills nontrivial characters in H^0,H^2, leaving V_b^{\otimes j}. A geometric interchange of two degree-one factors is minus ordinary tensor interchange. The permutation invariants are exactly the alternating tensors.□

Proof. The invariant assertion follows from the finite quotient formula. The pullback of a base hyperplane is n times the root hyperplane. For d\geq1, Lemma 5.3 makes all degrees outside d ambient and phase-invariant. For d=0, degree zero is the regular representation of the finite torsor.

For d\geq1, compute the rank at an evaluation plane. The curve C\to\PP^1 is branched at the zeros of its m coordinate forms. Pass to the effective cyclic quotient for b. Exactly the s=s(b) points in its support have nontrivial inertia character. Its character summand of the finite pushforward of the constant sheaf is a rank-one local system off those points, extended by zero at them. Its compactly supported Euler characteristic is 2-s, the rank-one Euler characteristic of the punctured sphere. The connected cyclic cover has no nontrivial character in degrees zero or two. Hence \dim V_b=s-2. The effective quotient makes this valid for nonfaithful characters and nonunit entries as well.

Lemma 5.5 gives the binomial rank there. The phase idempotent cuts out a local system on the connected \U_{d,m}, so the rank is constant. In dimension zero every character has rank one in the regular representation, agreeing with the formula since a nonzero sum-zero tuple has s\geq2.

For s\geq d+2, any d+1 selected coordinates detect P. Its projection is injective with coordinate-general image, and the selected root coordinates cannot all vanish. Forgetting the others defines p_S. Its pullback of \mathcal O(1) is \mathcal O_{X_P}(1), so the ample-line-bundle argument proves finiteness. It is surjective by taking roots of the remaining coordinate values of the determined base point.

The fibers are exactly the orbits of K_S=\ker(G_{m,n}\to G_{s,n}). For d\geq1, the induced finite map X_P/K_S\to X_{P_S} is birational and its target is smooth, so it is an isomorphism. For d=0, it is directly the quotient of finite torsors. The character b is inflated from G_{s,n}, so Lemma 3.1 proves the eigenspace assertion. Universal coordinate projections and constant phase actions give the asserted family compatibility.□

Proof. We verify the morphism and its character, compute full rank at an evaluation plane, transport that rank across the connected parameter space, and then handle the zero-dimensional endpoints.

The maximal minors of the bilinear orthogonal complement are, up to a common nonzero scalar and signs, the complementary maximal minors of P. This follows by comparing top exterior vectors under the standard volume pairing on \C^m. Hence Q is coordinate-general.

A point of X_P has at most d zero coordinates and a point of X_Q at most d'. Their product vector has at most m-2 zeros, so it defines a morphism. Its image is on the Fermat hypersurface since \sum_i(x_i y_i)^n=\sum_i x_i^n y_i^n=0. Equivariance for multiplication G_{m,n}\times G_{m,n}\to G_{m,n} sends pullback character b to (b,b). Proposition 5.6 leaves only the two middle degrees. Applied to the hyperplane \sum_i z_i=0, it also gives \dim L_b=\binom{m-2}{m-2}=1.

First suppose d,d'>0, and put N=m-2=d+d'. Choose the evaluation planes of Lemma 5.5, and set

Lagrange interpolation gives

by comparing the coefficient of z^{m-1} in the interpolation formula for z^e. Products of polynomials of degrees at most d,d' have degree at most m-2; the identities and dimension give

Choose \xi_i^n=c_i, and let s_\xi([z_i])=[\xi_i z_i] be coordinate scaling. It restricts to phase-equivariant isomorphisms

The actual evaluation maps of Lemma 5.5 give the commutative square

Indeed, both composites send an ordered tuple (x^{(1)},\ldots,x^{(N)}) to [\xi_i\prod_{\nu=1}^N x_i^{(\nu)}]_{i=1}^m. With V=H^1(C,\kk)_b, pullback in this square and Lemma 5.5 identify the three character spaces as

Full support gives \dim V=N. Use the unnormalized alternating inclusion

More precisely, Lemma 5.5 says that \mu_j^* identifies its character space with \operatorname{im}\iota_j. The exact equality of coordinate-product maps shows that the induced pullback from the Fermat line is the exterior coproduct

after \iota_d\otimes\iota_{d'} it is \iota_N, and these inclusions are injective. For a basis v_1,\ldots,v_N, it is

Each wedge uses increasing order. Complementary wedge bases are paired bijectively with nonzero coefficients, proving full rank \binom{N}{d}.

