arXiv:2109.03987v1 [math.AG] 9 Sep 2021

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THE DUAL LAGRANGIAN FIBRATION OF KNOWN HYPER-KAHLER

MANIFOLDS

YOON-JOO KIM

Abstract. Given a Lagrangian fibration π : X → Pn of a compact hyper-Kahler manifold ofK3[n], Kumn, OG10 or OG6-type, we construct a natural compactification of its dual torusfibration. Specifically, this compactification is given by a quotient of X by Aut(X/Pn), thegroup of automorphisms acting trivially on the second cohomology and respecting the Lagrangianfibration. It is a compact hyper-Kahler orbifold with identical period mapping behavior as X.

1. Introduction

Mirror symmetry is a highly active research area in modern mathematics. Among various ap-proaches to it, the Strominger–Yau–Zaslow (SYZ) conjecture predicts mirror symmetry can berealized by dualizing certain Lagrangian torus fibrations. More specifically, a compact Calabi–Yaumanifold X approaching a large complex structure limit point is expected to admit a special La-grangian torus fibration, and its mirror dual space X should be an appropriate compactification ofthe dual of this torus fibration. One case where this picture is better understood is for K3 surfaces;K3 surfaces are hyper-Kahler, and so admit a 2-sphere family of complex structures. This allowsone to hyper-Kahler rotate the complex structure and transform a special Lagrangian fibration intoa holomorphic elliptic fibration. Since elliptic curves are self-dual, one concludes that K3 surfacesare topologically self-mirror dual. This becomes the starting point of mirror symmetry for K3surfaces; the interested reader may consult [Gro99, §7], [GW00] or [Huy04] for more details.

A compact hyper-Kahler manifold is a higher dimensional generalization of a K3 surface. It isa simply connected compact Kahler manifold with a unique global holomorphic symplectic form.In [GTZ13, §2], mirror symmetry for K3 surfaces is generalized to certain compact hyper-Kahlermanifolds (see [Ver99] for a different approach). As for K3 surfaces, a hyper-Kahler rotation cantransform a special Lagrangian fibration into a holomorphic Lagrangian fibration π : X → B.Gross–Tosatti–Zhang studied such holomorphic Lagrangian fibrations π under the following twoconditions: (1) π admits at least one section, and (2) all the torus fibers of π are principally polar-ized. In this setting, they showed X is topologically self-dual and described its mirror symmetry atthe level of the period domain. In this paper, we are interested in the duality of Lagrangian fiberedhyper-Kahler manifolds without the two conditions above. Our main result is the construction ofa compactification of the dual torus fibration for all currently known examples of hyper-Kahlermanifolds. However, we do not attempt to describe their mirror symmetry further; we hope toreturn to this topic in the future.

Let X be a compact hyper-Kahler manifold of dimension 2n. By a Lagrangian fibration of Xwe mean a holomorphic surjective morphism π : X → B such that it has connected fibers and Bis a complex manifold with 0 < dimB < 2n. By [Hwa08] and [GL14], the base B is necessarilyisomorphic to the projective space Pn. It is well-known that any smooth fiber of π is a complex

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http://arxiv.org/abs/2109.03987v1

2 Y.-J. KIM

Lagrangian subtorus of X , so by restricting the Lagrangian fibration π to its smooth locus B0 ⊂ Bwe get a torus fibration π0 : X0 → B0, a smooth proper family of complex tori.

We want to construct the dual torus fibration π0, and ultimately its compactification π. Aconstruction of the dual torus fibration as the relative Picard scheme was proposed in [Saw04].While this definition behaves well when π0 admits a section, we would like to propose a slightlydifferent approach to handle the case when π0 has no section. Specifically, our starting point is thestandard fact that all the torus fibers of π0 are canonically polarized (e.g., Voisin’s argument in[Cam06, Prop 2.1]). That is, each torus fiber F admits a natural isogeny F → F to its dual torus.Denoting the kernel of this isogeny by G, we obtain an isomorphism F ∼= F/G. The idea is tomake this discussion global over the entire base B0. In Theorem 3.1, we will attach a canonicallypolarized abelian scheme P0 → B0 to π0 so that X0 becomes a P0-torsor (this combines the resultsof Arinkin–Fedorov and van Geemen–Voisin). Let K be the kernel of the canonical polarizationP0 → P0 over B0. The group scheme K then acts on both P0 and X0. After taking the associatedquotients, we recover on the one hand the dual abelian scheme P0

∼= P0/K, and on the other handobtain a new space

X0 = X0/K, π0 : X0 → B0.

By construction, X0 is a P0-torsor. Therefore, it is a smooth proper family of complex tori whichare fiberwise dual to the original fibration π0. This is our definition of the dual torus fibration π0.

It is important to notice that K is only a finite etale group scheme over B0. One can think of thisas the total space of a local system on B0; there is a monodromy issue hiding on the background,and a priori K may not be a constant group scheme. We are now ready to state the main result ofthis paper.

Theorem 1.1. Let π : X → B be a Lagrangian fibration of a compact hyper-Kahler manifold.

Assume X is of K3[n], Kumn, OG10 or OG6-type. Then

(1) The group scheme K is a constant group scheme with fiber

Aut(X/B) = f ∈ Aut(X) : π f = π, f∗ acts as the identity on H2(X,Z).

(2) The quotient map

X = X/Aut(X/B), π : X → B

compactifies the dual torus fibration π0.(3) X is a compact hyper-Kahler orbifold and π is a Lagrangian fibration. Moreover, X has

the same period mapping/deformation behavior as X.1

If X is of K3[n] or OG10-type, then in fact the group Aut(X/B) is trivial and thus these hyper-Kahler manifolds are self-dual, as studied in [GTZ13]. On the other hand, if X is of Kumn orOG6-type, then Aut(X/B) is nontrivial and X is not even homeomorphic to X . Still, one caninterpret the third item as a certain self-duality at the level of period domains. Note that X is aglobal quotient of X by a specific group Aut(X/B) acting trivially on H2(X,Z). As a result, thesecond rational cohomology of X andX are canonically isometric as Beauville–Bogomolov quadraticspaces. The higher cohomology of X may be strictly smaller than that of X by [Ogu20], but theyare still tightly connected via their Looijenga–Lunts–Verbitsky (LLV) structures (see [LL97], [Ver95]and [GKLR]). Finally, the singularities of X are quotient singularities of high codimension (≥ 4),

1(Note added in proof) While circulating the preliminary version of our manuscript, we have learned about thepaper [Saw20]. In Theorem 24 of the paper, an alternative construction for the dual Lagrangian fibration of certainKumn-type hyper-Kahler manifolds is announced (without details).

DUAL LAGRANGIAN FIBRATION OF KNOWN HK 3

so they do not admit any symplectic resolutions. Thus, X is truly a singular hyper-Kahler variety.For the reader’s convenience, we briefly recall the notion of a singular hyper-Kahler variety and ofa Lagrangian fibration for such a variety in Appendix A.

Remark 1.2. [MT07] and [Men14] introduced an explicit geometric construction of certain 4-dimensional Lagrangian fibered hyper-Kahler orbifolds, and realized their dual Lagrangian fibra-tions using the same construction. It would be interesting to find a connection between their resultsand our perspective.

There are two key ingredients for our proof of Theorem 1.1: the group Aut(X/B) and thenotion of a polarization type. The definition of the group Aut(X/B) is inspired by the similargroup Aut(X), which has already played an important role in the theory of hyper-Kahler manifolds.The two main properties of Aut(X) are its finiteness [Huy99] and deformation invariance [HT13].Furthermore, Aut(X) is computed for all known deformation types of hyper-Kahler manifolds (see[Bea83a], [BNWS11] and [MW17]). We prove similar results for the group Aut(X/B): it is finiteabelian (Proposition 3.29) and deformation invariant (Theorem 2.3). We also compute Aut(X/B)for all known deformation types in Theorem 5.1. The idea of considering the polarization type ofthe fibers of π0 has long been used, but only recently comprehensively studied in [Wie16, Wie18].We relate the polarization type to the study of our group scheme K in this work.

