Infinitesimal Torelli for weighted complete intersections and certain Fano threefolds

We generalize the classical approach of describing the infinitesimal Torelli map in terms of multiplication in a Jacobi ring to the case of quasi-smooth complete intersections in weighted projective space. As an application, we prove the infinitesimal Torelli theorem for hyperelliptic Fano threefolds of Picard rank 1, index 1, degree 4 and study the action of the automorphism group on cohomology. The results of this paper are used to prove Lang-Vojta's conjecture for the moduli of such Fano threefolds in a follow-up paper.


Introduction
The Torelli problem asks the question if given a family of varieties, whether the period map is injective, i.e., if the variety is uniquely determined by its Hodge structure.This question has first been studied for curves; see [2].The infinitesimal Torelli problem is the more general question that asks whether the period map has an injective differential.The problem can be formulated very concretely for a smooth projective variety X over C of dimension n.Namely, we say that X satisfies infinitesimal Torelli if the map induced by the contraction map is injective.In addition to curves, whether this holds has been studied among others for the following types of varieties: • hypersurfaces in projective space [7,11] • hypersurfaces in weighted projective space [40] • complete intersections in projective space [38,43,45] • zerosets of sections of vector bundles [16] • certain cyclic covers of a Hirzebruch surface [30] • complete intersections in certain homogeneous Kähler manifolds [29] • some weighted complete intersections [46] • certain Fano quasi-smooth weighted hypersurfaces [14] • some elliptic surfaces [27,28,41] The methods used in many of these studies have in common that they describe the cohomology groups relevant for the infinitesimal Torelli map as components of a so-called Jacobi ring and argue that the map can be interpreted as multiplication by some element in this ring.We generalize this method to the case of quasismooth complete intersections in weighted projective space.Following [10], we introduce the terminology: Definition 1.1.Let k be a field.For W = (W 0 , . . ., W n ) ∈ N n+1 a tuple of positive integers, let S W = k[x 0 , . . ., x n ] be the graded polynomial algebra with deg(x i ) = W i .We define weighted projective space over k with weights W to be P(W ) = Proj S W .Given d = (d 1 , . . ., d c ) ∈ N c , c ≤ n, a closed subvariety X ⊆ P(W ) is a complete intersection of degree d if it has codimension c and is given as the vanishing locus of homogeneous polynomials f 1 , . . ., f c ∈ S W with deg(f i ) = d i .A weighted complete intersection X = V (f 1 , . . ., f c ) ⊆ P(W ) is quasi-smooth if its affine cone A(X) = Spec S W /(f 1 , . . ., f c ) \ {0} is smooth.
Main results.Our first main result can be interpreted as giving an explicit description of the differential of the period map associated to a quasi-smooth weighted complete intersection in terms of its Jacobi ring.
Theorem 1.2.Let X = V + (f 1 , . . ., f c ) ⊆ P C (W 0 , . . ., W n ) be a quasi-smooth weighted complete intersection of degree (d 1 , . . ., d c ) with tangent sheaf Θ 1 X of dimension dim(X) = n − c > 2. Let R be the associated Jacobi ring.Let ν = W i − d j .For all integers p ∈ Z with 0 < p < n − c and p = n − c − p, there are isomorphisms . Under these isomorphisms, the contraction map Our second main result is an application of this theorem to prove the infinitesimal Torelli theorem for (smooth) Fano threefolds of Picard rank 1, index 1 and degree 4. By Iskovskikh's classification, there are two types of such varieties; see [20,Table 3.5].The varieties of the first type are smooth quartics in P 4 .For smooth hypersurfaces in projective space, the infinitesimal Torelli problem is completely understood.In particular, smooth quartic threefolds satisfy infinitesimal Torelli; see [7].The second type of Fano threefolds with Picard rank 1, index 1, degree 4 are called hyperelliptic; each such Fano threefold X is a double cover of a smooth quadric Q ⊆ P 4 ramified along a smooth divisor of degree 8 in Q.Such a double cover comes naturally with an involution ι associated to the double cover.It turns out that such hyperelliptic Fano threefolds do not satisfy infinitesimal Torelli, i.e., the period map on the moduli of Fano threefolds of Picard rank 1, index 1 and degree 4 does not have an injective differential.However, the following result says that the "restricted" period map on the locus of hyperelliptic Fano threefolds does have an injective differential.