The map P\mapsto P^\perp is algebraic; the bilinear, not Hermitian, orthogonal complement is essential here. The maps f_P form a morphism from the smooth projective product family over \U_{d,m} to the fixed Fermat variety. Cohomological pullback is a morphism of local systems. In flat character frames the tensor is constant, so its matrix rank is locally constant. Connectedness extends the computed rank to every P.

If d=0, choose a root point x above \PP(P). Every x_i\ne0, and multiplication by x identifies X_Q with F_n^{m-2}. The restriction of f_P to \{x\}\times X_Q is this isomorphism. Its pullback on L_b is nonzero. A nonzero eigenfunction on the finite torsor X_P is nonzero at every point, so evaluation at x detects a nonzero (b,b) tensor. Both character ranks are one, hence it has full rank. The case d'=0 is symmetric, including m=2 when all the varieties are finite.□

Proof. First target X=X_{P_S}. Put h=h_S, F=F_n^{s-2}, and Y=Y_S. Thus \dim F=d+h, \dim X=d, and \dim Y=h. The graph embedding

defines \Gamma_S\in\CH^{d+h}(F\times Y\times X)_\Q. The source dimension is d+2h, so the shift is -2h. On a Künneth tensor its action is

For a character b with support S, put b_S=b|_S and L_{b_S}=H^{s-2}(F,\kk)_{b_S}. The invariant Poincaré pairing on Y pairs only opposite characters and is perfect on each opposite pair, by its full nondegeneracy and the direct character decomposition. This also holds for the sum pairing on a finite torsor. It identifies H^h(Y,\kk)_{-b_S}(h) with the dual of H^h(Y,\kk)_{b_S}. Theorem 6.1 therefore makes the displayed graph action an isomorphism of Hodge-graded spaces

Composing with p_S^* fills W_{P,b}.

Both the graph cycle and the transpose graph for p_S^* have rational coefficients. Apply them to the entire source degree, without character projectors. After scalar extension their images contain all the character pieces just described. Hyperplane cycles from a point supply the invariant cohomology, including every degree outside the middle. The sum of these finite rational maps is thus surjective after extension to \kk; its rational cokernel is already zero. No algebraic Künneth projector is required.□

Proof. For b\ne0, put S=S(b) and s=|S|. If W_{P,b}\ne0, then s\geq d+2, and \dim Y_S=s-d-2. If c=1, m=d+2, so s=m is the only possibility and the source dimension is zero. If c=2, m=d+3, so s=m gives dimension one and s=m-1 gives dimension zero. The full-support source is X_{P^\perp}, connected by Lemma 5.4.

For a missing-one support, zero extension identifies

This is the coordinate-general line cut from the dual two-plane by its ith coordinate hyperplane. Its root torsor is precisely the reduced fiber \{y_i=0\} on C after forgetting that coordinate. The phase group has size n^{m-2}. Distinct boundary fibers are disjoint because a root-curve point has at most one zero coordinate.□

Proof. The ratios are row-basis independent and their denominators are units. The Plücker relations give

so the values lie in M_m.

Let A_0(z) have columns

Set t_1=1 and

All are nonzero regular functions of Q. With B=(a_1,a_2), left multiplication by

gives, by Cramer’s rule,

Hence Q=\operatorname{row}(A_0(z)\operatorname{diag}(t_i)). Conversely, substituting this form into the ratios recovers z,t_2,\ldots,t_m. These are inverse regular maps for the product isomorphism. The effective coefficient torus acts freely and transitively on the second factor, giving the quotient claim.□

Proof. The map is the n-power map on a torus times the identity. It is finite locally free of degree n^{m-1} and étale because n h_i^{n-1} is a unit. Its source is irreducible. The coordinate scalings satisfy the root equations by the preceding formulas for P,Q; their inverses and phase equivariance are immediate.