1.1. Structure of the paper. In Section 2, we prove the group Aut(X/B) is deformation in-variant on the Lagrangian fibration π. This is inspired by Hassett–Tschinkel’s proof of deformationinvariance of Aut(X) in [HT13, Thm 2.1]. In Section 3, we start by attaching an abelian schemeP0 to any Lagrangian fibration of a hyper-Kahler manifold: P0 is the identity component of therelative automorphism scheme of π. There exists a unique primitive polarization λ on P0 so thatwe can define its kernel group scheme K. We then reinterpret the automorphisms in Aut(X/B)as certain global sections of K. This section also discusses the notion of a polarization type. Inessence, the polarization type is the study of a single fiber of the group scheme K.

The goal of Section 5 is twofold. First, we compute the group Aut(X/B) for all currently knowndeformation types of hyper-Kahler manifolds. Second, we prove the equality K = Aut(X/B) inthe special case of a certain construction of Kumn-type hyper-Kahler manifolds. The material herewill be mostly concrete computations. Still, a systematic approach would be beneficial and reducethe amount of computation. We briefly discuss this approach in Section 4. In Section 6, we provethe main result of this article: there exists a natural compactification of the dual torus fibrationfor all currently known deformation types of hyper-Kahler manifolds. In Section 7, we give anillustration of the geometry and cohomology of X when X is of Kum2-type.

We provide two appendices. Appendix A contains various definitions of singular hyper-Kahlervarieties appearing in the literature. In Appendix B, we discuss certain special quotients of compacthyper-Kahler manifolds. The quotient X = X/Aut(X/B) is a special instance of this more generalset-up.

1.2. Notation and conventions. In this paper, a hyper-Kahler manifold X will be assumed com-pact, but not necessarily projective unless stated explicitly. When X further admits a Lagrangianfibration π : X → B, it is helpful to keep in mind that the following three conditions are equivalent:

(1) X is projective;(2) ρ(X) ≥ 2; and(3) π admits at least one rational multisection.

4 Y.-J. KIM

Indeed, the first two conditions are equivalent by a direct application of Huybrechts’s projectivenesscriterion in [Huy99, Huy03]. The implication (3) ⇒ (1) is shown in [Saw09, Lem 2], and the impli-cation (1) ⇒ (3) is a general scheme-theoretic fact that any smooth morphism between algebraicvarieties admits an etale local section.

Any Lagrangian fibration π : X → B in this paper will always have base B = Pn since we areassuming B is smooth and 0 < dimB < 2n (see [Hwa08] and [GL14]). It is conjectured that onlythe normalness of B would imply B ∼= Pn. Starting from Matsushita’s foundational observation,much progress has been made towards this conjecture as well. To the best of our knowledge,

the conjecture is known to hold when π admits a section [CMSB02, Thm 7.2], X is of K3[n] orKumn-type [Mar14] [Mat15], or when dimX ≤ 4 [Ou19] [HX20].

Assume X has dimension 2n. The Beauville–Bogomolov form and the Fujiki constant of X area unique primitive symmetric bilinear form q : H2(X,Z) ⊗H2(X,Z) → Z and a positive rationalnumber cX satisfying the Fujiki relation

∫

X

x2n = cX ·(2n)!

2n · n!· q(x)n for x ∈ H2(X,Z). (1.3)

The Fujiki constant is computed for all currently known deformation types of hyper-Kahler mani-

folds: (1) cX = 1 for K3[n] or OG10-type, and (2) cX = n+1 for Kumn or OG6-type (see [Bea83b]and [Rap07, Rap08]). In practice, we will mostly need a stronger version of the Fujiki relation,which follows from the polarization process

∫

X

x1 · · ·x2n = cX∑

σ

q(xσ(1), xσ(2)) · · · q(xσ(2n−1), xσ(2n)) for xi ∈ H2(X,Z).

Here σ ∈ S2n runs through all the 2n-permutations but up to 2n · n! ambiguities inducing thesame expression in the summation. The divisibility of x ∈ H2(X,Z) is defined to be the positiveinteger div(x) = gcdq(x, y) : y ∈ H2(X,Z). To study the full cohomology H∗(X,Q), we willuse the notion of the LLV algebra g and its action on the cohomology, which are introduced byLooijenga–Lunts [LL97] and Verbitsky [Ver95]. For the concrete computations we will follow therepresentation theoretic notation used in [GKLR, §2–3].

Throughout, group schemes will be used both in algebraic and analytic context. A group schemeis a morphism G→ S equipped with an identity section S → G, group law morphism G×S G→ Gand inverse G → G satisfying the usual axioms (either in the algebraic or analytic setting). Inparticular, by an abelian scheme P → S we mean an analytically proper connected commutativegroup scheme over S with complex torus fibers. Any abelian scheme P admits a dual abelianscheme P . A polarization of an abelian scheme P is a finite etale homomorphism λ : P → P overS such that for each fiber F , the restriction λ|F : F → F is of the form x 7→ [t∗xL ⊗ L−1] for anample line bundle L on F . Given a group scheme G → S, an analytic torsor under G (or analyticG-torsor) is a morphism Y → S equipped with a G-action, such that there exists an analytic

covering S =⊔

α Uα → S where the base change Y = Y ×S S and G = G×S S are G-equivariantly

isomorphic over S. In the algebraic setting, one can use a different topology, e.g., etale topology todefine an etale torsor. Our reference for the theory of abelian schemes is [MFK94], [BLR90] and[FC90]. For the notion of torsors, see [Mil80] or [BLR90].

1.3. Acknowledgment. I would like to thank my advisor Radu Laza for his invaluable commentsand suggestions. Without his guidance, I wouldn’t have completed this paper. I would also like tothank Mark Gross, Brendan Hassett, Klaus Hulek, Daniel Huybrechts and Qizheng Yin for somehelpful suggestions to improve the original manuscript of this paper. Special thanks go to Mohamed


El Alami, Olivier Martin and Tobias Shin for their careful proofreading and encouragement for thisproject. This work was partially supported by NSF Grant DMS-1802128 (PI Laza).

2. Deformation invariance of the H2-trivial automorphisms

Let X be a compact hyper-Kahler manifold. Consider the group of H2-trivial automorphisms

Aut(X) = ker(

Aut(X) → O(H2(X,Z), q), f 7→ (f∗)−1)

.

Here Aut(X) is the group of biholomorphic automorphisms of X . Huybrechts [Huy99, Prop 9.1]together with Hassett–Tschinkel [HT13, Thm 2.1] proved that Aut(X) is a finite group which isinvariant under deformations of X .

Now let us further assume X admits a Lagrangian fibration π : X → B. Then we can restrictour attention to H2-trivial automorphisms that respect the Lagrangian fibration

Aut(X/B) = Aut(X/B) ∩ Aut(X). (2.1)

Since Aut(X) is finite, so is Aut(X/B). In fact, we can further prove Aut(X/B) is abelian: thiswill be showed later in Proposition 3.29. Notice that Aut(X/B) not only depends on X but alsoon the Lagrangian fibration π : X → B. Hence, if X admits two different Lagrangian fibrationsthen they can have different Aut(X/B). In Section 5, we will compute Aut(X/B) for all currentlyknown deformation types of hyper-Kahler manifoldsX . In Section 3, we will reinterpret Aut(X/B)as a subgroup of the group of global sections of the “polarization scheme” K. These results willlead us to conjecture that the polarization scheme K coincides with the group Aut(X/B).

But before doing so, here we establish a more basic fact in this section; we prove Aut(X/B)is deformation invariant on π. To make this more precise, we first need to define the notion of afamily of Lagrangian fibered hyper-Kahler manifolds.

Definition 2.2. A family of Lagrangian fibered compact hyper-Kahler manifolds is a commutativediagram

X

B

S

π

p

q

with the following conditions.

(1) p : X → S is a smooth proper family of compact hyper-Kahler manifolds of dimension 2nover a complex analytic space S.

(2) q : B → S is the projectivization of a rank n+ 1 holomorphic vector bundle on S.(3) For all t ∈ S, the fiber π : Xt → Bt is a Lagrangian fibration.