Theorem 1.3 (Infinitesimal Torelli for hyperelliptic Fano threefolds).Let X be a hyperelliptic (smooth) Fano threefold of Picard rank 1, index 1 and degree 4 over C. Let ι ∈ Aut(X) be the involution.Then the ι-invariant part of the infinitesimal Torelli map As explained in [26, Section 3.5], among the Fano threefolds of Picard number 1 and index 1, infinitesimal Torelli is satisfied if the degree is 2, 6 or 8, and it is known to fail for degrees 10 and 14.Our work deals with one of the remaining cases, namely that of degree 4.
Note that the failure of infinitesimal Torelli for Fano threefolds of Picard number 1, index 1 and degree 4 is analogous to the failure of infinitesimal Torelli for curves of genus g ≥ 2. Such a curve satisfies infinitesimal Torelli if and only if it is hyperelliptic [8], but the period map restricted to the hyperelliptic locus is an embedding [32].
It is natural to study the action of the automorphism group of a variety on its cohomology group; see for example [6,22,31].As an application of the explicit description of the cohomology groups of a Fano threefold with Picard rank 1, index 1, and degree 4 given by Theorem 1.2, we get the following result about the action of the automorphism group.
Theorem 1.4.Let X be a (smooth) Fano threefold of Picard rank 1, index 1 and degree 4 over C. Then the following statements hold.
(2) If X is a smooth quartic, then Aut(X) acts faithfully on H 3 (X, C).
(3) If X is hyperelliptic, then the kernel ker Aut(X) → Aut(H 3 (X, C)) is isomorphic to Z/2Z and generated by the involution ι.
Ingredients of proof.For smooth complete intersections X = V (f 1 , . . ., f c ) in usual projective space, similar results to Theorem 1.2 have been achieved by relating the IVHS of X to the IVHS of the hypersurface V (F ) ⊆ P(E), with E = O P n (d i ); see [43].To avoid problems of this geometric approach arising from the singular nature of the surrounding weighted projective space in our case, we will use another purely algebraic approach inspired by the calculations of Flenner; see [15,Section 8].We will construct resolutions of the sheaves Ωp X that will give us spectral sequences converging towards the cohomology groups of interest.The difficult part will be to make sure that the identification of the cohomology parts with the homogeneous components of the Jacobi ring is done in such a way that the contraction map can be identified with the ring-multiplication.To do this, we will extend the contraction pairing to a pairing of the resolutions and then to a pairing of the spectral sequences.
We were first led to investigate the infinitesimal Torelli problem for these Fano threefolds when studying the arithmetic hyperbolicity of the moduli stack F of Fano threefolds of Picard rank 1, index 1, degree 4; see [26,Section 2] for a definition of this stack.The property of a stack being arithmetically hyperbolic, i.e., having "only finitely many integral points" is formalized in [25].
In [37], we prove the arithmetic hyperbolicity of this stack by first proving that the period map p : F an C → A an 30 is quasi-finite and then using Faltings's theorem [12] which says that the stack of principally polarized abelian varieties A 30 is arithmetically hyperbolic.For the cases of Fano threefolds of Picard rank 1, index 1 and degree 2, 6 or 8, the quasi-finiteness of the period map is deduced from it being unramified, i.e. its differential, the infinitesimal Torelli map, being injective; see [26].However, by our result in the degree 4 case, the infinitesimal Torelli map is not injective.We overcome this difficulty in [37] by showing that the moduli stack F has a natural two-step "stratification" and that on each stratum, the "restricted" period map is unramified.This then suffices to deduce the desired quasi-finiteness of the above period map.
Acknowledgements.I would like to thank Ariyan Javanpeykar.He introduced me to Lang-Vojta's conjecture.The work presented here was done under his supervision during my phd project.I am very grateful for many inspiring discussions and his help in writing this article.I gratefully acknowledge the support of SFB/Transregio 45.

Multigraded differential modules
In this section, we introduce multi-graded differential modules, which is a notion used for example in [5].In particular, this notion describes single and double complexes and pages of spectral sequences.
Let R be a (commutative) ring or, more generally, the structure sheaf O T of a scheme T .A differential d on an n-graded R-module E = p∈Z n E p for us is always considered to be an R-linear self map that is homogeneous of a certain degree with d • d = 0. Whenever we consider a module together with multiple differentials defined on it, we require the differentials to commute pairwise.