On the normalized base [s:t], the coordinate forms are

Their ordered zeros are the stated branch points. On t\ne0, put v=s/t, w_i=y_i/y_1. The function field is

Valuations at the distinct finite points show that these m-1 functions are independent in \C(v)^*/\C(v)^{*n}. Kummer theory gives degree n^{m-1}. Equivalently, adjoin the roots successively and use an Eisenstein test at the new branch point, where the earlier functions are units with local roots. Thus the cover is connected and its phase group is its full deck group.

At a finite point indexed by i\geq2, a positive meridian multiplies w_i by \zeta_n and fixes the other local roots, giving \bar e_i. At infinity all right sides have simple poles, so a meridian multiplies every w_i, i\geq2, by \zeta_n^{-1}. This is \bar e_1=-\sum_{i\geq2}\bar e_i. The entries have order n, sum to zero, and generate the group.

For *\in\PP^1\setminus D_z, positive meridians generate the punctured-sphere fundamental group with their product as the single relation. The displayed epimorphism is precisely abelianization modulo n:

A pure orientation-preserving mapping class conjugates each positive peripheral meridian, so it fixes this epimorphism. The standard classification of connected covers by epimorphisms shows that the ordered tuple determines the single fixed cover type here. For each complex structure on the pointed base, pullback charts off the punctures and the local ramification power charts give the unique complex structure on its compact covering surface. The root equations realize it for every configuration in M_m.□

Proof. We use the planar Schoenflies theorem in its annulus form: the region between two nested Jordan curves is an annulus. It gives the following disk extension fact. If D_0,D_1 are closed disks in the interior of a closed disk U, then an orientation-preserving homeomorphism D_0\to D_1 extends to a homeomorphism of U which is the identity on \partial U. Indeed, the two complementary regions in U are annuli. In annulus coordinates, the prescribed maps on their two boundary circles extend across the annulus: orientation-preserving circle homeomorphisms are isotopic, as is seen by taking increasing lifts to \mathbb R and linearly interpolating the lifts. Gluing this annulus map to the given disk map proves the extension fact.

Let g be the given sphere homeomorphism. Choose p\in S^2, a point q different from p and g(p), and a small closed disk D about p for which both D and g(D) avoid q. Their compact union lies in the interior of a closed disk U\subset S^2\setminus\{q\}. The disk extension fact gives a homeomorphism f of U, fixed on its boundary, which agrees with g on D. Extend f by the identity outside U. The Alexander trick isotopes f to the identity, supported in U: in unit-disk coordinates a boundary-fixed map F is deformed by

Now f^{-1}g fixes D pointwise, so it is a boundary-fixed homeomorphism of the complementary closed disk. A second Alexander trick isotopes it to the identity. Composing the two isotopies proves the assertion.□

Proof. Away from D_z, q_z is finite étale with free transitive phase fibers. This proves the torsor assertion except for connectedness. The total \cY_0 is irreducible: it is smooth over the irreducible M_m, and every fiber is connected. Indeed its irreducible components, being disjoint open components of a smooth variety, would have nonempty open images on the base; two such images meet, contradicting connectedness of that fiber. Removing the boundary leaves a nonempty irreducible open. Thus B_{m,n} is connected.

Fix (z_0,y_0)\in B_{m,n}, put b_0=q_{z_0}(y_0), and identify the fixed labelled topological sphere with the normalized base at (z_0,b_0). Also fix a G_{m,n}-equivariant identification of its topological cover with the root cover, taking the fixed upstairs point to y_0. The based epimorphisms agree because both take the ordered positive meridians to \bar e_i. A change of the chosen lift conjugates the epimorphism and hence leaves it unchanged for this abelian group.

