Lagrangian constant cycle subvarieties in Lagrangian fibrations
aa r X i v : . [ m a t h . AG ] D ec LAGRANGIAN CONSTANT CYCLE SUBVARIETIES IN LAGRANGIAN FIBRATIONS by Hsueh-Yung Lin
Abstract . —
We show that the image of a dominant meromorphic map from an irreducible compact Calabi-Yaumanifold X whose general fiber is of dimension strictly between 0 and dim X is rationally connected. Using thisresult, we construct for any hyper-Kähler manifold X admitting a Lagrangian fibration a Lagrangian constant cyclesubvariety Σ H in X which depends on a divisor class H whose restriction to some smooth Lagrangian fiber is ample.If dim X =
4, we also show that up to a scalar multiple, the class of a zero-cycle supported on Σ H in CH ( X ) does notdepend neither on H nor on the Lagrangian fibration (provided b ( X ) ≥ This note is devoted to the construction of some subvarieties Y in a projective hyper-Kähler manifold X admitting a Lagrangian fibration such that all points in Y are rationally equivalent in X . These subvarieties,called constant cycle subvarieties in [ ], depend on the choice of a Lagrangian fibration and a divisorclass H ∈ Pic( X ) whose restriction to some smooth Lagrangian fiber is ample, thus the image of the Gysinmap CH ( Y ) → CH ( X ) depends a priori on these choices as well. The result of this note is motivated bythe conjectural picture on the splitting property of the conjectural Bloch-Beilinson filtration of projectivehyper-Kähler manifolds due to Beauville [ ] and Voisin [ ] as we will explain below.The study of constant cycle subvarieties in hyper-Kähler manifolds was initiated by Huybrechts in thecase of K3 surfaces [ ]. Our motivation for studying these subvarieties comes rather from the attempt togeneralize the Beauville-Voisin canonical zero-cycle of a projective K3 surface [ ] to higher dimensionalcases. For a K3 surface S , recall that there are at least two ways to characterize the canonical zero-cycle o S :i) o S is the degree-one generator of the image of the intersection product [ ] ⌣ : CH ( S ) ⊗ CH ( S ) → CH ( S );ii) o S is the class of a point supported on a constant cycle curve in S [ , Lemma 2.2].Each characterization gives a priori di ff erent generalization of o S . The first one is related to Beauville’sconjecture on the weak splitting property of the Chow ring of projective hyper-Kähler manifolds [ ]: Conjecture 1.1 (Beauville [ ] ) . — Let X be a projective hyper-Kähler manifold. The restriction of the cycle classmap CH • ( X ) Q : = CH • ( X ) ⊗ Z Q → H • ( X , Q ) to the Q -sub-algebra generated by divisor classes is injective. The reader is referred to [
1, 20, 9, 14, 22 ] for recent developments of this conjecture. In particular, since H n ( X , Q ) = Q , Beauville’s conjecture contains as a sub-conjecture the statement that the intersection of any AGRANGIAN CONSTANT CYCLE SUBVARIETIES IN LAGRANGIAN FIBRATIONS n divisor classes in CH • ( X ) Q is proportional to the same degree one zero-cycle o X ∈ CH n ( X ) Q where 2 n isthe dimension of X , which generalizes property i ) of o S .The generalization of property ii ) is formulated in [ , Conjectures 0.4 and 0.8]: for 0 ≤ i ≤ n , let S i CH ( X )denote the subgroup of CH ( X ) generated by the classes of points whose rational orbit is of dimension ≥ i (1) . One hopes that this decreasing filtration S • CH ( X ) would define a splitting of the conjecturalBloch-Beilinson filtration F • BB in the sense that the inclusion S i CH ( X ) ֒ → CH ( X ) induces an isomorphism S i CH ( X ) ∼ −→ CH ( X ) / F n − i + BB CH ( X ) . Using the axioms of the Bloch-Beilinson conjecture, the surjectivity of the above map is proven in [ ]to be predicted by the following conjecture: Conjecture 1.2 ( [ ] ) . — Let X be a projective hyper-Kähler manifold of dimension n. For any ≤ i ≤ n, thedimension of the set of points S i X ⊂ X whose rational orbit has dimension ≥ i is n − i. We refer to [ ] for more details on Voisin’s circle of ideas for studying the splitting property of theBloch-Beilinson filtration on CH ( X ). Note that by [ , Theorem 1.3], we always have S i X ≤ n − i . So when i = n , Conjecture 1.2 is equivalent to the existence of constant cycle subvarieties of X of dimension n (whichare necessarily Lagrangian, by Roitman-Mumford’s theorem [ , Proposition 10.24]) and one would expectto recover the conjectural canonical zero-cycle for any projective hyper-Kähler manifold X by taking theclass of a point in any of these constant cycle Lagrangian subvarieties, hence the second generalization of o S .In general it is di ffi cult to construct Lagrangian constant cycle subvarieties. However, if X admits aLagrangian fibration π : X → B ( i.e. a surjective holomorphic map to a projective variety B whose generalfiber is a connected Lagrangian submanifold), we prove in Section 3 the following theorem, which provesin particular Conjecture 1.2 in the case i = n for every Lagrangian fibration. Theorem 1.3 . —
Let X be a projective hyper-Kähler manifold admitting a Lagrangian fibration π : X → B. Foreach divisor class H ∈ Pic( X ) whose restriction to some smooth Lagrangian fiber is ample, there exists a Lagrangianconstant cycle subvariety Σ π, H ⊂ X dominating B and all of whose points are rationally equivalent to H n · [ F ] · H n · [ F ] in CH ( X ) (2) where [ F ] the class of a fiber of F of π . The fact that [ F ] ∈ CH n ( X ) is independent of F is a direct consequence of the following general result. Theorem 1.4 . —
Let X be a Calabi-Yau manifold and f : X d B a dominant meromorphic map over a Kähler baseB. If < dim B < dim X, then B is rationally connected.
Here a
Calabi-Yau manifold is an irreducible (in the sense of Riemannian geometry) compact Kählermanifold with finite fundamental group and trivial canonical bundle. The Riemannian holonomy groupof a Calabi-Yau manifold associated to its Kähler metric is either SU( n ) or Sp( n ). Hyper-Kähler manifoldsand Calabi-Yau manifolds in the strict sense are examples of Calabi-Yau manifolds. Strictly speaking, wewill only apply Theorem 1.4 to projective Calabi-Yau manifolds and in this case, Theorem 1.4 is alreadyknown to be true by [ , Theorem 14]. However our proof is di ff erent from the proof of [ , Theorem 14]and works also for non-projective Calabi-Yau manifolds. (1) The rational orbit of a point z ∈ X is the set O z ⊂ X of points in X which are rationally equivalent to z . The set O z is a countable unionof Zariski closed subsets of X and we define dim O z to be the maximum of the dimension of all irreducible components of O z . (2) Since h , ( X ) =
0, by Roitman’s theorem [ ] CH ( X ) is torsion free. Thus H n · [ F ] · H n · [ F ] is well-defined. AGRANGIAN CONSTANT CYCLE SUBVARIETIES IN LAGRANGIAN FIBRATIONS Theorem 1.4 will allow to rephrase Theorem 1.3 replacing the fiber F by the cycle L n ∈ CH n ( X ). In the casewhere dim X =
4, under the mild assumption that a very general projective deformation of the Lagrangianfibration π : X → B with ρ ( X ) ≥ (3) (for instance when b ( X ) ≥ ]),Theorem 1.3 allows us to define for such a variety X a canonical zero-cycle o X ∈ CH ( X ) by taking the classof a point supported on any Lagrangian constant cycle subvariety Σ π, H defined above, whose class is alsoproportional to the product of any 4 divisors: Theorem 1.5 . — i) Let f : X → B be a projective hyper-Kähler manifold admitting a Lagrangian fibration of dimension n.Let L : = f ∗ c ( O B (1)) where O B (1) is an ample line bundle on B and let D be a divisor in X. Assume that [ D ] | F , ∈ H ( F , Q ) for some fiber F of π or dim X = , then the zero-cycle L n · D n is proportional to the classof a point x ∈ X which belongs to a Lagrangian constant cycle subvariety constructed in Theorem 1.3.ii) Assume that dim X = . If a very general projective deformation of X preserving the Lagrangian fibration with ρ ( X ) ≥ satisfies Matsushita’s conjecture (in particular if b ( X ) ≥ ] ), then the class of a point in Σ π, H modulo rational equivalence is independent of the divisor class H.iii) Under the same hypothesis as in ii ) , the class of a point in Σ π, H modulo rational equivalence is independent ofthe Lagrangian fibration. We will prove Theorem 1.4 in this section.
