A conditional abelianity theorem for special fourfold pairs with a half-weight divisor

Proof. Adjoin simultaneous mth roots of the finitely many sections in copies of the total space of L. The equations t_a^m=s_a are monic and define a finite space over V. Choose a reduced irreducible component dominating V and take a smooth projective resolution q:V'\to V of that component [Hir64]. The tautological roots are holomorphic sections of q^*L. Their differential images

are holomorphic, and \alpha_0 is generically nonzero because q is generically finite in characteristic zero.

Holomorphic forms on a compact Kähler manifold are closed [Dem12]. On the open set where t_0\ne0, we have \alpha_a=(t_a/t_0)\alpha_0, so

Taking the mth power of the ratio gives (2.1) upstairs; generic injectivity of differential pullback gives it downstairs.

If r independent covectors each wedge to zero with a nonzero p-form, expansion in a basis containing those covectors shows that every nonzero term of the form contains all r of them. Hence r\le p. In characteristic zero the image dimension of a section map is the generic rank of its ratio differentials, proving the bound. If that dimension is p, choose p algebraically independent ratios from one section map. The same exterior-algebra argument gives (2.2).□

Proof. In a local frame clear the denominators of a meromorphic generator and divide out all common irreducible factors of its entries. Smooth local rings are factorial, so the resulting vector v has holomorphic entries with no common height-one factor. If av is holomorphic for a meromorphic scalar a, a pole of a at a prime divisor would force that prime to divide every entry of v. Thus a has no divisorial pole and is holomorphic. This proves that the saturated module is freely generated by v.

At every prime divisor some entry of v is a unit at its generic point. The corresponding entry of v^{\otimes m} has the same property. The identical pole test proves the tensor-power assertion. Holomorphic extension across codimension two gives the statements on the whole chart. The construction is intrinsic to the meromorphic direction and hence respects equivariance and changes of chart. The form-valued generator is allowed to vanish in codimension two; the assertion is about an invertible sheaf with a nonzero map, not about a subbundle everywhere.□

Proof. Let Z\subset X\times\PP^1 be the closure of the joint image of g and the function, on its regular locus. The actual image is constructible and dense in Z, so it contains a dense open of Z. Its fibres over general points of X have at most one point. The fibre-dimension theorem therefore makes Z\to X generically finite. After shrinking X, its generic degree is realized by distinct points, since the characteristic is zero, and all those points belong to the actual-image open. The degree is consequently one. Hence \CC(Z)=\CC(X), giving the descent. Connectedness of the general fibre ensures that the constant is a single value on that fibre.□

Specialness. The map to the root orbifold described above will be denoted by f. Suppose that i:L\to\Omega_Y^p contradicts specialness of Y. We first descend the plurisection ratios, then the associated differential tensors, and finally test their poles over base divisors. Choose sections s_0,\ldots,s_p of L^{\otimes m} as in Lemma 2.1, and put u_a=s_a/s_0.

For a smooth general fibre F over B, the exact cotangent sequence

gives a filtration of \Omega_Y^p|_F by subbundles with trivial graded pieces. Choose F for which both i|_F and s_0|_F are nonzero. The first nonzero projection of i|_F to a graded piece, followed by a suitable coordinate projection, gives a nonzero section t of L^{-1}|_F. Then

is nonzero. It is a nonzero constant because F is compact and connected. In particular t and s_0|_F have no zeros, and every u_a is constant on F. Lemma 2.4 gives

Their independent differentials imply p\le n and

By Lemma 2.1, the generic image of L is the line of g^*\sigma.