Consider any prescribed marked complex sphere. A complex curve of genus zero is \PP^1, and its first three branch labels give a unique Möbius normalization to \infty,0,1. The marking is then represented by an orientation-preserving self-homeomorphism g of the reference sphere which carries all reference labels to the normalized target labels. It fixes the first three labels at the endpoint. By Lemma 7.4, choose an unmarked isotopy g_t from the identity to g. For each t, let a_t be the unique Möbius transformation taking g_t(p_1),g_t(p_2),g_t(p_3) to \infty,0,1. The formulas for the transformation of a distinct triple show that a_t is continuous, and a_0=a_1=\id. Thus a_tg_t keeps the first three images fixed. Its remaining branch images and its auxiliary-point image give a path \gamma in V_m, starting at (z_0,b_0), whose endpoint marking is the prescribed g, not only a marking of the same unmarked fiber.

Lift \gamma uniquely to B_{m,n} from (z_0,y_0); write y_t for its tautological upstairs point. Naturality of oriented labelled meridians under a_tg_t and the abelianization description in Proposition 7.3 give equality of the exact based epimorphisms at every t. The covering-space lifting criterion therefore gives a unique G_{m,n}-equivariant lift of the base marking which sends the fixed upstairs point to y_t. Covering homotopy lifting and uniqueness make these lifts continuous along \gamma. They extend over the compactification through the fixed local ramification power charts, giving a continuous trivialization of the compact root family pulled back to the path interval. They therefore induce exactly its cohomological parallel transport along the projection of \gamma to M_m.

At the endpoint, this lifted covering identification is holomorphic for the complex structure of the prescribed marked cover: off the branch points both complex structures are pulled back from the normalized base, and the same statement holds in the local power charts. Hence it identifies the endpoint Hodge filtrations. This proves the asserted tensor consequence. No loop in V_m is required to lift to a loop in B_{m,n}; its endpoint lift may differ by a deck phase, which is precisely retained by the transported point y_t.

Finally V_m\to M_m is smooth and surjective, with fiber the punctured line. Its composite with the finite étale torsor is smooth and surjective. The product description of \widetilde\D_{2,m,n} gives the displayed fiber product; it is connected and its projection is the torus projection.□

Proof. The tensor variation \V^{\otimes 2a}(a), with the standard tensor/Tate polarization rescaled by a positive integer if needed, is integral, polarized, and of weight zero. Denote its polarization by Q_a. Multiplying a denominator makes a rational tensor integral. For an integral flat tensor u on the universal cover, let H_u\subset S be the set where some determination of u has type (0,0). Cattani–Deligne–Kaplan, Corollary 1.3 on p. 2 of the preprint [4], says that H_u is algebraic. Its proof writes H_u, for K=Q_a(u,u), as a union of images of connected components of the algebraic variety S^{(K)} finite over S in their Theorem 1.1. There are finitely many such components, and finite morphisms are closed, so H_u is closed.

If u is not Hodge throughout the universal cover, H_u is proper. On a simply connected analytic chart, its determinations form a countable set of integral tensors. Each has a closed analytic Hodge locus: for a real weight-zero tensor, type (0,0) is equivalent to membership in the holomorphic subbundle F^0. None contains an open subset, since analytic continuation on the connected universal cover and deck equivariance would then make u globally Hodge. If H_u=S, the chart would be covered by countably many nowhere-dense closed analytic subsets, contrary to Baire’s theorem. Hence H_u is proper.

There are countably many integral tensors in all powers. Remove H_u for the ones which are not globally Hodge. A rational Hodge tensor at a remaining point, after clearing a denominator, must be globally Hodge on the universal cover, equivalently Hodge after parallel transport along every path from that point.□

Proof. Write \pi:\cY_0\to M_m. Its integral polarized variation is \V=R^1\pi_*\Z. Apply Lemma 7.7. For a remaining z, choose an auxiliary point and one of its lifts, hence a point of B_{m,n}, and fix the initial based identification. A rational tensor Hodge in this marked fiber corresponds to a rational tensor in the root fiber. By the lemma, it remains Hodge under transport along every path in M_m. For any prescribed marked fiber, Lemma 7.5 supplies a path in V_m and its based lift whose endpoint filtration is that of the prescribed cover. Applying root-family globality to the projected path shows that the tensor is Hodge in that marked fiber. The fiber was arbitrary, so this marked parameter belongs to \mathcal T_\phi^{\mathrm{Hg}}. Theorem 1.1 applies.