Note that the second condition ensures B is projective over S and admits a relative ample linebundle OB/S(1). It is also possible to consider a weaker version of this definition which only assumesB → S to be a Pn-bundle. The obstruction for a Pn-bundle to be the projectivization of a vectorbundle lies in the analytic Brauer group H2(S,O∗

S). Thus, if H2(S,O∗

S) = 0 (for example, when Sis a complex open ball) then the second axiom is in fact equivalent to the weaker one. Notice thatthe pullback H = π∗OB/S(1) can be considered as a family of line bundles Ht on Xt. Conversely,starting from the line bundle H we can recover B ∼= PS(p∗H). Therefore, Definition 2.2 is in somesense equivalent to the family of pairs (X,H), where H = π∗OB(1).

As usual, two Lagrangian fibrations π : X → B and π′ : X ′ → B′ are deformation equivalent ifthere exists a family of Lagrangian fibered compact hyper-Kahler manifolds X/B/S over a connected

6 Y.-J. KIM

union of 1-dimensional open disks S, realizing them as two fibers at t, t′ ∈ S. Matsushita in [Mat16]proved such a deformation problem admits a local universal deformation.

We can now state the main theorem of this section.

Theorem 2.3. The group Aut(X/B) is invariant under deformations of π : X → B.

The rest of this section will be devoted to the proof of Theorem 2.3. The sketch of the proofis as follows. First, we descend the Aut(X)-action on X to B so that the Lagrangian fibrationπ : X → B becomes an equivariant morphism. This means we have a group homomorphismAut(X) → Aut(B) whose kernel is precisely Aut(X/B). Descending such an action is a nontrivialproblem (this is quite similar to the result of [Bri11]), so we need to overcome this issue using thenotion of G-linearizability of a line bundle. Next, we need to sheafify the discussions as we areinterested in the deformation behavior of them. The result will follow from formal properties of thekernel of a sheaf homomorphism.

2.1. G-linearizability of a line bundle. Before we get into the proof of Theorem 2.3, let usrecall the notion of G-linearizability of a line bundle on a complex manifold. For simplicity we onlyconsider finite group actions. Our references are [Bri18, §3], [Dol03, §7] and [MFK94], but we needto take some additional care since these references only consider the algebraic setting.

Let G be an arbitrary finite group and X be a complex manifold equipped with a holomorphicG-action. A G-linearized line bundle on X is a holomorphic line bundle L together with a collectionof isomorphisms Φg : g∗L → L for g ∈ G, satisfying the condition Φgg′ = Φg′ g′∗Φg for g, g′ ∈ G.A G-invariant line bundle on X is a holomorphic line bundle L such that g∗L ∼= L for all g ∈ G(without any condition). We denote by PicG(X ) and Pic(X )G the groups of G-linearized linebundles and G-invariant line bundles on X up to isomorphisms, respectively. The second group isprecisely the G-invariant subgroup of Pic(X ).

There is a forgetful homomorphism PicG(X ) → Pic(X )G, which is neither injective nor surjectivein general. To understand the obstruction to its surjectivity, one considers an exact sequence ofabelian groups ([Dol03, Rmk 7.2] or [Bri18, Prop 3.4.5])

PicG(X ) Pic(X )G H2(G,Γ) , Γ = H0(X ,O∗X ).

Both Dolgachev and Brion’s discussions are for algebraic varieties, but their proofs can be adaptedto our analytic setting as well. With this exact sequence in hand, we have:

Lemma 2.4. Every G-invariant line bundle H on X is G-linerizable up to a suitable tensor power.

Proof. It is a general fact in the theory of group cohomology of finite groups that all the higherdegree cohomologies H≥1(G,Γ) are |G|-torsion for any G-module Γ (e.g., [Ser79, Cor VIII.1]).Hence by the previous lemma, the |G|-th tensor H⊗|G| vanishes in H2(G,Γ) and hence comes from

PicG(X ).

For us, the importance of the G-linearizability of a line bundle comes from the induced G-actionon the higher direct images of a linearized line bundle. If L is a G-linearized line bundle on X andp : X → S is a G-invariant holomorphic map, then we have a contravariant G-action on all thehigher direct image sheaves

g∗ : Rkp∗L → Rkp∗L, (g g′)∗ = g′∗ g∗.

Now assume further L is globally generated over S and p∗L is a vector bundle on S. Then wehave a G-action on PS(p∗L) making the holomorphic map X → PS(p∗L) G-equivariant over S. See[MFK94, Prop 1.7].


2.2. The automorphism sheaves and deformation invariance of the H2-trivial automor-

phisms. Suppose we have a smooth proper family of hyper-Kahler manifolds p : X → S. LetU ⊂ S be an analytic open subset and denote by p : XU = p−1(U) → U the restricted family overU . We define the sheaf of H2-trivial automorphism groups AutX/S on S by

AutX/S(U) = f : XU → XU : U -automorphism such that f∗ : R2p∗Z → R2p∗Z is the identity.

By the work of Huybrechts and Hassett–Tschinkel, this sheaf is a local system of finite groups. Wecan consider it as a family of groups Aut(Xt) for t ∈ S. Similarly, given a family of Lagrangianfibered hyper-Kahler manifolds, we can define a family of groups Aut(Xt/Bt):

Definition 2.5. Given a family of Lagrangian fibered hyper-Kahler manifolds p : Xπ−→ B

q−→ S, we

define a sheaf of groups AutX/B/S on S by

AutX/B/S(U) = f : XU → XU : BU -automorphism such that f∗ : R2p∗Z → R2p∗Z is the identity.

Equivalently, we may define AutX/B/S = q∗AutX/B ∩AutX/S.

As mentioned, the sheaf of H2-trivial automorphisms AutX/S is a local system. The sheaf

AutX/B/S is a subsheaf of AutX/S, and our goal is to prove it is locally constant as well. The

question is certainly local on the base S, so we may assume S is a small open ball. Then AutX/Sbecomes a constant sheaf, so we may consider it as an abstract finite group

G = Aut(X)

acting on X → S fiberwise.Consider the automorphism sheaf AutB/S of the Pn-bundle B → S. It is the sheaf of analytic

local sections of the PGL(n+1,C)-group scheme AutB/S → S. Our first step is to realize the sheafAutX/B/S as the kernel of a certain homomorphism AutX/S → AutB/S .

Proposition 2.6. Assume S is an open ball. Then there exists a homomorphism of sheaves

G = AutX/S → AutB/S (2.7)

whose kernel is AutX/B/S.

Equivalently, the proposition states that there exists a G-action on B making π : X → B aG-equivariant morphism over S. To prove the proposition, we need to use the G-linearizabilityof line bundles in the previous subsection. The following lemma proves every line bundle on X isG-invariant.

Lemma 2.8. G acts trivially on Pic(X ).

Proof. We first claim G acts trivially on H2(X ,Z). Apply the Leray spectral sequence

Ep,q2 = Hp(S,Rqp∗Z) ⇒ Hp+q(X ,Z).

Noticing that R0p∗Z = Z, R1p∗Z = 0, and S is an open ball, we obtain an isomorphism H2(X ,Z) ∼=H0(S,R2p∗Z). This isomorphism respects the G-action as the Leray spectral sequence is functorial.Now G acts on H2(Xt,Z) trivially for any fiber Xt, so G acts on R2p∗Z trivially and the claimfollows.

It is enough to prove the first Chern class map Pic(X ) → H2(X ,Z) is injective. This homomor-phism is induced by the exponential sequence 0 → Z → OX → O∗

X → 0, so it suffices to proveH1(X ,OX ) = 0. Again, use the Leray spectral sequence

Ep,q2 = Hp(S,Rqp∗OX ) ⇒ Hp+q(X ,OX ).

8 Y.-J. KIM

This time, we have R0p∗OX = OS and R1p∗OX = 0. This implies H1(X ,OX ) = 0.

Consider the line bundle H = π∗OB/S(1) on X . Since Pic(X ) is G-invariant, we can applyLemma 2.4 to H and conclude H⊗m is G-linearizable for some positive integer m. As a result, wehave a G-equivariant morphism πm : X → Bm where Bm = PS

(

p∗H⊗m)

is the dual of the complete

linear system associated to H⊗m. Consider the diagram

X

B Bm

S

p

π

πm

qqm

. (2.9)

Lemma 2.10. The m-th relative Veronese embedding B → Bm makes the diagram (2.9) commute.

