Triangle varieties and surface decomposition of hyper-K¨ ahler manifolds

Triangle varieties and surface decomposition of hyper-K¨ ahler manifolds

Claire Voisin

Pour Miles Reid, avec estime et amiti´e Abstract

We introduce and study the notion of “surface decomposable” variety, and dis- cuss the possibility that any projective hyper-K¨ahler manifold is surface decomposable, which would produce new evidence for Beauville’s weak splitting conjecture. We show that surface decomposability relates to the Beauville-Fujiki relation, a constraint on the cohomology ring of the variety, and that general varieties with h^2,0 ̸= 0 are not surface decomposable. We also formalize the notion of triangle variety that is useful to produce surface decomposition. We show the existence of these geometric structures on most explicitly constructed classes of projective hyper-K¨ahler manifolds of Picard number 1.

0 Introduction

LetX be a complex manifold of dimension 2nequipped with a holomorphic 2-formσXwhich is everywhere of maximal rank 2n. Locally for the Euclidean topology on X, Darboux’s theorem tells that one can write, for an adequate choice of holomorphic coordinates,

σX=

∑n

i=1

dzi∧dzn+i,

that is,σ_X is the sum of nclosed holomorphic 2-forms of rank 2.

A natural question is whether this statement can be made more global, particularly in the projective case:

Question 0.1. Does there exist a generically finite coverϕ:Y →X, such thatϕ^∗σX is the sum ofnclosed holomorphic 2-forms of rank2 onY?

Our goal in this paper is to study a geometric variant of this question, namely the possibility that any projective hyper-K¨ahler manifold issurface decomposable (or admits a surface decomposition) in the following sense:

Definition 0.2. A smooth projective variety X of dimension 2n will be said to be surface decomposable if there exist a smooth variety Γ, smooth projective surfaces S₁, . . . , S_n, and generically finite surjective morphismsϕ: Γ→X, ψ: Γ→S₁×. . .×S_n such that for any holomorphic2-form σ∈H⁰(X,Ω²_X),

ϕ^∗σ=ψ^∗(∑

i

pr^∗_iσi) (1)

for some holomorphic2-forms σ_i onS_i.

We will show that surface decomposability is restrictive for general projective varieties of dimension 2n≥4 (see Theorem 1.6). The reason is that it essentially implies the Beauville- Fujiki formulas describing the top self-intersection on cohomology of degree 2 as the power of a quadratic form, or at least a product of quadratic forms (see Proposition 1.5).

Remark 0.3. Surface decompositions are natural in the hyper-K¨ahler context which is the one of this paper, but one can of course introduce in general decompositions into summands of other dimensions. For example, we can consider curve decompositionsof any variety X of dimensionn, given by the data of generically finite surjective morphisms

ϕ: Γ→X, ψ: Γ→C1×. . .×Cn, (2) such that for any 1-formα∈H^1,0(X),

ϕ^∗α=ψ^∗(∑

i

pr^∗_iα_i) (3)

for some formsα_i ∈H^1,0(C_i). Abelian varieties are clearly curve decomposable. One can show similarly that this notion puts strong restrictions on the structure of the intersection pairing∧2n

H¹(X,C)→H²ⁿ(X,C) =C, whenh^1,0(X) is large compared ton.

The first examples of hyper-K¨ahler manifolds were constructed by Beauville [4] and Fujiki [14] as punctual Hilbert schemes ofK3 surfaces or abelian surfaces and hence were rationally dominated by products of surfaces. They were thus obviously surface decomposable. How- ever, it follows from deformation theory that theseK3 or abelian surfaces disappear under a general deformation to a projective hyper-K¨ahler manifold with Picard number 1. Indeed, the parameter space forK3 surfaces is too small to parameterize also general deformations with Picard number 1 of their punctual Hilbert schemes. The starting point of this paper is the observation that on many explicitly described general deformations as above, a surface decomposition still exists.

Let us make several remarks concerning Definition 0.2. First of all, the condition (1) has been asked only for holomorphic 2-forms, but by an elementary argument involving Hodge structures (see Section 1), it then follows that it is satisfied for any transcendental classη ∈H²(X,Q)tr, the latter space being defined as the smallest Hodge substructure of H²(X,Q) whose complexification containsH^2,0(X).

Next, if we allow an arbitrarily large number N of summands Si and only ask that ϕ is surjective and ψ is generically finite, then the definition is (at least conjecturally) not restrictive since the property is satisfied by any smooth projective varietyX satisfying the Lefschetz standard conjecture in degree 2 (see Proposition 1.2). Similarly, we could consider, instead of (1), the weaker condition thatϕis surjective generically finite and

σ=ϕ_∗(ψ^∗(∑

i

pr^∗_iσi)) inH^2,0(X), (4)

but, as before, this is implied by Lefschetz standard conjecture, and we even can takeN = 1 to achieve (4), as shows the case of the (2,0)-forms on the symmetric product or rather punctual Hilbert schemeX =S^[k] of a simply connected surfaceS: they are all obtained starting from a (2,0)-form on S by a formula like (4).