We next descend the plurisections through their images as differential tensors. Write

Shrink the smooth base open so that \sigma is holomorphic and nowhere zero. Because g is a submersion there, g^*\sigma is nowhere zero at every point of the full inverse image. Equation (3.2) makes each h_a holomorphic there: locally one divides by a nonvanishing coefficient of the reference tensor. Compact connected fibres force h_a to be constant on each such fibre. Therefore

Let \tau=\pi^*\sigma be the meromorphic form on the smooth root charts. Lemma 2.2 gives an orbifold line bundle K\to\Omega_{\cX}^p with generic direction \tau. We claim that every tensor

is holomorphic. Its differential pullback to Y is holomorphic by (3.2). At the general point of any ordinary base prime, condition 3 supplies a local section of a submersive lift. At D, condition 4 does the same in the root chart. Pulling back along such a local section proves that (3.3) has no pole at that divisorial point. Any polar chart divisor belongs to the invariant global polar locus of this rational tensor. Its finite image is a base divisor, so equivariance and the preceding tests exclude every polar component. Extension across codimension two proves the claim everywhere.

By Lemma 2.2, the tensors in (3.3) are sections of K^{\otimes m}. Their ratios are the independent v_a, so \kappa(K)\ge p. Conversely, f^*K\to\Omega_Y^p is a nonzero line map: on the dense ordinary open it is the differential pullback by a dominant map. Pullback preserves independence of section ratios, and Lemma 2.1 gives

Thus K contradicts specialness of \cX. This proves specialness of Y.□

Fundamental groups and conclusion. Set Y^\circ=Y\setminus g^{-1}(D) and X^\circ=X\setminus D. For a smooth manifold and a divisor, the inclusion of the complement surjects on fundamental groups, with kernel normally generated by meridians of the irreducible divisor components. Indeed, perturb loops off the divisor and perturb null-homotopy discs transversely to its smooth strata, avoiding the strata of complex codimension at least two. Puncturing a disc at its transverse intersections gives the normal generators. This applies even if the reduced support of g^*D is singular.

Consider \pi_1(Y^\circ)\to\pi_1(X^\circ)\to G. At the general point of a component of g^*D, its meridian maps to a conjugate of \gamma^{2e}, where 2e is that component’s multiplicity. The winding calculation uses a smooth local equation for D and remains valid when the component maps into a smaller subset of D. Positive meridians of D are conjugate because D is smooth and connected. Thus all the meridians killed on filling Y^\circ die in G, and the map factors through \pi_1(Y).

Over B the map is proper and smooth, hence a differentiable fibre bundle [Ehr52]. Connectedness of its fibres gives \pi_1(g^{-1}B)\twoheadrightarrow\pi_1(B). General position gives \pi_1(B)\twoheadrightarrow\pi_1(X^\circ). The map just factored is therefore surjective, proving (3.1). Finally, Y is a connected smooth projective special manifold. Assumption 1.2 applies, and the image of a finite-index abelian subgroup of \pi_1(Y) is finite-index abelian in G.□

Proof. Take a finite algebraic trivializing cover of X for the bundles involved, with smooth root charts

where t is the local divisor equation. These charts are smooth since D is smooth and reduced; away from D they are étale. Their products with the four projective spaces are smooth. At any point of such a product, each equation in (4.1) has nonzero evaluation as a linear functional on its own parameter block: at least one coordinate in that projective factor is nonzero. The eight evaluations are independent. Thus the universal incidence over the chart product is the kernel of a surjective vector-bundle map from the parameter bundle to the eight equation lines, and is smooth.

Generic smoothness in characteristic zero makes its fibres over a dense parameter open smooth. Indeed, each component of the critical locus has image of dimension smaller than \dim\mathsf A, since a dominating component would have generically surjective differential on its smooth locus, contradicting criticality. Images of nondominating components may likewise be discarded. There are finitely many algebraic charts, so one common dense open of parameters proves 1.

Only points over D can have nontrivial stabilizer. There the involution on a factor sends

Its fixed locus is the union of the two eigenspace lines. If a fixed point satisfies both equations, either the first pair of columns or the last pair of columns of C_j(x) is dependent. Each of these determinant conditions is a proper hypersurface in matrix space. A fixed point in the fourfold product therefore requires one such condition in each of the four independent matrices. At fixed x\in D its parameter codimension is four. The bad incidence over D has dimension at most

General parameters avoid its image. This excludes stabilizers above every point of D, including degenerate fibres, and proves 2.