Proof of Theorem 1.4 . — Up to a bimeromorphic modification, we suppose that B is smooth. If B has anon-trivial holomorphic 2-form α , then 2 ≤ dim B < dim X and f ∗ α , B is projective.By Graber-Harris-Starr’s theorem [ ], it su ffi ces to show that if B satisfies the condition in Theorem 1.4,then B is uniruled. Indeed, suppose that B is not rationally connected and let B d B ′ be the MRC-fibrationof B , then the composition X d B d B ′ is dominant with 0 < dim B ′ < dim X . So B ′ would be uniruled,contradicting [ , Corollary1 . B is not uniruled. By [ ], the canonical class c ( K B ) is pseudo-e ff ective. This meansthat the class c ( K B ) ∈ H ( B , R ) is a limit of e ff ective divisor classes. Let X p ←− ˜ X q −→ B be a resolution of f : X d B with ˜ X smooth. As p ∗ q ∗ maps e ff ective divisor classes to e ff ective divisor classes, the class p ∗ q ∗ c ( K B ) is also pseudo-e ff ective.Since q : ˜ X → B is surjective, the induced map q ∗ K B → Ω k ˜ X is non-zero where k = dim B . As X is smooth,this map determines a non-zero morphism L → Ω kX where L is a line bundle such that c ( L ) = p ∗ q ∗ c ( K B ).Let ω be a Kähler form in X . Since the holonomy group Hol( X ) of X with respect to the Kähler metric iseither SU( n ) or Sp( n /
2) where n = dim X , there exists a Kähler-Einstein metric on T X whose correspondingKähler form is cohomologous to ω [ ], which further implies that Ω kX is ω -polystable by the Donaldson-Uhlenbeck-Yau theorem [
8, 16 ]. Precisely, Ω kX = E ⊕ · · · ⊕ E m where E i is ω -stable of slope µ ω ( E ) = X )-module Ω kX | x intoirreducible representations over any x ∈ X described as follows ( cf. [ , §13, n o , ChapterVI.3]). (3) We say that a Lagrangian fibration π : X → B satisfies Matsushita’s conjecture if either π : X → B is isotrivial or the induced modulimap B d A n to some suitable moduli space of abelian varieties is generically injective. AGRANGIAN CONSTANT CYCLE SUBVARIETIES IN LAGRANGIAN FIBRATIONS Lemma 2.1 . — i) If
Hol( X ) = SU( n ) , then the Hol( X ) -module Ω kX | x is irreducible.ii) If Hol( X ) = Sp( n / , then Ω kX | x = M k ≥ k − r ≥ η r | x ∧ P k − r , where η is a holomorphic symplectic 2-form on X and, P k − r is an irreducible Sp( n / -submodule of Ω k − rX | x .Moreover, dim C P k − r = nk − r ! − nk − r − ! . By the above lemma, if Hol( X ) = Sp( n /
2) and k is odd or Hol( X ) = SU( n ), then dim E i > = rank( L ) for all i (since 0 < k < n ). As c ( L ) is pseudo-e ff ective (so µ ω ( L ) ≥ = µ ω ( E )) and E i is stable, there is no non trivialmorphism from L to E i for all i , contradicting the non-vanishing of L → Ω kX . Finally if Hol( X ) = Sp( n / k is even, then m ≥ i such that rank( E i ) =