The reason why (4) is much weaker than (1) is the fact that pull-back maps are compat- ible with cup-products, while push-forward maps are not. More precisely, we will show (see Proposition 1.5) that the surface decomposability implies and provides a geometric expla- nation for Beauville-Fujiki’s famous formula for the self-intersection of degree 2 cohomology on a hyper-K¨ahler manifold: ∫

X

α²ⁿ=λq(α)ⁿ,

whereq is the Beauville-Bogomolov quadratic form onH²(X,Q)tr (see [4]). To prove this implication, we have to assume that the Mumford-Tate group of the Hodge structure on H²(X,Q)tris large enough to guarantee that all the quadratic forms onH²(X,Q)trinduced from (,)S_ivia the morphism of Hodge structuresψi :η7→ηi(see Section 1) are proportional, but this is automatic if X is the general member of a family of polarized hyper-K¨ahler manifolds.

This observation suggests that surface decomposability could be a way to approach the weak splitting property of hyper-K¨ahler manifolds (see Conjecture 1.7) conjectured by Beauville in [6]. It says that cohomological polynomial relations between divisors on hyper- K¨ahler manifolds X are satisfied on the Chow level. A weaker version asks that X has a canonical 0-cycle oX ∈ CH0(X) such that D²ⁿ is proportional to oX in CH0(X) for any divisorD onX.

In this direction, we prove Theorem 1.8 which has the following consequence:

Theorem 0.4. Assume that the general member X_t of a family (X_t)_t∈B of hyper-K¨ahler manifolds with given Picard latticeΛ has a surface decomposition with a simply connected Γ. Then the Beauville weak splitting property holds for the divisor classes belonging to NS(X_t)^⊥Λ if and only if there exists a0-cycleo_Xt ∈CH₀(X_t)such thatD²ⁿ is proportional tooX_t inCH0(Xt)for any divisor D∈NS(Xt)^⊥Λ.

The importance of Theorem 0.4 is that it reduces the weak splitting property to checking the weak version, namely in top degree. The weakness of the result is that it applies only to divisors of class perpendicular to Λ, which means, in practice, primitive. One has to understand separately what happens with the powersh^k of the polarizing class. In all the geometric examples we have, the natural surface decomposition that we exhibit provides (1) only on primitive cohomology.

Remark 0.5. This raises the question whether hyper-K¨ahler manifolds could admit a sur- face decomposition such that formula (1) holds on the whole cohomologyH², instead of only primitive or transcendental cohomology. This question is very interesting in the case of the Hilbert schemeS^[n] of a K3 surfaceS, where the natural surface decomposition provides (1) only on the part of H² which is orthogonal to the class of the exceptional divisor over the diagonal. Combined with the theorem above, this could be a way to reproving the weak splitting conjecture forS^[n](proved by Maulik and Negut [22]), as the weak version is known for them by [30].

Remark 0.6. The statement above is empty for the very general member of the family since it has NS(X_t)^⊥Λ = 0. The statement above is interesting for special hyper-K¨ahler manifolds with higher Picard rank ρ≥ ρ_gen+ 2, which are parameterized by a countable union of closed algebraic subsets in the baseB, which is dense inB if dimB≥2.

The second geometric notion that will play an important role in this paper is the follow- ing.

Definition 0.7. [29] An algebraically coisotropic subvariety of a hyper-K¨ahler manifold X of dimension 2n is a subvariety Z ⊂ X of codimension k ≤ n which admits a rational fibrationϕ:Z99KW, whereW is smooth anddimW = 2n−2k, such that

σ_X|Zreg =ϕ^∗_regσW, (5) where ϕ_reg : Z_reg 99K W is the restriction of ϕ to the regular locus of Z and σ_W is a holomorphic2-form on W.

This notion is to be distinguished from the notion of coisotropic subvariety, which just asks that the restrictionσ_X|Zreg has rank 2n−2kat any point, or equivalently that

T_Z^⊥σX

reg,x ⊂T_Zreg,x (6)

at any pointxofZreg. Equation (6) defines then a foliation onZreg and Z is algebraically coisotropic when the leaves of this foliation are algebraic. The two notions coincide in the case n =k of Lagrangian varieties. Fork = 1, any divisor is coisotropic but smooth ample divisors are not algebraically isotropic (see [2]). It is not easy a priori to construct algebraically coisotropic divisors in a projective hyper-K¨ahler manifolds. Examples are given by uniruled (singular) divisors. Indeed, starting from a singular uniruled divisor

D ⊂X, consider a desingularization De → D with induced morphism j : De →X. Then by assumption De is uniruled, hence its maximal rationally connected fibration (or MRC fibration, see [18]) is nontrivial, producing, possibly after changing the birational model of D, a morphisme

f :De →B

with rationally connected fibers of positive dimension. Any holomorphic form onDe is pulled- back viaf from a holomorphic form onB. We apply this to the pull-backj^∗σ_X ∈H^2,0(D)e and conclude that

j^∗σX =f^∗σB in H^2,0(D).e (7) Formula (7) implies that the fibers off are tangent to the kernel ofj^∗σX. As the generic rank ofj^∗σX is 2n−1,n= dimX, we conclude that the fibers off are 1-dimensional (so in fact D is ruled), andD is algebraically coisotropic. Unfortunately, the uniruled divisors in a hyper-K¨ahler manifold X are rigid, because rational curves cannot coverX (indeed, the MRC fibration ofX is trivial as shows the argument above). One open question is whether a projective hyper-K¨ahler manifold can always be swept out by algebraically coisotropic divisor. This is certainly true ifX has a surface decomposition (see Proposition 0.11 below).