Proof. Notice that the morphism π : X → B associated to H has connected fibers. Hence, forany t ∈ S, the fiber π : Xt → Bt becomes the Iitaka fibration of the line bundle Ht (e.g., [Laz04,§2.1.B]). This in particular implies that any morphism πm : Xt → (Bm)t associated to H⊗m

t factorsthrough the Iitaka fibration π. The factorization is precisely the Stein factorization of πm andBt → (Bm)t is precisely the m-th Veronese embedding. In other words, the m-th relative Veroneseembedding makes the diagram commute.

Now the equivariance of πm with the diagram (2.9) implies π is equivariant. This completesthe proof of Proposition 2.6. As mentioned, the result is quite similar to [Bri11] (both of themuse the connectedness of fibers) but we do not know the precise relation between them. WithProposition 2.6 at hand, the proof of the main theorem is immediate.

Proof of Theorem 2.3. Let X → B → S be a family of Lagrangian fibered hyper-Kahler manifoldsover an open ball S. The sheaves AutX/S and AutB/S are represented by the constant groupschemes

AutX/S∼=

⊔

f∈Aut(X)

S, AutB/S ∼= PGL(n+ 1)× S.

The homomorphism (2.7) becomes a morphism α : AutX/S → AutB/S . The desired sheaf AutX/B/Sis representable by kerα by Proposition 2.6.

To prove kerα is a constant subgroup scheme, it is enough to show the following: let S′ be aconnected component of AutX/S. Consider the restriction of α followed by the projection

β : S′ → PGL(n+ 1).

Then we claim that either β(S′) = id or β(S′) 6∋ id. Notice that the image β(S′) consists of|G|-torsion matrices in PGL(n + 1). Since the set of |G|-torsion matrices is a disjoint union ofPGL(n+ 1)-adjoint orbits (classified by eigenvalues), the connected set β(S′) has to lie in a singleorbit. The adjoint orbit containing the identity matrix is a singleton set id. Hence the claimfollows.

3. Abelian schemes associated to Lagrangian fibrations

The aim of this section is to associate a polarized abelian scheme to every Lagrangian fiberedcompact hyper-Kahler manifold, and to discuss consequences. The following is the first maintheorem of this section.


Theorem 3.1. Let π : X → B be a Lagrangian fibration of a compact hyper-Kahler manifold andB0 ⊂ B its smooth locus. Set X0 = π−1(B0) so that it becomes a smooth proper family of complextori over B0.

(1) There exists a unique projective abelian scheme ν : P0 → B0 making π : X0 → B0 ananalytic torsor under ν.

(2) Moreover, the abelian scheme is simple and has a unique choice of a primitive polarization

λ : P0 → P0. (3.2)

Here P0 → B0 is the dual abelian scheme of P0 → B0.

Definition 3.3. The abelian scheme ν : P0 → B0 in Theorem 3.1 is called the abelian schemeassociated to π.

Our statement is motivated by Arinkin–Fedorov’s result in [AF16, Thm 2], van Geemen–Voisin’sargument in [vGV16], and Sawon’s result in [Saw04]. The theorem combines and slightly generalizesthese results. Before discussing the applications of this theorem, let us first present some examples.

Example 3.4. Let X be a smooth projective moduli of torsion coherent sheaves on a K3 surface

with a fixed Mukai vector, so that it becomes a hyper-Kahler manifold of K3[n]-type equipped witha Lagrangian fibration π : X → B (see, e.g., [dCRS19]). In this case, it is known that the torus

fibration π : X0 → B0 is isomorphic to a relative Jacobian PicdC/B0associated to a certain universal

family C/B0 of smooth curves on the K3 surface. Now PicdC/B0is a torsor under the numerically

trivial relative Jacobian Pic0C/B0[BLR90, Thm 9.3.1]. By the uniqueness assertion of Theorem 3.1,

this is the associated abelian scheme P0.

Example 3.5. When π : X → B = P1 is an elliptic K3 surface, Theorem 3.1 is a weaker versionof the relative Jacobian fibration construction of π (e.g., [Huy16, §11.4]). In this case, one mayeven construct a semi-abelian scheme P → B over the entire base (Neron model) so that thesmooth locus of π becomes a torsor under P . Arinkin–Fedorov generalized this result to certainhigher dimensional projective hyper-Kahler manifolds. A stronger version of Theorem 3.1 wouldpotentially improve the arguments in this paper, but we will not discuss this further.

Example 3.6. For higher dimensional compact hyper-Kahler manifolds, one may still consider therelative Picard scheme Pic0X0/B0

→ B0. However, in general this is the dual of the abelian scheme

P0. This means we can consider P0 as the “double Picard scheme” of the original X0 [Saw04].However, for us it will be more useful to consider P0 as the identity component of the relativeautomorphism scheme of X0/B0. We will show this in Proposition 3.18.

Example 3.7. When π admits at least one rational section, then the abelian scheme P0 is in factisomorphic to X0. This is because the rational section must be defined over B0 by Remark 3.27, sothat X0 becomes a trivial P0-torsor. In some sense, Theorem 3.1 is thus a generalization of certainproperties of X0 to the case where π does not have any rational section. For example, one canstudy the Mordell–Weil group of ν, generalizing the study of the Mordell–Weil group of π.

One application of Theorem 3.1 is a more systematic study of the polarization type of thetorus fibers arising in π. The study of the polarization type of the torus fibers goes back to at least[Saw03], which in turn references an earlier idea of Mukai (see Proposition 5.3 in loc. cit.). However,to our knowledge, Wieneck’s series of papers [Wie16, Wie18] were the first work to consider thepolarization type as an invariant attached to a Lagrangian fibration and study them in great details

10 Y.-J. KIM

for K3[n] and Kumn-type hyper-Kahler manifolds. Using Theorem 3.1, we can given an alternativedefinition of the polarization type.

Definition 3.8. (1) The polarization scheme of π is the kernel

K = kerλ

of the polarization (3.2).(2) The polarization type of π is an n-tuple of positive integers (d1, · · · , dn) with d1 | · · · | dn

such that the fibers of the polarization scheme are isomorphic to (Z/d1 ⊕ · · · ⊕ Z/dn)⊕2.

The polarization scheme K is a finite etale commutative group scheme over B0. Hence itsfibers are all isomorphic and the polarization type is well-defined. The polarization type will be animportant ingredient for our method. We devote a short subsection 3.2 to collect its properties.

The second theme of this section is a relation between the group Aut(X/B) and the polarizationschemeK. We will see in Proposition 3.28 that every automorphism f ∈ Aut(X/B) defines a globalsection of the abelian scheme P0 → B0. If we consider Aut

(X/B) as a constant group scheme overB0, this means we have a closed immersion of group schemes

Aut(X/B) → P0. (3.9)

We expect the image of this injective map will be precisely the polarization scheme K ⊂ P0. Thisis a nontrivial claim; for example, this would imply the polarization scheme is a constant groupscheme. We were not able to prove this claim in general, and a large part of this paper will bedevoted to showing this for known deformation types of hyper-Kahler manifolds. The followingproposition will become the first step to our approach.

Proposition 3.10. Let π : X → B be a Lagrangian fibration of a compact hyper-Kahler manifold.Let (d1, · · · , dn) be its polarization type and h ∈ H2(X,Z) be the associated cohomology class of theline bundle π∗OB(1). If either one of the following two conditions holds, then (3.9) factors throughthe polarization scheme K and yields a closed immersion

Aut(X/B) → K. (3.11)

(1) The claim (3.11) holds for a deformation of π.(2) We have equalities cX = d1 · · · dn and div(h) = 1.