1. Moreover, E i ≃ O X and E i ֒ → Ω kX is given by the multiplication by η k / . We deduce that if U is a Zariski open subset of X restrictedto which f is well-defined, then locally the pullback under f | U of a non-zero holomorphic k -form α on f ( U )is proportional to η k / , which contradicts the fact that η is non-degenerate. (cid:3) As an immediate consequence,
Corollary 2.2 . —
The class of a fiber π − ( t ) of a Lagrangian fibration π : X → B modulo rational equivalence isindependent of t ∈ B. In particular, there exists µ ∈ Z \{ } such that L n = µ [ F ] in CH n ( X ) where L : = π ∗ c ( O B (1)) for some ample divisor O B (1) over B and [ F ] is the class of any fiber of π . Let X be a variety. A Zariski locally closed subset Y of X is called constant cycle if all points in Y arerationally equivalent in X . Note that the property of being constant cycle for a subvariety is birational inthe following sense: Lemma 3.1 . —
A subvariety Y of X is constant cycle if and only if there exists a Zariski open subset U of Y suchthat all points in U are rationally equivalent in X.Proof . — This follows from the fact that every zero-cycle in Y is rationally equivalent to a zero-cyclesupported in U . (cid:3) Lemma 3.2 . —
Let Y ⊂ X be a connected Zariski locally closed subset. If the image of the Gysin map i ∗ :CH ( Y ) Q → CH ( X ) Q is generated by an element o Y in CH ( X ) Q , then Y is constant cycle. In this case, we say thatY is represented by the zero-cycle o Y . If h , ( X ) = (e.g. when X is a projective hyper-Kähler manifold), the sameconclusion holds without Y being connected.Proof . — It su ffi ces to show that if every point supported on Y is torsion in CH ( X ), then Y is a constantcycle subvariety. Let α : Y ֒ → X → Alb( X )be the composition of the inclusion map Y ֒ → X with the Albanese map X → Alb( X ). If the image of i ∗ : CH ( Y ) → CH ( X ) consists of torsion classes, then by Roitman’s theorem [ ] the map α factorizes AGRANGIAN CONSTANT CYCLE SUBVARIETIES IN LAGRANGIAN FIBRATIONS through CH ( X ) tors ≃ Alb( X ) tors ֒ → Alb( X ) via the cycle class map Y → CH ( X ) tors . If Y is connected ordim Alb( X ) = h , ( X ) =
0, then α is constant, hence Y → CH ( X ) tors ֒ → CH ( X ) is constant. (cid:3) Now we restrict ourselves to constant cycle subvarieties on projective hyper-Kähler manifolds. Let X be a projective hyper-Kähler manifold of dimension 2 n and let η be a holomorphic symplectic 2-form on X .The following result is a direct consequence of Mumford-Roitman’s theorem [ , Proposition 10 . Proposition 3.3 . —
If Y is a constant cycle subvariety of X, then Y is isotropic for η . In particular, dim Y ≤ n andif dim Y = n, then Y is a Lagrangian constant cycle subvariety. The rest of Section 3 is devoted to the proof of Theorem 1.3 and Theorem 1.5.