In the other direction, we show in Proposition 1.10 that the existence of a 1-parameter family of algebraically coisotropic divisors for X implies a decomposition, on a generically finite cover ofX, of the holomorphic 2-form of X as a sum of one rank 2 and one rank 2n−2 holomorphic forms.

The theory of coisotropic subvarieties of higher codimension is more complicated. In the paper [29], we discussed the constraints on the cohomology classes of coisotropic subvarieties of higher codimension and asked whether the space of coisotropic classes, namely those satisfying these constraints, are generated by classes of algebraically coisotropic subvarieties.

We also proposed the construction of algebraically coisotropic subvarieties as total spaces of 2n−2k-dimensional families of constant cycles varieties (in the sense of Huybrechts [16]) of dimensionk.

The third notion that will be introduced and studied in this paper is that of triangle variety:

Definition 0.8. A triangle variety for X (equipped with a holomorphic 2-form σX) is a subvariety of X ×X ×X which dominates X by the three projections, maps in a generi- cally finite way to its image in X ×X via the three projections and is Lagrangian for the holomorphic formσ₁+σ₂+σ₃ on X³, whereσ_i= pr^∗_iσ_X.

The following example will be generalized in Section 2.5.

Example 0.9. LetS →B be an elliptic surface with a section. Then the graph of minus the relative sum mapS×BS99KS, which is naturally contained inS³, is a triangle variety for any holomorphic 2-formσonS.

Triangle varieties seem to exist for most explicitly constructed classes of projective hyper- K¨ahler manifolds. In fact, the simplest example of them, namely actual triangles in the Fano variety F1(Y) of lines of a smooth cubic fourfold Y, is studied with detail in [25] by Shen and Vial, who use them to study a decomposition (Beauville splitting) of the Chow groups ofF1(Y). The main geometric examples, including this one, will be presented in Section 2.

Remark 0.10. More generally, we can also introduce (and we will use)k+ 1-angle sub- varietiesT_k+1 ⊂ X^k+1 which are Lagrangian for the 2-form ∑

i±pr^∗_iσ_X. They are easily constructed by iteration starting from a triangle variety.

The first link between triangle varieties, surface decompositions and algebraically coisotropic subvarieties is the following obvious implication:

Proposition 0.11. If X has a surface decomposition, then it has mobile algebraically coisotropic subvarieties of any codimension k ≤ n. If the surfaces appearing in a surface decomposition of X have triangle varieties, (for example, if they are elliptic,) then so does X.

Proof. Indeed let ϕ: Γ → X, ψ: Γ → S1×. . .×Sn be surjective generically finite maps such that

ϕ^∗σX =ψ^∗(pr^∗₁σS₁+. . .+ pr^∗_nσS_n) inH^2,0(Γ).

For any integer k≤n, let Ci ⊂Si, i = 1, . . . , k, be very ample curves in general position.

Thenϕ(ψ⁻¹(C1×. . . Ck×Sk+1×. . .×Sn)) is an algebraically coisotropic subvariety ofX of codimensionk. ItTi⊂S_i³are triangle varieties, thenϕ³(T1×. . .×Tn) is a triangle variety forX.

We will show in the paper how conversely triangle (or n+ 1-angle) subvarieties for X and algebraically coisotropic subvarieties ofX of codimension n−1, where dim = 2n, can be used to construct surface decompositions and algebraically coisotropic varieties ofX of any codimension 1≤k≤n. We prove the following result (see Theorem 3.1).

Theorem 0.12. Let X be a projective hyper-K¨ahler variety of dimension 2n. Assume X has an+ 1-angle variety (see Remark 0.10)T_n+1 ⊂Xⁿ⁺¹ and an algebraically coisotropic subvariety τ :Z 99K Σ of dimension n+ 1. Let F ⊂X be the general fiber of τ. Then if the intersection number Fⁿ·p1...n(Tn+1) of cycles in Xⁿ is nonzero, X admits a surface decomposition.

Finally we will prove (see Theorem 3.3), as an application of Theorem 0.12 or variants of it, that most hyper-K¨ahler manifolds that have been explicitly constructed from algebraic geometry admit surface decompositions.