The first item says the desired property (3.11) is deformation invariant on π. The second itemproves (3.11) directly in certain cases. We note that the condition cX = d1 · · · dn is satisfied forall known deformation types of hyper-Kahler manifolds by Theorem 3.20. However, the condition

div(h) = 1 is more serious and sometimes fails for K3[n] or Kumn-type hyper-Kahler manifolds.Finally, we would like to mention a special consequence of the above discussion when π admits

a rational section. This may give the readers some more ideas on these objects. Recall fromExample 3.7 that the existence of a rational section implies X0

∼= P0. Therefore, (3.9) implies theexistence of certain torsion rational sections of the Lagrangian fibration

Aut(X/B) ⊂ MW(X/B).

For example, let us consider the case when X is of Kumn-type. We will prove in Theorem 5.1 thatthe order of Aut(X/B) is (n+ 1)2. Thus any Lagrangian fibration of a Kumn-type hyper-Kahlermanifold must have at least (n+1)2 torsion rational sections (once it admits a single torsion rationalsection), and the dual hyper-Kahler orbifold X is precisely the quotient of X by these special torsionrational sections. To our knowledge, this phenomenon has not been observed before and it became


one of our original motivations. See also [Sac20, §3–5] for some related ideas on the Mordell–Weilgroup and birational automorphisms defined by torsion rational sections.

3.1. Abelian scheme associated to a Lagrangian fibration. In this subsection, we presentthe proof of Theorem 3.1. Note again that we are assuming neither X is projective nor π has arational section.

Recall that every smooth closed fiber F of π is a complex torus (holomorphic Arnold–Liouvilletheorem). In fact, F is necessarily an abelian variety as observed by Voisin [Cam06, Prop 2.1]. Itwould be helpful for us to review this fact. The key idea is the following cohomological lemma, whichhas been discovered several times independently in [Voi92, Ogu09, Mat16] and recently generalizedinto higher degree cohomologies by Shen–Yin and Voisin [SY18].

Lemma 3.12. Let F be any smooth fiber of π and h ∈ H2(X,Z) the cohomology class of π∗OB(1).Then the restriction map

−|F : H2(X,Z) → H2(F,Z)

has ker(−|F ) = h⊥. Consequently, it has im(−|F ) ∼= Z.

Corollary 3.13 (Voisin). The image of the restriction map −|F is generated by an ample class ofF . As a result, F is an abelian variety.

Proof. Say y is an integral generator of Lemma 3.12. Choose any Kahler class ω ∈ H2(X,R) andconsider its restriction ω|F , a Kahler class on F . It has to be a nonzero real multiple of y. Thismeans, up to sign, y has to be a Kahler class on F . Hence y is an integral Kahler class, so it isample.

We caution the reader to be aware that the ample generator y of the restriction map need notbe primitive (see Proposition 3.21). One reasonable choice of a polarization on an abelian varietyfiber F is a unique primitive ample class in H2(F,Z) parallel to y. Theorem 3.1 is essentially amore global way to formulate this over the whole base B0.

We divide the proof of Theorem 3.1 into three parts: (1) an explicit construction of the polarizedabelian scheme P0, (2) proving such a construction makes X0 a torsor under P0, and finally (3) itsuniqueness. The uniqueness should be a more general fact about arbitrary torsors, at least in thealgebraic case (see Moret-Bailly’s answer in [MB]). The construction of P0 works for any properfamily of complex tori. The uniqueness of the polarization is the only part that needs the fact X0

is obtained from a Lagrangian fibered hyper-Kahler manifold X .The proof of the construction part closely follows [vGV16], but for completeness we reproduce

their argument here.

Proof of Theorem 3.1, construction. Apply the global invariant cycle theorem (for proper mapsbetween compact Kahler manifolds [Del71]) and Lemma 3.12 to obtain

H0(B0, R2π∗Q) = im(H2(X,Q) → H2(F,Q)) ∼= Q.

Hence, there exists a unique homomorphism (R2π∗Q)∨ → Q of local systems on B0 up to scalar.This is a homomorphism of Q-VHS: fiberwise, Corollary 3.13 proves the image of H2(X,Q) →H2(F,Q) is an ample class. Restrict it to the morphism of Z-VHS (R2π∗Z)

∨ → Z. The morphismcan be uniquely determined once we assume it to be primitive and represents an ample class oneach fiber. Finally, use the fact that π : X0 → B0 is a family of complex tori (abelian varieties) andobtain an isomorphism R2π∗Z = ∧2R1π∗Z. The result is a primitive polarization

(R1π∗Z)∨ ⊗ (R1π∗Z)

∨ → Z. (3.14)

12 Y.-J. KIM

We have constructed a weight −1 Z-VHS (R1π∗Z)∨ equipped with a polarization (3.14). Now use

a formal equivalence of categories between polarized weight −1 Z-VHS and that of polarized abelianschemes (e.g., [Del72, §5.2] [Del71, §4.4]). This constructs our desired abelian scheme ν : P0 → B0

with a unique primitive polarization λ : P0 → P0 over B0. To prove P0 is simple, we may prove thecorresponding VHS R1π∗Q is simple. This is tacitly proved in [vGV16] and later explicitly statedin [Voi18, Lem 4.5]. The idea is that if R1π∗Q splits as a direct sum V1⊕V2 of two VHS, then eachof them has their own polarizations, forcing h0(B0, R

2π∗Q) ≥ h0(B0,∧2V1) + h0(B0,∧2V2) ≥ 2.We omit the details here.

Proof of Theorem 3.1, torsor. Consider an analytic open covering Bi : i ∈ I of B0 so that overeach Bi, the restriction of the Lagrangian fibration π : Xi → Bi admits at least one holomorphicsection si : Bi → Xi. Considering si as a zero section, π : Xi → Bi becomes an abelian scheme.Hence by the equivalence of abelian schemes and (R1π∗Z)

∨, ν and π are isomorphic over Bi byφi : Xi → Pi sending si to the zero section of Pi.

Now use the isomorphism φi to transform the group law + : Pi ×BiPi → Pi into a Pi-action on

Xi. That is, we define a group action morphism by

ρi : Pi ×BiXi → Xi, (pi, xi) 7→ φ−1

i (φi(xi) + pi).

We want to patch ρi together to define a group action ρ : P0 ×B0X0 → X0 over the entire B0.

To do so, we need to check whether the definitions of ρi and ρj coincides over the intersectionBij = Bi ∩Bj , i.e.,

φ−1i (φi(xij) + pij) = φ−1

j (φj(xij) + pij) for all (pij , xij) ∈ Pij ×BijXij . (3.15)

Over Bij , one has a transition function φj φ−1i : Pij → Xij → Pij , an automorphism of Pij .

Recall that the isomorphisms φi and φj are constructed by choosing the zero sections si and sj , andthe corresponding isomorphisms φi : Xij

∼= Pij and φj : Xij∼= Pij are as abelian schemes. From it,

we notice the automorphism φj φ−1i : Pij → Pij is a translation automorphism. The translation is

by φj φ−1i (0), the difference of the two zero sections. With this, we have a sequence of identities

φj(xij) + pij = φj φ−1i (φi(xij)) + pij =

(

φi(xij) + φj φ−1i (0)

)

+ pij

=(

φi(xij) + pij)

+ φj φ−1i (0) = φj φ

−1i

(

φi(xij) + pij)

.

This proves (3.15). Hence ρi patches together and defines a morphism ρ : P0 ×B0X0 → X0. The

group action axioms are all easily verified. Also, X0 is clearly a P0-torsor by construction.

Proof of Theorem 3.1, uniqueness. Let ν : P0 → B0 be a (not necessarily projective) abelian schemeso that π becomes a torsor under ν. We claim R1ν∗Z ∼= R1π∗Z as VHS over B0. Consider thegroup scheme action map

P0 ×B0X0 X0

B0

ρ

µπ .

From the diagram, we have a pullback morphism between the VHS ρ∗ : R1π∗Z → R1µ∗Z. Thelatter VHS is isomorphic to the direct sum R1ν∗Z ⊕ R1π∗Z by the decomposition theorem forsmooth proper morphisms. Hence composing with the first projection, we obtain a morphismR1π∗Z → R1ν∗Z. Now over a small analytic open subset U ⊂ B0, fix any holomorphic sectionof π : XU → U so that we can identify PU and XU . Hence ρ becomes the addition operation ofthe abelian scheme XU ×U XU → XU . With this description, the pullback morphism is fiberwise


ρ∗ : H1(F,Z) → H1(F,Z) ⊕ H1(F,Z), x 7→ (x, x). Hence the morphism R1π∗Z → R1ν∗Z is anisomorphism over U , and the claim follows.