Proof of Theorem 1.3 . — First we prove the following
Lemma 3.4 . —
Let A be an abelian variety of dimension and H an ample divisor on A. Let D : = deg( H ) andchoose o ∈ A to be the origin of A with respect to which H is symmetric. Then for every x ∈ A the following holds:H = D [ x ] in CH ( A ) if and only if x is a D-torsion point. In particular, there exist exactly D points x ∈ A suchthat H = D [ x ] in CH ( A ) .Proof . — Since H is symmetric with respect to o , by Poincaré’s formula [ , Corollary 16 . . H = D [ o ] inCH ( A ). Let alb : A ∼ −→ Alb( A ) be the Albanese map of A with respect to o . Recall that alb factorizes throughthe Deligne cycle class map α : CH ( A ) hom → Alb( A ), where CH ( A ) hom denotes the subgroup of CH ( A )homologous to zero and the morphism A → CH ( A ) hom is given by x [ x ] − [ o ]. If H = D [ x ] in CH ( A ),then D · alb( x ) = α ( D [ x ] − D [ o ]) = α ( H − D [ o ]) = o . (3.1)in Alb( A ). Conversely if D · alb( x ) = o , then α ([ x ] − [ o ]) is D -torsion. Since the restriction of α to the torsionpart of CH ( A ) hom is an isomorphism onto the torsion part of A [
4, 15 ], we conclude that [ x ] − [ o ] is also D -torsion. Hence D [ x ] = D [ o ] = H in CH ( A ). (cid:3) Let U ⊂ B be a Zariski open subset of B parametrizing smooth fibers of π such that H | π − ( b ) is ample forany b ∈ U . Set X U : = π − ( U ). By a standard argument (see for example the proof of [ , Theorem 10 . H over U , parametrizing the data of a point t in U and a point x ∈ X t such that D [ x t ] = H | X t in CH ( X t ), is a countable union of irreducible subvarieties of X U .Let p : H → U be the natural projection. Since the X t ’s are abelian varieties, p is finite and dominant byLemma 3.4, so there exists an irreducible component Z of H such that p | Z is finite and dominant as well.By construction, viewing Z as a subvariety of X U , Z is Zariski locally closed in X of dimension n ; we define Σ π, H as the closure of Z in X , which is also of dimension n . Finally, for every x ∈ Z , let j : X t ֒ → X be theinclusion of the fiber of π containing x , then D [ x ] = ( j ∗ H ) n in CH ( X t ) thus D [ x ] = H n · [ F ] = c ( O B (1)) H n · π ∗ c ( O B (1)) n , (3.2)for some ample line bundle O B (1) over B where the last equality follows from Corollary 2.2. Hence Z is aZariski open subset U of Σ π, H whose points are rationally equivalent to a scalar multiple of H n · π ∗ c ( O B (1)) n .We conclude by Lemma 3.2 that U is constant cycle, and then by Lemma 3.1 that Σ π, H is a constant cyclesubvariety of X . (cid:3) AGRANGIAN CONSTANT CYCLE SUBVARIETIES IN LAGRANGIAN FIBRATIONS Before we start proving of Theorem 1.5, let us recall the following result of Matsushita and Voisin whichwill be useful later. Let π : X → B be a Lagrangian fibration and let L be the pullback of an ample divisorclass from the base. Let j : F ֒ → X be the inclusion map of a smooth Lagrangian fiber in X . Lemma 3.5 (Matsushita [ ] + Voisin [ ] ) . — If j ∗ : H ( X , Q ) → H ( F , Q ) denotes the restriction map and µ [ F ] : H ( X , Q ) → H n + ( X , Q ) the cup product with [ F ] , then ker µ [ F ] = ker j ∗ = ker q ( L , · ) where q : Sym H ( X , Q ) → Q is the Beauville-Bogomolov-Fujiki form associated to X. In particular, the image ofj ∗ : H ( X , Q ) → H ( F , Q ) is of rank one.Proof . — The first equality is exactly the statement of [ , Lemme 1 . ,Lemma 2.2], we have the inclusion ker q ( L , · ) ⊂ ker j ∗ . It follows that b ( X ) − = dim Q ker q ( L , · ) ≤ dim Q ker j ∗ ≤ b ( X ) − j ∗ (on any ample divisor class). Hence ker q ( L , · ) = ker j ∗ for dimensional reasons. (cid:3) Proof of Theorem 1.5 . — Let π : X → B be a Lagrangian fibration on a polarized hyper-Kähler manifold( X , H ) and let D be a divisor in X . Let F denote a fiber of π . If [ D ] | F = H ( F , Q ), then [ D ] · [ L ] n ∈ H n + ( X , Q )is proportional to j ∗ [ D ] =