1 The decomposition problem for hyper-K¨ ahler vari- eties

LetXbe a smooth projective manifold of dimension 2n(we will later on focus on the hyper- K¨ahler case). We wish to study the existence and consequences of a surface decomposition of the form described in the introduction (see Definition 0.2), namely the existence of smooth projective surfacesSi, i= 1, . . . , n, and an effective correspondence Γ (which can be assumed to be smooth and projective)

ϕ: Γ→X, ψ: Γ→S1×. . .×Sn,

withϕandψdominant generically finite, such that for anyσX ∈H^2,0(X), ϕ^∗σX=ψ^∗(

∑n

i=1

pr^∗_iσS_i) in H^2,0(Γ), (8) for someσS_i ∈H^2,0(Si), where the pr_i:S1×. . .×Sn→Si are the various projections. Let us spell-out the proof of the following

Lemma 1.1. Condition (8) implies more generally that, for any η∈H²(X,Q)_tr,

ϕ^∗η =ψ^∗(pr^∗₁η₁+. . .+ pr^∗_nη_n) in H²(Γ,Q), (9) for someη_i∈H²(S_i,Q)_tr.

Proof. Indeed, the formσS_i ∈H^2,0(Si) in (8) can be reconstructed fromσX by the action of the morphism of Hodge structures

ψi:H²(X,Q)tr →H²(Si,Q), η7→ηi

ψi(η) = 1 N Ni

pr_i∗(d²ⁿ⁻²∪ψ_∗(ϕ^∗η)), (10) whereN is the degree ofψandd=∑

ipr^∗_idi is the first Chern class of an ample divisor on

∏

iSiwith the property that pr_i∗(d²ⁿ⁻²) =Ni1Si inH⁰(Si,Q) for all i. The last condition indeed guarantees that

1 N Ni

pr_i∗(d²ⁿ⁻²·ψ_∗ψ^∗(∑

j

pr^∗_jη_j)) =η_i

for any cohomology classesη_jonS_j such thatη_j∪d_j= 0 for allj. The morphisms of Hodge structuresψ_ibeing defined as in (10), condition (8) then rewrites as

ϕ^∗σX =ψ^∗(∑

i

pr^∗_i(ψi(σX))) inH^2,0(Γ).

This equality defines a Hodge substructure of H²(X,Q). Hence, once it is satisfied on H^2,0(X), it is satisfied onH²(X,Q)tr.

A variant of this definition assumes thatS1=. . .=Snand the correspondence Γ is sym- metric with respect to the symmetric group action onSⁿ, but this is not essential. Another variant asks that condition (9) is satisfied for any η ∈ H²(X,Q)_prim, where the subscript

“prim” refers to the choice of an ample line bundleL on X, and primitive cohomology is primitive with respect tol=c₁(L). If we work with very general hyper-K¨ahler manifolds of Picard number 1, the two notions coincide. In the hyper-K¨ahler case, equation (8) decom- poses the smooth projective manifoldX in the sense that the rank 2nholomorphic 2-form onX gets decomposed as the sum ofn(generically) rank 2 holomorphic 2-formsψ^∗(pr^∗_iσS_i) on the generically finite cover Γ.

If we now relax the conditions onϕ, ψin Definition 0.2 and just ask thatϕis surjective and ϕ^∗σX = ψ^∗(∑N

i=1pr^∗_iσS_i) for any holomorphic 2-form on X, allowing an arbitrarily large number of summands, then a decomposition as in (8) should always exist, for any smooth projective varietyX. More precisely:

Proposition 1.2. Let X be a smooth projective variety. Assume X satisfies the Lefschetz standard conjecture for degree2cohomology. Then there is a generically finite coverϕ: Γ→ X, surfaces S₁, . . . , S_N, and a morphismψ: Γ→S₁×. . .×S_N, such that any (2,0)-form σon X satisfies

ϕ^∗σ=ψ^∗(∑

i

pr^∗_iσ_i) inH^2,0(Γ) (11)

for some(2,0)-forms σi on Si.

Remark 1.3. The proof will even show that we can takeS1=. . .=SN and Γ symmetric.

Proof of Proposition 1.2. The Lefschetz standard conjecture for degree 2 cohomology onX provides a codimension 2-cycleZ onX×X such that, ifn= dimX,Z^∗:H²ⁿ⁻²(X,Q)→ H²(X,Q) is the inverse of the Lefschetz isomorphism lⁿ⁻² : H²(X,Q) ∼= H²ⁿ⁻²(X,Q) induced by the first Chern classl of a very ample line bundleL onX. Letj:S→X be a smooth surface which is the complete intersection ofn−2 ample hypersurfaces in|L|. Then by the Lefschetz theorem on hyperplane sections, the Gysin morphism j_∗ : H²(S,Q) → H²ⁿ⁻²(X,Q) is surjective, so that, denoting byZ_S the restriction ofZ to X×S, we find that

Z_S^∗=Z^∗◦j_∗:H²(S,Q)→H²(X,Q)

is also surjective. We can make ZS is effective by replacing if necessary its negative com- ponents−ZS,i by effective residual cyclesZ_i^′ of classH²−ZS,i, whereH = pr^∗₁H1+ pr^∗₂H2

is a sufficiently ample line bundle onX×S. The cycleH²acts trivially on transcendental

cohomology, so this change does not affect Z_S^∗ :H²(S,Q)tr →H²(X,Q)tr. Because ZS is effective, it is given by a rational map