Let (E 1 , d 1 ) and (E 2 , d 2 ) be differential n-graded R-modules with homogeneous differentials of the same degree a ∈ Z n .Then the tensor product E 1 ⊗ E 2 comes with an induced (2n)-grading and the two homogeneous differentials d 1 ⊗ id and id ⊗d 2 , giving us a bidifferential 2n-graded R-module.
Definition 2.1.Let n ∈ Z >0 be a positive integer and let (E, d 1 , d 2 ) be a bidifferential 2n-graded R-module.Write the degree of where be a double complex.Then K = p,q∈Z K p,q is a bigraded module and d 1 , d 2 define differentials of degree (1, 0), (0, 1) on K, thus giving K the structure of a bidifferential bigraded module.In fact, giving the data of a double complex is equivalent to defining a bigraded module with differentials of degree (1, 0) and (0, 1).Similarly, a complex (L • , d) can be identified with the differential graded module (L = p∈Z L p , d).Under these identifications, the total single complex associated to K •,• and the total differential graded module associated to K •,• are the same.
Example 2.3.For us, the total differential bigraded module associated to a tensor product of bigraded differential modules with differentials of the same degree a ∈ Z 2 is of particular interest.So let and (E 2 , d 2 ) be differential bigraded modules.Then the differentials d 1 ⊗ id and id ⊗d 2 on the quadgraded module E 1 ⊗ E 2 have degrees (a 1 , a 2 , 0, 0) and (0, 0, a 1 , a 2 ).For p, q ∈ Z, we have On E s,u ⊗ E t,v the differential is given as

Pairings of filtered complexes
In this section, we explain how a pairing of filtered complexes induces a pairing of the associated homology complexes that respects the induced filtration.Let R be a ring or, more generally, the structure sheaf R = O T of a scheme T .All modules are considered to be R-modules and all single (resp.double) complexes are considered to be single (resp.double) complexes of R-modules.
Let (K, d), (K • 1 , d 1 ) and (K • 2 , d 2 ) be complexes.The total complex of the tensor product of K • 1 and K • 2 , as introduced in Section 2, is given by with the differential given by For p, q ∈ Z, we let φ p,q denote the map φ p,q : K p 1 ⊗ K q 2 → K p+q induced by φ.From now on let R be a ring.Directly from the definitions follows: a pairing of complexes.Then φ induces a pairing of the associated homology complexes , where K • is a complex with differential d, and F is a decreasing filtration on K • compatible with the differential, i.e., for each n ∈ Z, we have a decreasing filtration Given a filtered complex (K • , d, F ), there is an induced filtration on the homology complex H • (K • , d) given by For the associated graded, we have From the definition, it is evident that for each p, q ∈ Z, such a pairing of filtered complexes induces a pairing of complexes φ : Tot Hence the induced pairing of the homology complexes 2 is compatible with the induced filtrations on the homology complexes.Therefore, for each p, q, i, j ∈ Z, we get induced maps φ p,q,i,j and gr p,q,i,j (φ) making the diagram (3.2) α p,i ⊗α q,j β p,i ⊗β q,j gr p,q,i,j (φ) commute, where the maps α a,b denote the natural injections and the maps β a,b denote the natural surjections.

Spectral pairing
In this section, we follow [42, Tag 012K] and explain how to construct the spectral sequence associated to a filtered complex.Building on this, we show that a pairing of filtered complexes induces a pairing of the associated spectral sequences.
Let R be a ring.All complexes are complexes of R-modules.A spectral sequence is given by the data where E r is an R-module and d r ∈ End(E r ) is a differential such that We call E r the r-th page of E. A bigrading on the spectral sequence E is given by a direct sum decomposition for each page E r = p,q∈Z E p,q r such that the differential d r decomposes into a direct sum of maps d p,q r : E p,q r → E p+r,q−r+1 r and we have E p,q r+1 = ker(d p,q r )/ im d p−r,q+r−1 r .We can associate a bigraded spectral sequence to a filtered complex (K • , d, F ) in the following way.We define r /B p,q r .Now set B r = p,q B p,q r , Z r = p,q Z p,q r and E r = p,q E p,q r .Define the map d r : E r → E r as the direct sum of the maps ).This defines the bigraded spectral sequence (E r , d r ) associated to the filtered complex (K • , d, F ). Definition 4.1.Let (E r , d r ), ( ′ E r , ′ d r ) and ( ′′ E r , ′′ d r ) be bigraded spectral sequences.Let φ = (φ r ) r∈Z ≥0 be a collection of morphisms of bigraded differential modules r that are homogeneous of degree 0. We denote by the map induced by φ.The collection φ is called a pairing of bigraded spectral sequences if φ s,t,u,v r+1 is induced by φ s,t,u,v r for all r, s, t, u, v ∈ Z, r ≥ 0.
