{'fraction_non_alphanumeric': 0.052259965530661034, 'fraction_numerical': 0.01651192527936843, 'mean_word_length': 3.8446000538647995, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 14, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
The transcendental lattice of the sextic Fermat surface

We prove that the integral polarized Hodge structure on the transcendental lattice of a sextic Fermat surface is decomposable. This disproves a conjecture of Kulikov related to a Hodge theoretic approach to proving the irrationality of the very general cubic fourfold.


Introduction
Proving the irrationality of a very general cubic fourfold X ⊂ P 5 over the complex numbers is a well-known problem in algebraic geometry. At present, not a single cubic fourfold is provably irrational. However, families of rational cubic fourfolds are described by Fano [13], Tregub [34], [35], and Beauville-Donagi [6]. Hassett [17] identifies, via lattice theory, a countably infinite number of subvarieties, of codimension 2 in the moduli space, consisting of rational cubic fourfolds. So far, all known rational cubic fourfolds lie on two divisors of the moduli space, corresponding to the existence of a plane or a quartic scroll marking. Even the construction of additional classes of rational cubic fourfolds is an open problem.
Recently, Kulikov [19] initiated a conjectural approach to the irrationality problem for cubic fourfolds. The strategy is modeled on that of Clemens and Griffiths [11] for cubic threefolds, with the role of the intermediate Jacobian played by the integral polarized Hodge structure T X on the transcendental part of the middle cohomology H 4 (X, Z). Assuming the existence of a birational map r : P 4 X, by Hironaka's resolution of singularities, we can resolve r to a birational morphism f : X ′ → X by a sequence X ′ = X n → X n−1 → · · · → X 0 = P 4 X of blow-ups X i → X i−1 along points, smooth curves, or smooth surfaces. Blow-ups along points, curves, and surfaces of p g = 0 do not contribute to the transcendental lattice, hence there is a decomposition of polarized Hodge structures where the sum is taken over all surfaces S j of p g ≥ 1 that are the centers of blowups in the resolution. Here, for S a smooth projective surface, T S denotes the transcendental part of the middle cohomology H 2 (S, Z). On the other hand, there is a decomposition T X ′ = f * T X ⊕ (f * T X ) ⊥ . Kulikov proves [19,Lemmas 2,3] that for X very general, comparing these two decompositions yields an index j 0 such that T S j 0 = f * T X (1) ⊕ T ′ for some nontrivial polarized Hodge substructure T ′ ⊂ (f * T X ) ⊥ . The nontriviality of T ′ follows from standard estimates on the 2nd Betti number of a minimal model of S j 0 . Then the irrationality of the very general cubic fourfold would follow from the following conjecture. [19, p. 59]). Let S be a smooth projective surface over the complex numbers. Then the integral polarized Hodge structure T S on the transcendental part of H 2 (S, Z) is indecomposable.
For curves, the integral polarized Hodge structure on H 1 is indeed indecomposable because of Riemann's theorem describing the theta divisor of a Jacobian in geometric terms and showing that it is irreducible. Hence the Jacobian is indecomposable as a polarized abelian variety and this is what is used substantially in the proof of the irrationality of cubic threefolds by Clemens and Griffiths [11]. An essential point in Conjecture 1.1 is the indecomposability over Z. In fact, counterexamples over Q abound; note also that Jacobians of curves are decomposable if one considers them within the category of abelian varieties up to isogeny.
In this paper we prove that Conjecture 1.1 is false integrally, too: Theorem 1.2. Let S ⊂ P 3 be the sextic Fermat surface. Then the integral polarized Hodge structure T S on the transcendental part of H 2 (S, Z) is decomposable.
After recalling some general theory in Section 2, we prove this in Section 3.
In our analysis, we make use of the description of the integral Hodge structure of Fermat varieties as a module over the group ring of the automorphism group in the formulation of Looijenga [23, §2], which in turn draws on many previous sources [16], [32], [30], [28].
After a discussion of the results of this paper with the second author at an Oberwolfach workshop in May 2013, V. Kulikov suggested that his conjecture could be modified to the effect that surfaces with decomposable integral polarized Hodge structure on the transcendental lattice are rigid with this property, and that this would still imply irrationality of the very general cubic fourfold. In Section 4, we work out the details of this.
One may also wonder if surfaces S for which Conjecture 1.1 fails are always defined over Q.
Recall that the sextic Fermat surface has maximal Picard number. It is not clear to us whether there exists nonisotrivial positive dimensional families of surfaces with maximal Picard number and p g > 1.