ϕZ_S :X99KS^(N), so that

Z_S^∗σS =ϕ^∗_ZSσ_S(N) inH^2,0(X),

for any holomorphic 2-form σ_S on S, where σ_S(N) denotes the induced 2-form on S^(N). Recall that, denoting byµ:S^N →S^(N)the quotient map,

µ^∗σ_S(N)=

∑N

i=1

pr^∗_iσ_S in H^2,0(S^N). (12) The finite cover µ induces a finite cover ϕ: Γ := X ×S^(N) S^N → X, ψ : Γ→ S^N and we have a commutative diagram

Γ ^ψ //

ϕ

S^N

µ

X

ϕ_ZS

//S^(N)

(13)

From the commutativity of (13), we deduce the equality of holomorphic 2- forms on Γ ϕ^∗(ϕ^∗_ZSσ_S(N)) =ψ^∗(µ^∗σ_S(N)). (14) Combining (12) and (14), we get (11) withσX =Z_S^∗σS, σi=σS for alli.

Remark 1.4. To be fully rigorous in the above proof, we should introduce desingularizations ofS⁽ⁿ⁾and Γ to write the equalities above. This is done in [23].

1.1 Surface decomposition and cohomology ring

It is a well-known and fundamental result (see [8], [14]) that for a hyper-K¨ahler manifoldX of dimension 2n, there exist a quadratic formqonH²(X,Q) and a positive rational number λsuch that for any η∈H²(X,Q)

∫

X

η²ⁿ=λq(η)ⁿ. (15)

Let us show how this property follows, at least on transcendental cohomology, from the existence of a surface decomposition for a smooth projective varietyX, assuming it satisfies the following property (*). Recall first that a quadratic formqon a rational weight 2 Hodge structure, consisting of aQ-vector spaceH and a Hodge decomposition ofH_C:=H⊗C

H_C=H^2,0⊕H^1,1⊕H^0,2, H^p,q =H^q,p,

is said to satisfy the first Hodge-Riemann relations if the Hodge decomposition is orthogonal for the Hermitian pairingh(α, β) =q(α, β) onH_Cor, equivalently,q(H^2,0, H^2,0⊕H^1,1) = 0.

We will say that it satisfies the weak second Hodge-Riemann relations if q(α, α) ≥ 0 for α∈H^2,0and q(α, α)≤0 forα∈H^1,1. Consider the condition

(*)There exists up to a coefficient a unique quadratic form q satisfying the first Hodge- Riemann relations onH²(X,Q)tr.

Property (*) is well-known to be satisfied by a very general lattice polarized projective hyper-K¨ahler manifold. Note that we need in any case to use transcendental cohomol- ogy, namelyH²(X,Q)^{⊥N S(X)}, instead of primitive cohomology, as (*) is never satisfied on H²(X,Q)primif it is different fromH²(X,Q)tr, that is, if it contains rational classes of type (1,1). We have the following.

Proposition 1.5. (i) If a smooth projective varietyX of dimension 2n admits a surface decomposition as in Definition 0.2, there exist quadratic formsq1, . . . , qn satisfying the first and weak second Hodge-Riemann relations onH²(X,Q)tr, such that, for anyη∈H²(X,Q)tr

∫

X

η²ⁿ=q₁(η). . . q_n(η). (16) (ii) If furthermore X satisfies property (*), there exists a rational number λ and a quadratic formqsatisfying the first and weak second Hodge-Riemann relations onH²(X,Q)_tr, such that, for anyη ∈H²(X,Q)_tr,

∫

X

η²ⁿ=λq(η)ⁿ. (17)

Proof. We have by assumption, for anyη∈H²(X,Q)_tr, an equality ϕ^∗η=ψ^∗(

∑n

i=1

pr^∗_iηi), (18)

where ψ, ϕ are as in (8). For each surface Si, we have the Poincar´e pairing (,)S_i on H²(Si,Q)tr which satisfies the first and second Hodge-Riemann relations, and, as the mor- phismψi which mapsη to ηi is a morphism of Hodge structures (see (10)), it provides an intersection formqi(η) := (ηi, ηi)S_i onH²(X,Q)tr, which satisfies the first and weak second Hodge-Riemann relations.

Let now N, M be the respective degrees of the maps ϕ, ψ. We deduce from (18) the following equality:

N

∫

X

η²ⁿ=M

∫

S₁×...×S_n

∑n

i=1

(pr^∗_iη_i)²ⁿ=M(2n)!

2ⁿn!(η₁, η₁)_S1. . .(η_n, η_n)_Sn. (19) Let q₁(η) := (η₁, η₁)_S1, . . . , q_n(η) := (η_n, η_n)_Sn, where the η_i’s are defined by (18). Then (19) gives (16) up to a multiplicative coefficient, which proves (i).