For one specified three-column submatrix, rank at most one has codimension two in the space of 2\times3 matrices. Surjective evaluation implies that its incidence over X has dimension at most \dim\mathsf A+2. For a general parameter its bad base locus has dimension at most two, by the fibre-dimension theorem. The finitely many choices of factor and deleted column can be treated simultaneously. Their union is closed and gives 3.

Finally choose a point x_0\in X\setminus D. By evaluation surjectivity, all pair minors can be made nonzero at x_0; for example the columns (1,\lambda_i)^{\mathsf t} with distinct \lambda_i have this property. This is a nonempty Zariski-open parameter condition and can be intersected with the preceding dense opens. Nonvanishing then holds on an open neighbourhood B of x_0, proving 4.□

Proof. Square the four coordinates in each factor. The identification T^2=\pi^*\OO_X(D) gives a globally defined invariant map to the ordinary projective variety

Its map on a root chart is the finite root map W\to U together with the finite projective coordinate-square maps. Restriction to the closed zero locus remains finite. Passing to the finite-group quotient preserves finiteness: over an affine target this takes an invariant subalgebra of a finite algebra, which is again finite. These invariant algebras glue under the bundle transitions, since the coordinate-square construction is invariant and global. Hence S\to Q is a finite analytic map.

For completeness, a finite analytic space over a complex projective variety algebraizes by projective GAGA [Ser56]. Its finite coherent direct-image algebra, including multiplication and unit, algebraizes, and relative spectrum recovers the space after analytification. A finite algebraic morphism is projective [TSPA26, Tag 01WG]. Thus S is projective and in particular compact.□

Proof. At x\in B, consider one matrix C=C_j(x). It has rank two. Coordinate squaring on \PP^3 is finite and surjective; the common quadric zero set is its inverse image of \PP(\ker C)\simeq\PP^1. Thus it is nonempty and has dimension one. At a solution, put b_i=z_i^2. A nonzero vector b\in\ker C cannot have support one or two, since every pair of columns is independent. The vertical derivative

therefore has rank two. It annihilates the radial direction at a solution, so it still has rank two on the projective tangent space. This proves both smoothness of the curve and submersivity of the family over B.

The curve is a smooth complete intersection of two quadrics in \PP^3. Its Koszul resolution is

The intermediate cohomology vanishings for line bundles on \PP^3 give H^0(C,\OO_C)=\CC, hence connectedness [TSPA26, Tag 01XS]. Adjunction gives K_C=\OO_C. Thus C is a genus-one curve and its cotangent line is trivial. Taking four factors proves the first assertion and triviality of the cotangent bundle of each product fibre.

Now let x\notin Z, possibly on D. In each matrix C_j(x), the three columns remaining after any deletion span \CC^2. Its kernel is consequently contained in no coordinate hyperplane: one can prescribe a chosen coordinate to be one and solve for the other three. A vector space over \CC is not the union of finitely many proper linear subspaces, so its kernel contains a vector b with all four coordinates nonzero. Choose square roots z_i of its entries. Formula (4.4) then has rank two. Choosing such a point in each factor makes the full vertical derivative of the eight equations surjective. The implicit function theorem gives a submersion from their zero set to the base root chart at this tuple. The argument works for either chosen chart point over x and in particular for w=0.□

Proof of Proposition 4.1. The smooth projective S has disjoint connected components. Its fibres over B are connected and nonempty, so there is exactly one component Y dominating X. More explicitly, the proper image of a dominating component is all of X; two such components would disconnect the fibres over B. Every submersive point constructed in Lemma 4.4 lies on a dominating component, since its local image has full dimension. Therefore all these points lie on Y.