Form the definitions, we see: ) be pairs of filtered complexes with their associated bigraded spectral sequences.Any pairing of filtered complexes induces a pairing of the associated spectral sequences φ = ( φr ) r∈Z ≥0 , is induced by φs,t,u,v r follows from the fact that both maps are induced by φ.For details see [36].
For a filtered complex (K • , d, F ), we define If we now suppose that the filtration is finite, i.e., for all n ∈ Z, there are l, m ∈ Z such that F l K n = K n and F m K n = 0, then the chains Z p,q 0 ⊇ . . .Z p,q r ⊇ Z p,q r+1 ⊇ . . .and B p,q 0 ⊆ . . .B p,q r ⊆ B p,q r+1 ⊆ . . .become stationary and assume Z p,q ∞ and B p,q ∞ after finitely many steps.We have If we now put n = p + q and compare with Equation (3.1), we get an identity (4.1) ) be pairs of filtered complexes with their associated bigraded spectral sequences such that all the filtrations are finite, and let φ : Tot a pairing of filtered complexes.The induced pairing of the associated spectral sequences induces a pairing of bigraded modules φ∞ : Tot such that for all i, j, p, q ∈ Z, the diagram Proof.By Lemma 4.2, the pairing of filtered complexes φ induces a pairing of spectral sequences ∞ after finitely many pages.Hence the maps φi,j,p,q r converge to a map φi,j,p,q It coincides with gr p+i,q+j,i,j (φ) as both maps are induced by φ.

The contraction pairing on the affine cone
Let k be a field of characteristic zero and let X = V + (f 1 , . . ., f c ) ⊆ P k (W 0 , . . ., W n ) be a quasi-smooth weighted complete intersection of degree (d 1 , . . ., d c ) with coordinate ring U be the sheaf of k-differentials on U and let Θ 1 U be its dual, namely the tangent sheaf.Let p be an integer satisfying 1 ≤ p ≤ n − c.Building on Flenner's calculations [15,Section 8], in this section we will construct free resolutions of the sheaves Ω p U and extend the contraction pairing to these resolutions and their associated total Čech cohomology complexes.
The resolutions.The conormal sequence associated to the closed immersion of the smooth complete intersection U into A n+1 \ {0}, namely U is locally free; see [18,Theorem II.8.17].This uses the smoothness of U .The O U -module Ω 1 A n+1 \{0} ⊗ O U is free of rank n + 1 and spanned by the elements dx 0 , . . ., dx n .The conormal sheaf I/I 2 of the complete intersection U is free of rank c and is generated by the elements f 1 , . . ., f c .Hence the conormal sequence is described by the exact sequence (5.1) of O U -modules, where the y i are basis elements, the morphism φ is the O U -linear map with and π is the natural surjection.Note, if we set deg(y i ) = d i and deg(dx i ) = w i , then the induces morphisms U ) are homogeneous of degree 0.
For any quasi-coherent O U -module N and r ∈ Z ≥0 , let S r (N ) denote the r-th symmetric power of N .As the O U -module F is free with a basis y 1 , . . ., y c , the symmetric power S r (F ) is free with a basis formed by the elements where λ ∈ Z c ≥0 with λ i = r.For the notation y λ , we will allow λ ∈ Z c .Namely, if λ i < 0 for some i, then we set y λ = 0. Similarly to [34, example (ii)], we define the complex (K • p , d • Kp ) of O U -modules with components K q p = S −q (F ) ⊗ p+q (G) for −p ≤ q ≤ 0 and K q p = 0 otherwise and differential given as the O U -linear map that sends y λ ⊗ ω, where λ ∈ Z c ≥0 with i=c λ i = −q and ω = where e i ∈ Z c denotes the i-th standard basis vector.