We next assume property (*) which implies that (ηi, ηi)S_i =µiq(η) for some rational numbersµi, sinceqi satisfies the first Hodge-Riemann relations. Equation (19) then gives:

N

∫

X

η²ⁿ =M(2n)!

2ⁿn!µ1. . . µnq(η)ⁿ, proving (ii).

Proposition 1.5 (i) now implies the following result, showing that having a surface de- composition is a restrictive condition:

Theorem 1.6. LetS₁, S₂, S₃be three smooth projective surfaces withh^1,0(S_i) = 0, h^2,0(S_i)̸= 0for all i, and letH = pr^∗₁H₁+ pr^∗₂H₂+ pr^∗₃H₃∈Pic (S₁×S₂×S₃)be a very ample divisor onS₁×S₂×S₃. Let Y ⊂S₁×S₂×S₃ be the smooth complete intersection of two general members of|H|. Then Y is not surface decomposable.

Proof. Ash^1,0(S_i) = 0, we have

H²(S1×S2×S3,Q) =H²(S1,Q)⊕H²(S2,Q)⊕H²(S3,Q)

and similarly for transcendental cohomology. By the Lefschetz hyperplane section theorem, we get, as dimY = 4:

H²(Y,Q)tr=H²(S1,Q)tr⊕H²(S2,Q)tr⊕H²(S3,Q)tr,

We now compute∫

Y α⁴ for α∈ H²(Y,Q)tr: For α=α1+α2+α3, using∫

S_iαi∪hi = 0, whereh_i:=c₁(H_i), we get

∫

Y

α⁴=

∫

S1×S2×S3

(pr^∗₁α1+ pr^∗₂α2+ pr^∗₃α3)⁴(pr^∗₁h1+ pr^∗₂h2+ pr^∗₃h3)²

=λ₁q₂(α₂)q₃(α₃) +λ₂q₁(α₁)q₃(α₃) +λ₃q₁(α₁)q₂(α₂), (20) whereqi(αi) :=∫

S_iα²_i, and the constantsλiare nonzero rational numbers. It is immediate to see that (20) is not of the form (16), namely the product of two quadrics inα=α1+α2+α3. Indeed, the hypersurface inP(H²(Y,C)tr) defined by (20) is irreducible, being fibered with irreducible fibers over the smooth conic inP²_Cwith equationλ1y2y3+λ2y1y3+λ3y1y2= 0, via the rational map

P(H²(Y,C)tr)99KP²_C,

α=α1+α2+α37→(q1(α1), q2(α2), q3(α3)).

1.2 Application to Beauville’s weak splitting conjecture

In the paper [7], it was observed that a projective K3 surface has the following property:

there is a canonical 0-cycleo_S ∈CH₀(S) of degree 1 (in fact, it can be defined as ^c2₂₄^(S)) such that for any divisorD∈PicS = CH¹(S) = NS(S), one has

D²=q(D)o_S in CH₀(S), (21) whereq(D) = ([D],[D])S. One can rephrase this result by saying :

(*)Any cohomological polynomial relation

Q([D1], . . . ,[Dk]) = 0 inH^∗(S,Q) involving only divisor classes is already satisfied inCH(S)_Q.

In [6], Beauville made the following conjecture, generalizing the result above:

Conjecture 1.7. Let X be a projective hyper-K¨ahler manifold. Then the cycle class map is injective on the subalgebra ofCH^∗(X)_Q generated by divisor classes.

This conjecture is called Beauville’s weak splitting conjecture. It is equivalent to the fact that property (*), that we will call the weak splitting property, is satisfied for any divisor classes on X. Let us discuss Conjecture 1.7 in relation with the notion of surface decomposition. Let X be a projective hyper-K¨ahler manifold, and let Λ ⊂ NS(X) be a lattice polarization (which means that Λ contains an ample class). The very general Λ- polarized deformation Xt ofX is the very general fiber of a family X →B parameterized by a quasiprojective base B and it satisfies H²(Xt,Q)tr =H²(Xt,Q)^⊥Λ. We assume that the general (or very general) Λ-polarized deformationXtofX has a surface decomposition.

Then, by standard spreading arguments involving relative Chow varieties, after passing to a generically finite coverB^′ of B, we have projective morphisms Γ → B^′, Si →B^′, with dimSi/B^′= 2, and morphisms overB^′

ϕ: Γ→ X, ψ: Γ→ S1×B^′. . .×B^′Sn

inducing a surface decomposition at the general point t ∈ B^′. After shrinking B^′, by desingularization of the general fiber, one can assume that the fibers ΓtandSi,t are smooth and we get by specialization a diagram

ϕt: Γt→Xt, ψt: Γt→S1,t×. . .×Sn,t (22)

such that

ϕ^∗_tσX_t =ψ_t^∗(∑

i

pr^∗_iσS_i,t) inH^2,0(Γt) (23) for some (2,0)-formsσ_Si,tonS_i,t. We proved in Lemma 1.1 that the relation (23) then holds in fact for any classα∈H²(X_t,Q)_tr=H²(X_t,Q)^⊥Λ and that there is for eachia (locally constant) morphism of Hodge structures

ψ_i,t:H²(X_t,Q)^⊥Λ→H²(S_i,t,Q) given by (10) such that

ϕ^∗_tα=ψ_t^∗(∑

i

pr^∗_i(ψi,t(α))) inH²(Γt,Q). (24) Let us now assume furthermore thatH¹(Γt,Z) = 0, or equivalently

NS (Γt) = Pic (Γt). (25)

Note that, since Γ_tdominates eachS_i,t, this implies the same equality for eachS_i,t. In the situation described above, we have the following result.