The variety Y is connected smooth projective, of dimension 4+4=8. Its map g:Y\to X is surjective and has the asserted fibres over B. Its map to \cX gives

The pullback of the root-divisor section is generically nonzero, so f^*\cD is an effective integral Cartier divisor. Thus every divisorial multiplicity over D is even. Since \codim_X Z\ge2, every base prime divisor has general point outside Z. Lemma 4.4 supplies the ordinary or root-chart submersions there. All four conditions of Proposition 3.1 follow.□

Proof of Theorem 1.3. Apply Proposition 4.1 to (X,D). Specialness of \cX and Proposition 3.1 make the resulting eightfold Y special and give \pi_1(Y)\twoheadrightarrow G. Assumption 1.2, in dimension eight, makes \pi_1(Y) virtually abelian. Its quotient G is virtually abelian as claimed.□

Proof. Choose a line bundle H such that H^{\otimes2}(-D) is very ample. Successive general members B_i of this system are smooth and irreducible and can be chosen transverse to all the preceding strata, including those in D. Thus the asserted simple normal crossing condition follows from Bertini’s theorem. In particular, at a point of D at most three of the B_i pass through the point.

Let s_D and s_i be the divisor sections for D and B_i. Form the cover by adjoining roots

where y_i takes values in H, and normalize. In a rational frame of H, the five right-hand sides have independent classes modulo squares in \mathbb C(X)^*. Indeed, the order along B_i of a product of these functions is odd exactly when its ith factor occurs an odd number of times. A product representing a square must therefore have every exponent even. The resulting field extension has degree 2^5, with the independent sign changes as its Galois group. Its normalization S is connected and finite over X, hence projective.

We compute this normalization locally analytically. Units in the equations have holomorphic square roots after shrinking the neighborhood. Off D, the equations give independent roots of the passing B_i coordinates; the other roots separate the local sheets. At x\in D, the equations become

There is an index j_0\notin I(x). On a component set w_0=y_{j_0} and, in its meromorphic function field, set w_i=y_i/w_0 for i\in I(x). These functions satisfy the displayed independent root equations. The other y_j equal \pm w_0, and their relative signs separate the components. The independent-root charts are finite and normal over the base chart and have exactly the prescribed generic sheets, so they are its normalization. They are smooth. The sign changes give the stabilizers and their actions stated in the lemma.

For completeness, every element of the latter stabilizer with nonzero \tau-coefficient has at least

nonzero coordinates in the basis e_1,\ldots,e_5. Consequently e_i belongs to a point stabilizer precisely over B_i. In the local charts its fixed locus is the root-coordinate hyperplane for B_i. This proves the assertion about S^{e_i}. Each of its components has dimension three, so its finite image in the irreducible divisor B_i has dimension three and equals B_i. The equation z_0=w_0^2 also proves the assertion about \pi^*D.□

Proof. Fix an elliptic curve E and a nonzero point c\in E[2]. Set

On the factor E_{ij} let e_i act by v\mapsto-v, let e_j act by v\mapsto c-v, and let the other generators act trivially. The two displayed involutions commute because 2c=0, and their product is translation by c. If an element has both i and j in its support, it therefore acts without fixed points on this factor and hence on F.

For a single e_i, its four moving factors each carry an involution of the form v\mapsto a-v. The equation 2v=a has solutions and a finite reduced solution set. Thus F^{e_i} is a nonempty disjoint union of translates of products of the six remaining elliptic factors. It is smooth of codimension four. Two different generators cannot fix the same point, since their product is fixed-point-free.□

Proof. Use the diagonal action of A on V=S\times F. By Lemma A.2, a nontrivial point stabilizer can only be generated by a single e_i. The entire nonfree locus is the union of the pairwise disjoint smooth subvarieties

They have codimension five. They are invariant under A, since A is abelian. At a point of Z_i the stabilizer is exactly \langle e_i\rangle. In local coordinates it fixes the tangent coordinates to Z_i and negates all five normal coordinates: one from S, and four from the moving elliptic factors.