Finally, we mention a few other conjectural approaches to prove the irrationality of the very general cubic fourfold. A derived categorical approach due to Kuznetsov [20] has seen recent activity [1], [4], [24]. Using the theory of semiorthogonal decompositions Kuznetsov constructs a triangulated category A X ⊂ D b (X) and conjectures that it encodes all the information concerning the rationality of X. The irrationality of the very general cubic fourfold would be a consequence. This approach runs into some difficulties due to certain pathologies that semiorthogonal decompositions of derived categories may exhibit, see [7], [8], [9], but which can possibly be overcome if complemented by new ideas.
There is also a cohomological invariant approach due to Colliot-Thélène. Many unramified cohomology groups of X vanish as a consequence of the integral Hodge conjecture [12, Thm.  [25,Thm. 2.11], the vanishing of all unramified cohomology groups arising from cycle modules is controlled by the vanishing of the Chow group A 0 (X F ) of 0-cycles of degree 0 on X F = X × C F over all field extensions F/C. Hence the detection of such a nontrivial 0-cycle on a cubic fourfold over a sufficiently complicated field F would imply irrationality.
We would like to thank A. Beauville, F. Bogomolov, L. Katzarkov, and especially V. Kulikov for discussions and suggestions concerning the present material.

Integral polarized Hodge structures on Fermat surfaces
Here we describe the integral polarized Hodge structure on the cohomology of Fermat varieties, especially Fermat surfaces, building on and developing further [23]. (1) An integral polarized Hodge structure (IPHS) of weight n ∈ N is a triple (H Z , H p,q , Q) where -H Z is a free Z-module of finite rank and Q : H Z × H Z → Z is a nondegenerate (i.e., nondegenerate over Q) bilinear form with symmetry property Q(x, y) = (−1) n Q(y, x) for all x, y ∈ H Z .
-The H p,q , 0 ≤ p, q ≤ n, are complex linear subspaces in the complexification H C = H Z ⊗ Z C with the property that and such that H p,q = H q,p , with the conjugation on H Z ⊗ Z C being induced by the conjugation on C.
-Extend Q to H C by linearity. Then we require the orthogonality condition Sometimes one also requires the positivity condition We choose not to make it part of the abstract notion of integral polarized Hodge structure for definiteness, but this is immaterial for everything that follows: most Hodge structures that occur in this article have this property as they are sub-Hodge structures of geometric Hodge structures on the primitive cohomology of smooth projective varieties.
(2) The notion of morphism of integral polarized Hodge structures ( . Such an f is necessarily an embedding since Q 1 , Q 2 are nondegenerate. (3) There is a natural notion of direct sum of two integral polarized Hodge struc- ). An integral polarized Hodge structure is indecomposable if it is not a direct sum of two nontrivial integral polarized Hodge structures.
Let X be an n-dimensional smooth projective variety. We assume that n = 2m is even. Consider the the middle cohomology H Z := H n (X, Z)/(torsion) and its Hodge decomposition H C = H n (X, C) = p+q=n H p,q into the spaces of harmonic (p, q)-forms. Consider the bilinear form Q defined as the restriction to H Z of Then this triple defines an integral polarized Hodge structure; the positivity condition in Definition 2.1(1) is not satisfied, but it holds if we pass to primitive cohomology.
a) The IPHS on the primitive cohomology of X, denoted by (H n 0 (X, Z), H p,q 0 , Q 0 ), is defined as follows: H n 0 (X, Z) ⊂ H n (X, Z)/(torsion) is the sublattice which is orthogonal (with respect to Q) to the middle power of the polarization class h m ∈ H m,m ∩ H n (X, Z) and H p,q and Q T the restriction of Q to T X . It is an integral polarized Hodge substructure of the primitive cohomology.
We will now assume n = 2 and describe this structure for the Fermat surface , taking our point of departure from [23], which we would like to simplify and amplify in several respects.
Looijenga's computation starts by considering homology. Poincaré duality gives an isomorphism P : with Q, then P is an isomorphism of integral lattices.
Proof. We have the following commutative diagram: where ∩ is the topological cap product, Q ′ the intersection product, and Q the bilinear form on cohomology as defined above. The assertion follows. By abuse of notation, we will also write Q (and not Q ′ ) for the pairing on homology in the following.