By composing it with the natural surjection Note for p = 1 this is Sequence (5.1).By dualizing the exact sequence (5.1) of locally free sheaves, we get an exact sequence where the elements δ 0 , . . ., δ n are the dual basis for dx 0 , . . ., dx n and the elements y * 1 , . . ., y * c are the dual basis for y 1 , . . ., y c .The differential φ * maps δ i to , so that the sequence becomes homogeneous of degree 0 on global sections.We define the complex and differential φ * .These complexes give the desired resolutions.
Theorem 5.1.In the situation above, for every p ∈ {1, . . ., n − c}, the complex of O U -modules Proof.We have already proven the second statement and the first statement for p = 1.Let p > 1 and let V = Spec B ⊆ U be any affine open such that the sheaf Ω 1 U restricted to V is free.Let M = Γ(V, Ω 1 U ) be a free B-module.Hence it is m-torsion-free (see [34,Introduction] for definition) for any positive integer m ∈ Z >0 .Hence by applying [34, Theorem 3.1] (note, since char(k) = 0, the ring B is a Q-algebra and hence the divided powers used in that reference are isomorphic to symmetric powers) to M with the free resolution U is locally free, we can cover U with affine opens V such that the restriction is free.So we are done.
The pairing of resolutions.There are O U -bilinear contraction maps γG : These contraction maps induce morphisms as γq = id G ⊗γ F ⊕ (−1) q id F ⊗γ G .These maps are compatible with the differentials and induce γ; for details on the proof see [36].
Lemma 5.2.The maps above define a pairing of complexes and induce the contraction pairing, i.e., given sections θ of Θ 1 U and ω ′ of p G with ω := ( p π) (ω ′ ), we have The pairing of the total Čech complexes.Let U be an open affine covering of U .For p ∈ {−1, 1, . . ., n− c}, let Č• (U, K • p ) be the Čech double complex (as defined in [42, Tag 01FP]) and let p ) be the associated total complex.We consider the cup product map of complexes )) as defined in [42, Tag 07MB] and compose it with the map ) induced by the pairing of complexes from Lemma 5.2 to obtain a pairing of complexes

Cohomology for weighted complete intersections
In this section, we explain how to calculate the cohomology of certain coherent sheaves on weighted complete intersections.We give an overview of results on that matter found in [10] and [15,Section 8].We start with weighted projective space, where a similar statement can be found in [10].The proof for the case of usual projective space, found in [18, Theorem III.5.1], also works in the general case.Lemma 6.1.Let k be a field, let W ∈ N n+1 be weights, let S W = k[x 0 , . . ., x n ] be the weighted polynomial algebra and let P = P k (W ) = Proj S W be weighted projective space.Then the following statements hold.
(1) The natural map S W → l∈Z H 0 (P, O P (l)) is an isomorphism of graded S W -modules.
To better handle the top cohomology, we introduce the k-dual module.Definition 6.2.Let k be a field, let A be a k-algebra and let M be a graded A-module.We define the k-dual module of M to be the graded A-module Here (S W ) −l is spanned by the monomials x α0 0 . . .x αn n with α i W i = −l.We denote the corresponding dual basis elements by φ α0,...,αn ∈ D(S W ) l .