Remark 3.16. (1) A posteriori, one has an interpretation of Corollary 3.13 in terms of Theo-rem 3.1. The abelian scheme ν is projective, and ν and π are fiberwise isomorphic. Hencethe smooth fibers of π are projective, even when the hyper-Kahler manifold X is not.

(2) Theorem 3.1 is also related to Oguiso’s result [Ogu09] in the following sense. Consider thegeneric fiber PL → SpecL of ν : P0 → B0. It is an abelian variety over L. Since P0 hasa unique polarization (up to scalar), PL has a unique polarization. Now ampleness is anopen condition in NS(PL)R, so the uniqueness of the polarization implies ρ(PL) = 1.

(3) If we further assume X is projective, then the discussion becomes algebraic and hence thesmooth morphism π : X0 → B0 admits etale local sections. Thus π becomes an etale torsorunder ν.

The unique abelian scheme P0 above should be considered as the identity component of the“automorphism scheme” of π (see [AF16, §8.3]). Unfortunately we do not know the existence ofan automorphism group scheme in the non-projective setting, so here we need a care (when X isprojective, this is a byproduct of the existence of a Hilbert scheme [Kol96, I.1.10.2]). Nevertheless,one may concentrate only on the translation automorphisms to avoid such a technical problem. Letus define a subsheaf AuttrX0/B0

⊂ AutX0/B0of the automorphism sheaf by

AuttrX0/B0(U) = f : XU → XU : U -automorphism acting by translation on each fibers. (3.17)

Proposition 3.18. The abelian scheme ν : P0 → B0 represents AuttrX0/B0.

Proof. This almost follows from the definition. Let us temporarily denote by P0 the sheaf of analyticlocal sections of ν. Since P0 acts on X0 by fiberwise translation, we have a sheaf homomorphismP0 → AuttrX0/B0

. The homomorphism is injective because the P0-action is effective. Now X0

was in fact a torsor under P0. Over a small analytic open subset U ⊂ B0, XU → U admitsa section so it becomes an abelian scheme isomorphic to PU → U . Hence the set of sectionsPU (U) = XU (U) consists of precisely the translation automorphisms of XU . This proves thehomomorphism P0 → AuttrX0/B0

is surjective stalkwise. Thus it is an isomorphism.

3.2. Polarization type and divisibility of π∗OB(1). The purpose of this subsection is to studytwo numerical invariants associated to a Lagrangian fibered hyper-Kahler manifold and study theirrelations: they are the polarization type of π in Definition 3.8 and the divisibility of the line bundleπ∗OB(1). Throughout, we will write h ∈ H2(X,Z) the first Chern class of π∗OB(1).

The polarization type of π is an n-tuple of positive integers (d1, · · · , dn) with d1 | · · · | dn suchthat the fiber of the polarization scheme K is isomorphic to (Z/d1 ⊕ · · · ⊕ Z/dn)

⊕2. Since we areassuming the polarization λ : P0 → P0 is primitive, we always have d1 = 1. The polarization typewas already computed for all currently known deformation types of hyper-Kahler manifolds. Thecomputations were based on its original definition in [Wie16]. Therefore, to use the previous resultswe first need to show our definition is equivalent to the original one.

Lemma 3.19. The polarization type in Definition 3.8 is equivalent to the definition in [Wie16].

Proof. Recall our definition of the polarization is constructed as the primitive morphism (R2π∗Z)∨ →

Z of local systems, or equivalently a primitive morphism Z → R2π∗Z. Fix any smooth fiber F of π.As a local system, R2π∗Z is identified with a Z-module H2(F,Z) with a monodromy π1(B0)-action.In this setting, the primitive morphism Z → R2π∗Z of local systems corresponds to a primitive

14 Y.-J. KIM

homomorphism Z → H2(F,Z) of π1(B0)-modules. In other words, our definition of the polariza-tion type is equivalent to the primitive polarization type of a single smooth fiber F coming fromthe image of H2(X,Q) → H2(F,Q) by the global invariant cycle theorem. This was precisely howWieneck defined the polarization type.

We can now use the previous results on computations of the polarization type of π. The followingtheorem collects all possible polarization types that can occur for known deformation types of hyper-

Kahler manifolds. For K3[n] and Kumn-types the computations are done by [Wie16, Wie18]. ForOG10 and OG6, the computations are contained in [MO20] and [MR21], respectively.

Theorem 3.20 ([Wie16, Wie18], [MO20], [MR21]). Let π : X → B be a Lagrangian fibered compacthyper-Kahler manifold. Then the polarization type of π is

(1, · · · , 1) if X is of K3[n]-type;

(1, 1, 1, 1, 1) if X is of OG10-type;

(1, · · · , 1, d1, d2) if X is of Kumn-type; and

(1, 2, 2) if X is of OG6-type.

When X is of Kumn-type, we set d1 = div(h) in H2(X,Z) and d2 = n+1d1

.

It is also important for us that the polarization type is deformation invariant on π (see [Wie16,Thm 1.1]). We will later recover this result in Corollary 3.33. Observe in Theorem 3.20 thatwe have an equality cX = d1 · · · dn for all known deformation types of hyper-Kahler manifolds.In this sense, we expect the polarization type should be considered as a refinement of the Fujikiconstant cX . When such an equality holds, we can explain why the restriction homomorphismH2(X,Z) → H2(F,Z) may be non-primitive in certain cases.

Proposition 3.21. Assume we have an equality cX = d1 · · · dn. Then the image of the restrictionhomomorphism H2(X,Z) → H2(F,Z) in Lemma 3.12 is generated by aθ, where a = div(h) and θis a primitive ample class representing the canonical polarization of F .

Proof. Choose a cohomology class x ∈ H2(X,Z) with q(h, x) = a. By Lemma 3.12, the classx|F ∈ H2(F,Z) must be a positive integer multiple of the primitive polarization class θ. Setx|F = bθ. Now the claim directly follows from the Fujiki relation

d1 · · · dn =1

n!

∫

F

θn =1

n!

∫

X

hn(

1bx)n

= cX · q(

h, 1bx)n

= cX

(a

b

)n

.

Though not used in this paper, the divisibility of h is also related to the existence of a rationalsection of π. We end this subsection with the following observation.

Proposition 3.22. Assume cX = d1 · · · dn and π admits at least one rational section. Thendiv(h) = 1 or 2.

Proof. If π admits a rational section, then X0∼= P0 becomes a projective abelian scheme (Exam-

ple 3.7). By the general theory of abelian schemes, twice a polarization is always associated to aline bundle (e.g., [MFK94, Prop 6.10] or [FC90, Def I.1.6]). This means 2θ ∈ H2(F,Z) is containedin the image of Pic(X) ⊂ H2(X,Z) → H2(F,Z). By Proposition 3.21, this implies div(h) = 1 or2.

If X is of K3[n] or Kumn-type then its Lagrangian fibration π : X → B may have div(h) > 2. Insuch cases, π (and any of its deformation) would never admit any rational section and the notionof the P0-torsor is necessary.


3.3. The polarization scheme and H2-trivial automorphisms. The goal of this subsection isto prove Proposition 3.10. In fact, we obtain the following more precise versions of it.

Proposition 3.23. Let π : X → B and π′ : X ′ → B′ be two deformation equivalent Lagrangianfibrations of compact hyper-Kahler manifolds. Let a be any positive integer. Then

Aut(X/B) → ker(aλ : P0 → P0) (3.24)

if and only if the same holds for π′.

Proposition 3.25. Let π : X → B be a Lagrangian fibration of a compact hyper-Kahler manifoldand (d1, · · · , dn) its polarization type. Assume we have an equality cX = d1 · · · dn. Then (3.24)holds for a = div(h).