0. Therefore if dim X =
4, then D n · L n = ( X ) by [ , Theorem 0 . D ] | F , H ( F , Q ), then since j ∗ : H ( X , Q ) → H ( F , Q ) is of rank one by Lemma 3.5, either D | F or − D | F isample on F . Suppose without loss of generality that D | F is ample on F , then by Theorem 1.3, there exists d ∈ Z \{ } such that D n · F = d [ x ] in CH ( X ) for any point x in the Lagrangian constant cycle subvariety Σ π, D .Since D n · L n is non-trivially proportional to D n · F in CH ( X ) by Corollary 2.2, this proves i ).To prove ii ), let H and H be two divisor classes such that H is ample and H satisfies the assumption ofTheorem 1.3. By Lemma 3.5, for a smooth fiber F b : = π − ( b ), there exist α, β ∈ Z \{ } such that ( α H − β H ) | F b iscohomologous to 0 on F b . It follows that ( α H − β H ) | F b is cohomologous to 0 on F b for all b ∈ U where U ⊂ B is the smooth locus of π . This implies by Lemma 3.5 that the product [ F b ] · ( α H − β H ), hence L · ( α H − β H ),is cohomologous to 0 on X . Since a very general deformation of X preserving the Lagrangian fibrationand H , H satisfies Matsushita’s conjecture, we can apply [ , Theorem 0.8] so that α H · L = β H · L inCH ( X ) ⊗ Q . It follows that α H · L − β H · L = ( α H − β H ) · ( α H + β H ) · L = ( X ) . Since the zero-cycles supported on the constant cycle subvarieties Σ π, H and Σ π, H constructed above areproportional to H · L and H · L in CH ( X ) respectively, the second statement of Theorem 1.3 follows.Now we prove iii ). Let π : X → B and π ′ : X → B ′ be two Lagrangian fibrations and let L : = π ∗ c ( O B (1))and L ′ : = π ∗ c ( O B ′ (1)). By Theorem 1.3, what we need to prove is reduced to the statement that H · L is proportional to H · L ′ in CH ( X ), so we can assume that L and L ′ are not proportional, otherwise theproof is finished. Since the restriction of q to NS( X ) Q is of signature (1 , − ρ ( X )), the restriction of q to thetwo-dimensional subspace generated by L and L ′ cannot be zero. Since q ( L , L ) = q ( L ′ , L ′ ) =
0, this impliesthat q ( L , L ′ ) ,
0. By Lemma 3.5, we have j ∗ L ′| F = L ′ · [ F ] , ∈ H ( X , Q ) where j : F ֒ → X is the inclusionof a smooth fiber of π , so L ′| F , ∈ H ( F , Q ). Again as j ∗ : H ( X , Q ) → H ( F , Q ) is of rank one, either L ′| F or( − L ′ ) | F is ample. Similarly either L | F ′ or ( − L ) | F ′ is ample where F ′ is a smooth fiber of π ′ . However as L and L ′ are pullbacks of ample classes in B and B ′ respectively, the classes ( − L ) | F ′ and ( − L ′ ) | F cannot be ample. AGRANGIAN CONSTANT CYCLE SUBVARIETIES IN LAGRANGIAN FIBRATIONS So necessarily, L ′| F and L | F ′ are ample. Now the second point of Theorem 1.5 implies that the class of a pointin Σ π, H equals the class of a point in Σ π, L ′ in CH ( X ), which is proportional to L · L ′ , Σ π ′ , H is also proportional to L ′ · L in CH ( X ), which finishes the proof of iii ). (cid:3) Acknowledgement
This note was written up as a part of the author’s PhD thesis at CMLS, École Polytechnique. I would liketo thank my thesis advisor C. Voisin for introducing me to this beautiful subject and for valuable discussions.I also thank O. Benoist, L. Fu, Ch. Lehn, and G. Pacienza for interesting remarks and questions, and alsothe referees for carefully reading the manuscript and suggestions.
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