Note that D defines a contravariant additive self-functor.If we assume A to be finitely generated over k (hence noetherian) and restrict D to the category of finitely generated graded A-modules, then it is exact.This is because in that case, the homogeneous components M l are finite-dimensional k-vector spaces.In particular, under the application of D, injections become surjections, kernels become cokernels and vice versa.Consider a complete intersection X = V (f 1 , . . ., f c ) ⊆ P(W ) of codimension c and degree d 1 , . . ., d c in P(W ).The surjection of coordinate rings S W → A = S W /(f 1 , . . ., f c ) naturally induces an embedding D(A) ⊆ D(S).For r ∈ {1, . . ., c} the scheme X r = Proj A r , A r = S W /(f 1 , . . ., f r ) is a weighted complete intersection of codimension r.We have a chain of closed immersions f r+1 is a regular sequence.Considering the associated long exact cohomology sequence and arguing inductively, we can prove the following lemma.(The induction starts with Lemma 6.1 and Remark 6.3.)For more details on the proof, we refer to [36].Lemma 6.4.Let l ∈ Z and let X be a weighted complete intersection of codimension c as above.Let l∈Z H n (X, O X (l)) ∼ = D(A) (ν).Remark 6.5.If A is the coordinate ring of a weighted complete intersection X with affine cone U = Y \ {0}, where Y = Spec A, and M is a graded A-module, then there is a natural isomorphism of graded A-modules where M (l) denotes the module M with grading shifted by l, and (_) ∼ denotes the functor that associates to an A-module its associated O Y -module (respectively its associated graded O X -module).This isomorphism can be established by compairing the Čech cohomology with respect to the coverings {D(x i )} for U and {D + (x i )} for X (see [37]) or with methods of local cohomology (see [15,Section 8]).We can use this identification to bring the results above in a more compact form.In particular, we have

The Jacobi ring of a weighted complete intersection
In this section, we will introduce the Jacobi ring of a weighted complete intersection and explain how cohomology can be expressed in terms of it.Our methods build on Flenner's calculation in [15,Section 8].We continue with notations and conventions from Section 5.As all components of the complexes K • p are free, the homology of the associated total complex of the Čech double complexes with respect to the affine covering U calculates the hypercohomology of these complexes, i.e., [42,Tag 0FLH].The total complex associated to a double complex comes with two filtrations F 1 and F 2 given by see [42,Tag 012X].The pairing γ is compatible with these filtrations.Hence, by Theorem 4.3, we get pairings of the associated spectral sequences, one for each filtration.We denote the spectral sequences associated to the filtered complex (L • p , F i ) by (E •,• i,p,r ) r∈Z ≥0 .See Section 4 or [42, Tag 0130] for formulas for the computation of the pages of these spectral sequences.We first compute the pairing of spectral sequences associated to the filtration F 1 .By Theorem 5.1, on the first page, we see Therefore, all spectral sequences converge on the second page with otherwise By Theorem 4.3, there is a pairing induced by γ Tot In particular, we obtain a pairing ).We note that it is the pairing induced by the contraction map γ : on cohomology, see Lemma 5.2.Now, we compute the pairing of spectral sequences associated to the filtration F 2 .On the first page, we see E s,t 2,p,1 = H t ( Č• (U, K s p )) = H t (U, K s p ).All modules involved in the complex K • p are free.So by Lemma 6.4, we see that the spectral sequence satisfies E s,t 2,p,1 = 0 if t = 0, n − c.We made the assumption that p < n − c.Hence, we see that the spectral sequences converge on page 2 since the differential never connects non-vanishing parts on later pages.We have The pairing is the one induced by the pairing on cohomology.Note that for both filtrations, all spectral sequences converge in such a way that for each integer m there is only one combination of (s, t) depending on m such that s + t = m and , the maps α m,s and β m,s are both isomorphisms.We combine Diagram (3.2) for suitable choices of i and j with the diagram from Theorem 4.3 to get a commutative diagram where the horizontal morphisms are isomorphisms.Thus, we have identified the pairings of spectral sequences for the filtrations F 1 and F 2 with each other.As shown above, the pairing is identified with the contraction map ) induced by γ.We now explicitly calculate all the cohomology groups involved in this pairing.The group ).We compute: We note that deg(y β ) = β i d i , and deg(dx i ) = W i .Hence, by Lemma 6.4 and Remark 6.5, we see that and that By Lemma ??, the kernel of the map ) is the k-dual of the cokernel of the map α : To describe the cokernel of this map, we introduce the Jacobi ring.We see that coker(α) is the part of R in which we fix the first degree to be p.In fact, if we view this part R p, * as a graded module via deg 2 , we get an isomorphism coker(α) ∼ = R p, * (−ν) of graded modules.This shows where the differential maps δ i to c j=1 ∂ xi (f j ) • y * j .By Lemma 6.4 and Remark 6.5, we can identify F * with H 0 (U, F * ) and G * with H 0 (U, G * ).Hence it can be identified with the deg 1 = 1 part of the Jacobi ring, namely There is the following vanishing result.
These results allow us to calculate the relevant cohomology groups.Lemma 9.3.In the situation above, the following identities hold.
Proof of Theorem 1.2.The statement is a combination of Lemma 9.3, Lemma 7.2 and Remark 7.3.