Proposition 3.10 is simply the case a = 1 for the above two propositions. The rough strategyof the proof of these propositions is as follows. First, we prove Aut(X/B) acts on π : X0 → B0

fiberwise by translations. Since P0 was the group scheme representing translation automorphismsof π (Proposition 3.18), this proves Aut(X/B) → P0. Proposition 3.23 will follow from a simpleobservation about torsion sections (Lemma 3.34). For Proposition 3.25, the reason why we needa = div(h) is because the image of the restriction map H2(X,Z) → H2(F,Z) is precisely a timesthe primitive polarization (Proposition 3.21). Such a polarization on F can be lifted to a line bundleon X , and the claim follows because Aut(X/B) acts trivially on Pic(X) ⊂ H2(X,Z).

The actual realization of the above idea will be quite lengthy due to the technical difficultiesarising in deformation arguments. The three main steps will be Proposition 3.28, 3.31 and 3.34.

Lemma 3.26. Any rational section of ν : P0 → B0 can be uniquely extended to an honest section.

Proof. Assume s : B0 99K P0 is a rational section undefined at b ∈ B0. Let S ⊂ P0 be the closure ofthe image of s, so that we obtain a proper birational morphism ν|S : S → B0. Since B0 is smooth

and s is undefined at b, the fiber Sb = (ν|S)−1(b) is a uniruled variety (e.g., [Kol96, Thm VI.1.2]).

This means an abelian variety ν−1(b) contains a uniruled variety Sb. Contradiction. See [BLR90,Cor 8.4.6] for an alternative proof.

Remark 3.27. The same argument applies to π and proves the following: any rational section of πis necessarily defined over B0.

Proposition 3.28. Every H2-trivial automorphism in Aut(X/B) defines a global section of P0 →B0. That is, we have a closed immersion of group schemes

Aut(X/B) → P0.

Proof. Recall from Proposition 3.18 that P0 is the abelian scheme representing the translation au-tomorphism sheaf AuttrX0/B0

. Hence our goal is to prove Aut(X/B) acts on π : X0 → B0 by

fiberwise translation automorphisms. Consider the quotient X = X/Aut(X/B) with a commuta-tive diagram

X

X

B

p

π

π

.

We first claim p is etale on general fibers over B. Let S ⊂ X be the ramified locus of p. It hascodimension ≥ 2 because p is quasi-etale by Proposition B.2. Let b ∈ B be a general point, so thatthe fibers F = Xb and F = Xb are both smooth. Observe the ramification locus of p : F → F

16 Y.-J. KIM

is precisely S ∩ F , which is of codimension ≥ 2 since b is general. The purity of the branch locustheorem forces p to be etale.

Now we have a finite etale quotient p : F → F = F/Aut(X/B) between smooth projectivevarieties. Its Galois group Aut(X/B) acts on F by fixed point free automorphisms. Since F isan abelian variety, this means Aut(X/B) acts on F by translations. The conclusion is that ona general fiber of π, the group Aut(X/B) acts by translation. Finally, by Proposition 3.18 thismeans Aut(X/B) defines a rational section of ν : P0 → B0. By Lemma 3.26, the rational sectionmust be defined over the entire B0 and becomes an honest section. Hence Aut(X/B) acts bytranslations over the entire B0.

An immediate byproduct is that Aut(X/B) is abelian.

Proposition 3.29. Aut(X/B) is a finite abelian group.

The next step is to understand the behavior of the polarization λ under deformations of π. Arelated result is Wieneck’s deformation invariance of the polarization type of π [Wie16, Thm 1.1].Recall that the polarization scheme K was defined to be the kernel of the polarization kerλ. To dealwith a more technical Proposition 3.23 and 3.25, we also need to consider the kernels Ka = ker(aλ)for each positive integer a:

0 Ka P0 P0 0aλ .

Since the abelian scheme P0 was associated to the VHS (R1π∗Z)∨, there is a VHS version of this

sequence

0 (R1π∗Z)∨ R1π∗Z Ka 0aλ∗ . (3.30)

The cokernel Ka is a local system of finite abelian groups on B0 and is related to Ka as follows: Ka

is a sheaf of analytic sections of the group scheme Ka → B0, and Ka is the total space of the localsystem Ka. Therefore, we can relate Ka to either abelian schemes or variation of Hodge structures.This technical flexibility will be useful to describe deformation behaviors of Ka.

Lemma 3.31. Let p : Xπ−→ B

q−→ ∆ be a family of Lagrangian fibered compact hyper-Kahler

manifolds over a complex open disc ∆. Set B0 ⊂ B the smooth locus of π. Then for each positiveinteger a, there exists a finite etale group scheme Ka over B0 parametrizing the group schemes Ka

over (B0)t for all t ∈ ∆.

Proof. Let X0 = π−1(B0) be the preimage of B0, so that the restriction π0 : X0 → B0 of π is asmooth proper family of abelian varieties. Consider the local system R2π0,∗Z on B0. Our firstclaim is H0(B0, R

2π0,∗Z) ∼= Z. Denoting by j : B0 → B an open immersion, it is enough to proveH0(∆, q∗j∗R

2π0,∗Z) ∼= Z. Notice that q∗j∗R2π0,∗Z is a constructible sheaf, because R2π0,∗Z is a

local system, its pushforward by j∗ is a constructible sheaf on B (e.g., [KS90, Ex VIII.10]), andagain its pushforward by q∗ is a constructible sheaf on ∆. For each t ∈ S, we have a Lagrangianfibered hyper-Kahler manifold π : Xt → Bt and we may apply our previous discussions

H0((B0)t, R2π∗Z) ∼= Z.

This proves every fiber of q∗j∗R2π0,∗Z is isomorphic to Z. In this setting, we will formally prove

the sheaf has Z global sections in Lemma 3.32. This proves the claim H0(B0, R2π0,∗Z) ∼= Z.

We have a unique primitive morphism (R2π0,∗Z)∨ → Z of local systems on B0. As π0 is a

family of abelian varieties, we have an isomorphism R2π0,∗Z = ∧2R1π0,∗Z. This gives us a uniqueprimitive morphism of local systems (in fact, a polarization of VHS by Corollary 3.13) (R1π0,∗Z)

∨⊗(R1π0,∗Z)

∨ → Z over B0. Consider the scalar multiple a of it and induce a morphism (R1π0,∗Z)∨ →


R1π0,∗Z whose cokernel Ka is a local system on B0, parametrizing the family of local systems Ka oneach (B0)t in (3.30). The total space of Ka gives our desired finite etale group scheme Ka → B0.

Lemma 3.32. Let F be a constructible sheaf on a complex open disc ∆. If every fiber of F isisomorphic to Z, then we have H0(∆, F ) ∼= Z.

Proof. Since F is constructible, F|U is a local system on a completement U of a finite set of pointst1, · · · , tk ∈ ∆. Let Ui ⊂ U be a small punctured disc around ti. The restriction F|Ui

is determinedby the representation

ρi : Z ∼= π1(Ui) → Aut(Z) = ±1.

We have only two possibilities ρi(1) = ±1 for each i. Suppose we have ρi(1) = −1 for some i.Consider the total space f : Et(F ) → ∆ of the entire constructible sheaf F (espace etale). Themap f is holomorphic and etale, i.e., a local isomorphism. The condition ρi(1) = −1 geometricallytranslates to the fact that f−1(Ui) consists of a single copy of Ui (the zero section) and infinitenumber of two-sheeted coverings of the punctured disc Ui. By the very assumption, the preimagef−1(ti) = p1, p2, · · · should be isomorphic to Z. Since f is a local isomorphism, there should bean open disc neighborhood of each pi ∈ Et(F ). Along the two-sheeted coverings of Ui in f

−1(Ui),this cannot happen. Therefore, the only possibility is that all pi are the non-Hausdorff points fillingin the unique punctured disc component in f−1(Ui) (i.e., the zero section). Hence we obtain atleast Z global sections around the zero section and we are done.

The remaining case is when ρi(1) = 1 for all i. This means F|U is a constant sheaf Z. Be awarethat this does not imply F is a constant sheaf Z on ∆. This can be again conveniently seen in thetotal space f : Et(F ) → ∆. Although f is a local homeomorphism, it is not a covering space unlessEt(F ) is Hausdorff. Indeed, the fibers f−1(ti) can consist of non-Hausdorff points in Et(F ) andthis gives us a classification of such a constructible sheaf F . In any case, there are always Z globalsections.