Theorem 1.8. For anyt∈B^′, the weak splitting property holds for divisor classes onX_t which are inH²(Xt,Q)^⊥Λ if and only if, for each surfaceSi,t, the Beauville-Voisin relation (21) holds onImψi,t for an adequate0-cycle oS_i,t ∈CH0(Si,t).

Proof. Using (25), we conclude that (24) holds in Pic (Γt)_Qforα∈Pic (Xt)^⊥Λ= NS (Xt)^⊥Λ (where the point t is now special in B^′, being in a Noether-Lefschetz locus), and more precisely, that the morphism of Hodge structuresψi,t induces for anyt∈B^′ aQ-linear map

ψi,t : Pic (Xt)^⊥_Q^Λ→Pic (Si,t)_Q such that, for anyD∈Pic (X_t)^⊥_Q^Λ:

ϕ^∗_tD=ψ^∗_t(∑

i

pr^∗_i(ψ_i,t(D))) in Pic(Γ_t)_Q = CH¹(Γ_t)_Q. (26) As in the cohomological setting which has been studied in the previous section, the important point here is the fact that the pull-back maps appearing on both sides are compatible with intersection product. Note also that they are injective since the maps ϕt andψt are dominant. For any point t ∈ B, let D1, . . . , Dk ∈ CH¹(X)_Q and let Q be a degree l homogeneous polynomial withQ-coefficients inkvariables. Then we get from (26):

ϕ^∗_tQ(D1, . . . , Dk) =ψ^∗_t(Q(D^′₁, . . . , D^′_k)) in CH^l(Γt)_Q, (27) whereD_j^′ :=∑

ipr^∗_i(ψi,t(Dj)). Assume thatXtsatisfies the weak splitting property, at least for divisor classesD ∈CH¹(X_t)^⊥Λ. There is then a 0-cycleo_X∈CH₀(X) of degree 1 such that

D²ⁿ= (degDⁿ)o_X in CH₀(X) (28)

for anyD∈CH¹(Xt)^⊥Λ. Pulling-back to Γtand using (27), we have now ϕ^∗_t(D²ⁿ) =(2n)!

2ⁿn!ψ^∗_t(

∏n

j=1

pr^∗_j(ψj,t(D)²)) in CH0(Γt). (29)

Note that any D ∈ CH¹(Xt)^⊥Λ satisfies q(D) ̸= 0 by the Hodge index theorem, where q is the Beauville-Bogomolov quadratic form onH²(Xt,Q), which can also be defined as the Lefschetz intersection pairing on H²(Xt,Q)^⊥Λ (see [4]). As we have degD²ⁿ =λq([D])ⁿ withλ̸= 0 by (15), we conclude that degD²ⁿ̸= 0. Let

oS_j,t := pr_j∗( 1

degϕ_t(ψt∗(ϕ^∗_toX))) ∈ CH0(Sj,t)_Q.

This cycle has degree 1 and we get from (29) by pushing-forward toS_j,t via pr_j◦ψ that ψj,t(D)²is proportional tooS_j,t. Indeed, (pr_j◦ψ)_∗(ϕ^∗_t(D²ⁿ)) is a 0-cycle of degree different from 0 on Sj,t, which by (29) is proportional to both oS_j,t and ψj,t(D)². This proves the

“only if” direction.

Conversely, assume each surfaceSi,thas a 0-cycleoS_i,tof degree 1 with the property that divisorsDi in Imψi,t ⊂NS(Si,t)_Q = Pic (Si,t)_Q satisfy D²_i = (Di, Di)S_i,toS_i,t in CH0(Si,t) or equivalently that for anyDi, D_i^′ ∈Imψi,t

Di·D_i^′= (Di, D_i^′)S_i,toS_i,t in CH0(Si,t). (30) We now use the fact that, at the very general point ofB^′, the Mumford-Tate group of the Hodge structure on H²(X_t,Q)^⊥Λ is the orthogonal group, and thus the intersection form ψ^∗_i,t((,)_Si,t) equalsµ_iq onH²(X_t,Q)^⊥Λ, for some coefficientµ_i. It then follows from (30) that a numerical relationq(D) = 0 forD∈ Pic (X_t)_Cproduces relations

D²_i = 0 in CH0(Si,t)_C, (31)

for anyi= 1, . . . , n, whereDi:=ψi,t(D).