Surjectivity of \lambda makes inverse images of the proper algebraic exceptional subsets proper. At a complex point above z, roots of its torus coordinates exist, and Proposition  7.3 identifies its dual curve with X_{Q_0(z)}. The Hodge conclusion is invariant under this algebraic isomorphism.□

Proof of Theorem 1.2. For c=0, products of hyperplane classes span every power of projective space. If d=0, the smooth target and all its powers are reduced finite sets, whose point cycles span cohomology. Assume henceforth d\geq1 and c=1 or 2.

Proposition 6.2 gives finitely many rational algebraic maps spanning each cohomology degree of X_P. For a nontrivial character with support S, their source is F_n^{s-2}\times Y_S, with shift -2h_S. Apply Proposition 3.3 to H^{s-2}(F_n^{s-2},\Q), the cohomology group that witnesses character spanning. Compose its finite CM-source maps with the Gale correspondence. These compositions are rational algebraic: pull the cycles to a triple product, intersect them, and push forward.

For a finite Y_S, use the inclusions of all its points to span its entire degree-zero cohomology. For the full-support curve C, choose an Abel map C\to\Jac(C). Its pullback on H^1 is an isomorphism induced by the transpose graph, and supplies the factor required in the contraction. Hyperplane cycles from a point supply the ambient target classes.

Suppose one Fermat-source map from a CM abelian variety B has shift 2j, and put h=h_S. Let A_S=\Jac(C) when Y_S=C, and let A_S be a point for one chosen component when Y_S is finite. The composite with the degree-preserving A_S-to-Y_S map and the Gale shift -2h has total shift 2(j-h):

Its Künneth subspace H^{d+h-2j}(B)\otimes H^h(A_S) is the part used to witness spanning. The map itself is applied to the whole source degree.

Put A_C=\Jac(C) in codimension two and let A_C be a point in codimension one. Combining these maps with the ambient point correspondences gives, in every degree k, a surjection

where v_\nu\in\{0,1\} and B_\nu is CM. In codimension one all v_\nu=0. The integers j_\nu include the negative Gale shifts and the CM-source shifts. All maps are rational algebraic correspondences. After scalar extension to \kk, the specified Künneth character pieces span by Proposition 6.2. Applying each rational correspondence to the whole source degree only enlarges its image, so the rational cokernel vanishes. No algebraic Künneth projector is used.

It remains to verify the finite-product Hodge hypothesis for these sources; Companion results 2.3 will then apply to (*) and give all self-powers of X_P at once. For c=1, take as source collection all CM abelian varieties, including a point. Their finite products are CM, so the CM Hodge theorem [17, Theorem 1.1] gives the required hypothesis at every parameter. Equivalently, coordinate scaling identifies each smooth one-equation diagonal hypersurface with a Fermat hypersurface.

For c=2, choose P^\perp outside the union in Corollary 7.8. For this fixed curve C, let

A finite product of these sources again has the displayed form. If a\geq1, it is an algebraic retract of (\Jac(C)\times M)^a: retain every Jacobian factor and one M factor, and set the other M coordinates to zero for the section. Pull a Hodge class on the retract to that power. Corollary 7.8 makes it algebraic there, and pullback along the section returns an algebraic representative on the retract. The corollary applies to every M at the same parameter. If a=0, the CM Hodge theorem applies. Thus \mathscr S_C satisfies the full finite-product hypothesis of Companion results 2.3 as well. Applying that transfer to (*), with target X_P, proves the assertion for every self-power in both cases.

The exceptional union depends only on the tensor variation of the single curve, not on the number of factors or the CM sources. The same parameter therefore works for every self-power. Proposition  5.2 identifies its ambient open with the full smooth diagonal parameter locus.□