Blow up their disjoint union and take the quotient:

The action lifts to the blowup. We verify that Y is smooth. If (v_0,\ldots,v_4) are the normal coordinates just described, then in the blowup chart with distinguished coordinate v_j we have coordinates

besides the tangent coordinates. The stabilizer acts by

Its quotient chart has smooth coordinates t^2,(a_k) and the tangent coordinates. Every stabilizer on \widehat V is contained in the stabilizer of its image in V; away from the exceptional divisors it is trivial. These charts therefore establish smoothness everywhere.

The blowup is connected, smooth, and projective. Its finite-group quotient is projective: tensoring the translates of an ample line bundle gives an invariant linearized ample line bundle, whose invariant section ring constructs the projective quotient. The quotient is connected and has the same dimension as V, namely 4+10=14. The morphism \pi\circ\operatorname{pr}_S\circ b is invariant and descends to a surjective morphism f:Y\to X.

Over U, the map S\to X is an étale A-torsor and the blowup has no centre. Hence

Analytically trivializing the finite étale torsor identifies this map locally with projection from a product with F. This proves the first assertion, including properness, smoothness, and connectedness of these fibres. Their cotangent bundles are trivial.

We next check the divisor D at all points, not only at generic points away from the auxiliary divisors. Let x\in D and use the coordinates of Lemma A.1. A stabilizer at a point of S\times F over x is either trivial or \langle e_i\rangle for some i\in I(x). Indeed an inertia element involving \tau has support of size at least two and has no fixed point on F. In particular, every possible product stabilizer fixes the function w_0, whose square is the pullback of z_0.

The pullback of w_0 to a blowup chart is still holomorphic and is fixed by the stabilizer of every point in that chart. A neighborhood of the corresponding point of the finite quotient is the quotient of a sufficiently small neighborhood by this stabilizer. Thus w_0 descends there to a holomorphic function whose square is f^*z_0. Together with the unchanged base coordinates it gives a local holomorphic lift to the root chart of D. This argument includes points on exceptional divisors and all intersections with the auxiliary branch divisors.

We can also determine the multiplicities exactly. Write \pi^*D=2R with R reduced. No component of a centre Z_i is contained in R\times F, because every component of S^{e_i} dominates B_i\ne D. Hence blowing up introduces no exceptional component in the pullback of R\times F, and

where \widehat R is its reduced strict transform. At a general point of every component of \widehat R the image in D avoids all B_i. There the only nontrivial inertia of S\to X is generated by \tau, which acts freely on F, so q is unramified. Every prime component of f^*D is dominated by a prime component of \widehat R, since q is finite. The ramification index there is one, and the displayed equality proves that its coefficient in f^*D is exactly two.

Over a general point of D, all the auxiliary divisors are absent. There are no blowup centres and the diagonal action is free. Consequently the quotient is locally isomorphic to S\times F, and projection to (w_0,z_2,z_3,z_4) is a submersive local lift to the root chart. This completes the second assertion.

Finally fix i and take a general point of B_i outside D and the other B_j. Choose a point of S^{e_i} above it and a point of F^{e_i}. Near the resulting point of Z_i, let w be the cover coordinate with w^2=z_i, where z_i defines B_i. The five normal coordinates to the centre are w,u_1,\ldots,u_4, with the u_j coming from the moving elliptic factors. The other three base coordinates are tangent to the centre and remain unchanged.

In the blowup chart distinguished by w, use

as its normal-direction coordinates. Passing to the local quotient replaces w by \xi=w^2 and leaves the ratios and all tangent coordinates unchanged. The map to X then has coordinates

It is a submersion, including at \xi=0 on the exceptional divisor. This provides the required point above each general point of B_i. Every other prime divisor distinct from D has its general point in U, where f is smooth. The third assertion follows.□