Looijenga now works with the primitive homology H 0 2 (X d , Z) defined as the orthogonal to h (the embedding hyperplane class from P 3 ), viewed as an element of H 2 (X d , Z) (so this is P −1 (h), to be precise). Hence Poincaré duality induces an isomorphism of lattices The intersection form is given as follows: abbreviating where we calculate in the group ring Conversely, all tuples K = (k 0 , k 1 , k 2 , k 3 ) ∈ {1, . . . , d − 1} 4 with zero sum mod d give a character, which we denote by χ K . Notice that the complex zeros Z of the ideal I d are precisely the points P K := (ζ k 0 d , . . . , ζ k 3 d ) with K as above.
We now have to describe how H 0 2 (X d , Z) ≃ H 2 0 (X d , Z), viewed as a sublattice of H 2 0 (X d , Z) ⊗ C = H 2 0 (X d , C), is positioned relative to the Hodge subspaces of H 2 0 (X d , C). Note that this will allow us to compute everything: the algebraic part, the transcendental part, and the induced integral polarized Hodge structure.
The Poincaré duality isomorphism P is equivariant for the natural actions of G on homology and cohomology (it is given by cap product with the fundamental class, which is invariant). Via P , we identify H p,q ⊂ H 2 0 (X d , C) with its image in H 0 2 (X d , C), which we denote by the same letter. Proposition 2.7. The Hodge subspace H p,q ⊂ L ⊗ C is where (L ⊗ C) χ K is the eigenspace of the character χ K and the sum runs over all characters with |K| := i k i = (q + 1)d.
In other words, for Z p,q = {P K | i k i = (q + 1)d}, Proof. We can identify the corresponding character spaces in L ⊗ Z C ≃ H 0 2 (X d , C) and H 2 0 (X d , C) via the diagram We then apply [23], section after Cor. 2.4 on p. 8.
Lemma 2.8. The quadratic form Q can be written as for some α P ∈ Q(ζ d ) * for each P ∈ Z.
Proof. First, note that Q is invariant under G and also satisfies Remark 2.9. The whole construction up to now is also invariant under the symmetric group S 4 acting by permutations on the u i . Therefore α P is constant on the orbits of the action of S 4 on Z.
Proposition 2.10. Consider the action of the Galois group Γ = Gal(Q(ζ d )/Q) on Z. Let Z A := P K | Γ · P K ⊂ Z 1,1 and Z T := Z\Z A . Then Proof. By [32], Theorem I(iii), we get the assertion about A X . The assertion about T X then follows from Lemma 2.8.
Direct computation gives the above matrix for Q (1,1,2,2) and its determinant. The existence of L ′ (1,2,1,2) and L ′ (1,2,2,1) with the analogous bases follows by symmetry. Proposition 3.4. Let 1,2,2) ⊕ L ′ (1,2,1,2) ⊕ L ′ (1,2,2,1) . Then we have an equality of lattices T ′ X = T X . In particular, always L ′ β = L β . Proof. It is clear that T ′ X is a sublattice of T X of finite index. Consider the basis of T ′ X consisting of the union of the basis vectors of the L ′ β constructed above. One can check that the reductions of the vectors of this basis modulo 2 and 3 are still linearly independent. Since the discriminant of T ′ X is only divisible by primes 2 and 3, this proves that there is no sublattice of L which contains T ′ X as a proper sublattice of finite index. In particular, T X = T ′ X . This completes the proof of Theorem 1.2.
Remark 3.5. Proposition 3.4 implies also that the discriminant of the transcendental lattice T X is 2 40 3 30 and consequently the discriminant of the Picard lattice is −2 40 3 30 (the sign is negative since the signature of Q on H 1,1 is (1, 85)). We could not find this number in the literature.
Remark 3.6. We found the matrices M (1,1,1,3) and M (1,1,2,2) as follows: using Proposition 2.7 we find a Q-basis of L (1,1,1,3) and of L (1,1,2,2) ⊕ L (1,2,1,2) ⊕ L (1,2,2,1) . Clearing denominators, we find vectors in the lattice L that form a basis over Q of L (1,1,1,3) and of L (1,1,2,2) ⊕ L (1,2,1,2) ⊕ L (1,2,2,1) , respectively. These vectors generate lattices M and N , which are not saturated, however. For each prime p dividing the discriminant of M , for example, we reduce a set of basis vectors mod p in the ambient L, and if the reductions happen to become linearly dependent, we lift the dependency relation to Z and find a vector divisible by p. Continuing in this way we arrive at a saturated sublattice M ′ spanning the same Q-subspace as M . Using the LLL-algorithm we find vectors in M ′ with small coefficients. Among these we choose one with small length; evaluating this on Z (1,1,1,3) gives the first column of M (1,1,1,3) . The remaining columns are obtained using the S 4 -symmetry and conjugation.