By [8] (or rather, the same arguments as in [8] using the fact that the Beauville- Bogomolov pairingqremains nondegenerate on NS(Xt)_Q), we know that the relations in the subalgebra ofH^∗(Xt,C) generated by NS (Xt)_C are generated by the Bogomolov-Verbitsky relations

dⁿ⁺¹= 0 if q(d) = 0. (32)

This is true as well (for the same reasons) if we restrict to the subalgebra generated by NS (X_t)^⊥_C^Λ= Pic (X_t)^⊥_C^Λ. Next, (27) provides for anyD∈Pic (X_t)^⊥_C^Λ

ϕ^∗_t(Dⁿ⁺¹) =ψ_t^∗((

∑n

i=1

pr^∗_iDi)ⁿ⁺¹) =∑

i

pr^∗₁D1·. . .·pr^∗_iD_i²·pr^∗_nDn (33) +. . . in CH(Γ_t)_C,

where the remaining term “. . .” involves products pr^∗_iD²_i ·pr^∗_jD²_j of two squares, then three squares pr^∗_iD²_i·pr^∗_jD²_j·pr^∗_kD²_ketc... Using (31) and (33), we getϕ^∗_t(Dⁿ⁺¹) = 0 in CHⁿ⁺¹(Γ_t)_C, hence Dⁿ⁺¹ = 0 in CHⁿ⁺¹(Xt)_C, whenever q(D) = 0. In other words, the Bogomolov- Verbitsky relations (32) are satisfied in CHⁿ⁺¹(Xt)_C, which concludes the proof.

We get the following corollary:

Corollary 1.9. (Cf. Theorem 0.4) Under the same assumptions as in Theorem 1.8, the weak splitting property holds for divisor classes onXtwhich are inH²(Xt,Q)^⊥Λ if and only if they hold in top degree, that is,

(*) there exists a canonical 0-cycle oX_t ∈ CH0(Xt) such that for any D ∈NS(Xt)^⊥Λ, D²ⁿ is proportional tooX_t inCH0(Xt).

Proof. The “only if” is clear. In the other direction, examining the proof of Theorem 1.8, we observe that we only used relations (29) in top degree 2n to conclude that, if (*) holds, definingo_Si,t ∈CH₀(S_i,t) := _deg¹_ϕ

tpr_i∗(ψ_t∗(ϕ^∗_to_Xt)]∈CH₀(S_i,t), the zero-cycle D²_i,t is proportional tooS_i,t in CH0(Si,t), for anyDt∈NS(Xt)^⊥Λ, whereDi,t:=ψi,t(Dt). Hence by Theorem 1.8, (*) implies the weak splitting property for NS(Xt)^⊥Λ.

1.3 Decomposition from families of algebraically coisotropic divi- sors

We study in this section a weaker notion of decomposition for a holomorphic 2-form into forms of smaller rank (see Question 0.1). The following is a weak converse to Proposition 0.11.

Proposition 1.10. Let X be smooth projective variety of dimension 2n equiped with a generically nondegenerate holomorphic 2-form σX. Assume X is swept-out by (possibly singular) algebraically coisotropic divisors. Then there exists a generically finite cover Φ : D^′→X such that

Φ^∗σX=η1+η2in H^2,0(D^′), (34) whererankη₁= 2, andrankη₂= 2n−2. More precisely,η₂is the pull-back of a holomorphic 2-form on a variety of dimension≤2n−1.

Here by the rank we mean the generic rank of the considered forms.

Proof of Proposition 1.10. By assumption, there exists a 1-parameter family D →C, D →X

of divisorsDt⊂X whose characteristic foliation (on the regular locus ofDt) is algebraically integrable, that is, there exists a rational map

ϕ_t:D_t99KB_t

with dimBt = 2n−2, such that the equality σ_X|Dt = ϕ^∗_tσB_t, for some holomorphic 2- form σB_t on the regular locus of Bt, holds on the regular locus of Dt. Note that, by desingularisation, we can assume Dt and Bt smooth, at least for general t. Indeed, the 2-formσB_t extends holomorphically on any smooth projective model Bet of Bt, because it can be constructed as

ϕ˜t∗(˜j^∗_tσX∧ω)

where ˜jt:Det→X is a smooth model ofDtsuch that ˜ϕt:Det→Betis a morphism, andω is a closed (1,1)-form on Detwhose integral over the fibers of ˜ϕtis 1.

As usual, the data above (namely the family of varieties Bt and morphismsϕt) can be put in a family, possibly after base change from the original familyD →Cof divisors onX and birational transformations. We thus get the following diagram

D^{′ J} //

Φ

X

B

(35)

where all the varieties are smooth and projective, the morphismJ is surjective generically finite, dimB = 2n−1 and B admits a morphism f : B → C such that, considering the induced diagram of fibers over a general pointt∈C

D^′t Jt //

Φt

X

Bt

, (36)

one has

J_t^∗σX= Φ^∗_tσB_t inH^2,0(Bt). (37)