Rigidity and transcendental lattice decompositions
By a family of surfaces π : S → B, we mean a flat surjective morphism of schemes or analytic spaces, all of whose fibers are projective surfaces. All families considered will actually have smooth fibers, so that π is even a smooth map. A family is isotrivial if the fibers S b for b ∈ B(C) are all isomorphic, equivalently, the family is locally trivial for theétale topology, equivalently, the classifying map to the coarse moduli space maps to a closed point (if it is defined for the isomorphism class of the fiber). If B is a scheme or analytic space, then by a "very general" point, we mean any point outside of a countable union of proper analytic subsets.  Since the integral polarized Hodge structures on the transcendental parts T X of the middle cohomology of cubic fourfolds X are uncountably many, and, moreover, in [19] it is proved that, if such X were rational, their T X must occur as proper summands of T S , for S a surface, it suffices to prove the following. Remark that surfaces with p g ≤ 1 have indecomposable T S , so one can restrict to surfaces with p g ≥ 2.
Proof of Theorem 4.3. We divide the proof into a series of technical steps.
Step 1. There are countably many families S i → B i of projective surfaces over irreducible base varieties B i , such that every isomorphism class of a projective surface is represented by a fiber of some such family. One can take for example the universal families over the Hilbert schemes of two-dimensional subschemes of P n , n ∈ N, since there are only countably many Hilbert polynomials.
Hence it suffices to prove the following. Step 2. We first prove a linear algebra result. Let M ⊂ L be a sublattice of L. Let Z ⊂Ď be the subset such that M is contained in the H 1,1 -part defined by the Hodge filtration corresponding to the points in Z (this is a closed algebraic subset). Then for each z ∈ Z ∩ D, we get an induced integral polarized Hodge structure on M ⊥ ⊂ L, by putting H p,q M ⊥ := H p,q ∩ (M ⊥ ⊗ C). Let Z M ⊂ Z ∩ D be the locus of points z where this induced integral polarized Hodge structure is decomposable. Then Z M is an intersection with D of countably many locally closed algebraic subvarieties ofĎ.
To prove this, note first that a given lattice, M ⊥ in our case, can decompose in at most a countable number of ways into two nontrivial summands. We fix such a decomposition of M ⊥ . We then have to prove that the points z ∈Ď such that the corresponding Hodge filtration decomposes in a way compatible with the fixed decomposition of M ⊥ , form a finite union of locally closed algebraic subsets ofĎ. This follows from the following. Lemma 4.6. Let F = Fl(k 1 , . . . , k r ; V ) be the flag variety of flags (F 1 , . . . , F r ) of type (k 1 , . . . , k r ) in a complex vector space V . Let W ⊂ V be a fixed subspace with a given direct sum decomposition W = W 1 ⊕W 2 into subspaces W 1 ⊂ W and W 2 ⊂ W . Then the subset Z W 1 ,W 2 of flags such that is a finite union of locally closed algebraic subsets.
The proof of this Lemma is straightforward: it suffices to prove it in case F = Fl(k; V ) is a Grassmannian. The set of k-subspaces of V intersecting W in a subspace of fixed dimension k ′ is a locally closed subset of the Grassmannian. We now fix k ′ ≤ k and also positive integers a and b with a + b = k ′ . We have a morphism Gr(a, W 1 ) × Gr(b, W 2 ) → Gr(k ′ , W ) (direct sum of subspaces), whose image is a closed subset G ⊂ Gr(k ′ , W ). Over Gr(k ′ , W ) we have the tautological bundle E whose fiber over a point is the given subspace of dimension k ′ of W . We consider the relative Grassmannian ψ : Gr(k − k ′ , V ⊗ O/E ) → Gr(k ′ , W ). Now Gr(k − k ′ , V ⊗ O/E ) is proper and has a natural morphism f to Gr(k, V ). Its image is closed (but consists also of subspaces whose dimension of intersection with W is strictly larger than k ′ of course). In any case, f (ψ −1 (G)), intersected with the locus of subspaces L of dimension k in V whose intersection with W has dimension exactly k ′ , is exactly the locus of such subspaces L such that L ∩ W decomposes into a direct sum L ∩ W 1 of dimension a and L ∩ W 2 of dimension b. Since there are only finitely many choices for a and b, the result follows.