Lemma 3.31 in particular recovers [Wie16, Thm 1.1].

Corollary 3.33. The polarization type of π is invariant under deformations of π.

The following final observation is elementary but nontrivial. We match the notation of the lemmato our original discussion.

Lemma 3.34. Let P0 → B0 be an abelian scheme over a complex manifold B0 and aλ : P0 → P0 apolarization with Ka = ker(aλ). Assume there exists a torsion section f : B0 → P0. If f(B0)∩Ka 6=∅ then f(B0) ⊂ Ka.

Proof. The statement is topological and local on the base B0, so we may assume B0 is a complex openball S and P0 → B0 is homeomorphic to the topological constant group scheme (R/Z)2n × S → S.In this setting, the kernel Ka is a constant subgroup scheme and the torsion section f is a constantsection. Hence f(S) ∩Ka 6= ∅ if and only if f(S) ⊂ Ka.

Proof of Proposition 3.23. Consider a one-parameter family of Lagrangian fibered hyper-Kahlermanifolds X → B → ∆ over a complex disc ∆. By Lemma 3.31, there exists a notion of a familyof abelian schemes P0 → B0 and a family of finite etale group schemes Ka ⊂ P0. Proposition 3.28proves we have a closed immersion Aut(X/B) → P0 for a single fiber. In fact, the argumentapplies to the entire family and produces Aut(X/B) global sections of P0 → B0, or equivalentlyan embedding

Aut(X/B) → P0.

18 Y.-J. KIM

Since Aut(X/B) is finite, the global sections are torsion. Suppose we had Aut(X/B) → Ka forthe original Lagrangian fibration over 0 ∈ ∆. Then this forces Aut(X/B) → Ka over the entire ∆by Lemma 3.34. Hence the claim follows.

Proof of Proposition 3.25. Recall from Proposition 3.21 that the restriction map H2(X,Z) →H2(F,Z) has a rank 1 image generated by the class aθ, where a = div(h) and θ is the primi-tive ample class corresponding to our polarization λ : F → F . The preimage of aθ ∈ H2(F,Z)under this restriction homomorphism is precisely S = x ∈ H2(X,Z) : q(x, h) = a. By Propo-sition 3.23, the claim is invariant under deformations of π. We may thus deform π and assumePic(X) ∩ S 6= ∅. In other words, we may assume the composition Pic(X) ⊂ H2(X,Z) → H2(F,Z)is generated by aθ.

The assertion Aut(X/B) → Ka = ker(aλ) is equivalent to aλ(Aut(X/B)) = 0. The latterequality may be verified fiberwise, so we may concentrate on a single fiber F = ν−1(b) = π−1(b).Let L be any line bundle on X such that its image under Pic(X) → H2(F,Z) is aθ. This meansthe polariztion aλ can be described as

aλ : F → F , tx 7→ [t∗x(L|F )⊗ L−1|F ].

If we assume tx = f|F is from a global H2-trivial automorphism f ∈ Aut(X/B), then we have asequence of identities

t∗x(L|F ) = (f|F )∗(L|F ) = (f∗L)|F ∼= L|F ,

where the last isomorphism follows from the fact f acts on Pic(X) ⊂ H2(X,Z) trivially. Thisproves aλ sends Aut(X/B) to 0 and the claim follows.

4. The minimal split covering and H2-trivial automorphisms

This section discusses an explicit construction of certain H2-trivial automorphisms. This will beconveniently used in the next section when we describe the Aut(X)-action explicitly for certainexamples of Kumn-type hyper-Kahler manifolds. Recall that the group Aut(X) is computed for all

known deformation types of hyper-Kahler manifolds; Beauville [Bea83a] for K3[n]-types, Boissiere–Nieper-Wißkirchen–Sarti [BNWS11] for Kumn-types, and Mongardi–Wandel [MW17] for OG10and OG6-types. The strategy is to compute the group for a specific choice of a complex structureand then use Hassett–Tschinkel’s deformation equivalence [HT13, Thm 2.1]. Unfortunately, thisargument doesn’t tell us how Aut(X) exactly acts on X for the deformations. The goal of thissection is to introduce Proposition 4.5 to partially resolve this problem.

Throughout the section, we stick to the following setting. Let M be a projective holomorphicsymplectic manifold, not necessarily irreducible. By Beauville–Bogomolov decomposition theorem,M must admit a finite etale covering X × T → M , called a split covering, where X is a finiteproduct of projective hyper-Kahler manifolds and T is an abelian variety. In fact, Beauville in[Bea83a, §3] also considered minimal such a covering. A minimal split covering ofM is the smallestpossible split covering of M , in the sense that every split covering factors through it. The minimalsplit covering of M always exists and is unique up to a (non-unique) isomorphism. Moreover, itis a Galois covering. We refer to Beauville’s original paper for more details about minimal splitcoverings.

Meanwhile, Kawamata [Kaw85, Thm 8.3] proved that if M is a K-trivial smooth projectivevariety then its Albanese morphism Alb : M → Alb(M) has to be an etale fiber bundle. Moreconcretely, there exists an isogeny φ : T → Alb(M) of abelian varieties such that the base change


of Alb becomes a trivial fiber bundle over T . We obtain a cartesian diagram

X × T M

T Alb(M)

Φ

pr2 Albφ

, (4.1)

where X is a fiber of the Albanese morphism. In particular, one sees Φ : X × T → M becomes asplit covering of M .

Combining the two discussions, we obtain:

Proposition 4.2. Let M be a projective holomorphic symplectic manifold and Alb :M → Alb(M)

its Albanese morphism, an etale fiber bundle by Kawamata. Assume X = Alb−1(0) is a projectivehyper-Kahler manifold. Then there exists a unique isogeny φ : T → Alb(M) such that the morphismΦ in the fiber diagram (4.1) becomes the minimal split covering of Beauville.

Proof. Use Kawamata’s result to construct an isogeny φ′ : T ′ → Alb(M) trivializing the Albanesemap as in (4.1). Since φ′ is a finite Galois covering, Φ′ is also a finite Galois covering with Gal(Φ′) ∼=Gal(φ′). The first lemma in [Bea83a, §3] claims Aut(X × T ′) = Aut(X) × Aut(T ′). Hence theGal(Φ′)-action on X × T ′ is by (f, a) where f and a are automorphisms on X and T , respectively.The isomorphism Gal(Φ′) → Gal(φ′) is by the second projection (f, a) 7→ a. Since Gal(φ′) is thekernel of the isogeny φ′, the automorphisms a must be translations of T ′.

Now consider the homomorphism Gal(Φ′) → Aut(X) by (f, a) 7→ f . Set H by the kernel ofit; it consists of elements of the form (idX , a). Under the isomorphism Gal(Φ′) ∼= Gal(φ′), we canconsider it as a subgroup of Gal(φ′), so there exists a Galois covering T ′ → T = T ′/H correspondingto it. Let φ : T → Alb(M) be the morphism factorizing φ′. We have a cartesian diagram

X × T ′ X × T M

T ′ T Alb(M)

pr2

Φ

pr2 Albφ

.

By construction, Gal(Φ) consists of automorphisms (f, a) with no (idX , a) (i.e., the Gal(φ)-actionon X is effective). But this means Φ is precisely Beauville’s minimal split covering [Bea83a, §3].The uniqueness of φ follows from the uniqueness of the minimal split covering.

The proposition in particular proves that the minimal split covering can be always realized byan isogeny φ : T → Alb(M) and the base change (4.1).

Definition 4.3. We call φ : T → Alb(M) in Proposition 4.2 the minimal isogeny trivializing theAlbanese morphism Alb :M → Alb(M). It is unique up to a (non-unique) isomorphism.

In fact, the proof of Proposition 4.2 is saying more about an arbitrary isogny φ′.

Corollary 4.4. Notation as in Proposition 4.2. Let φ′ : T ′ → Alb(M) be any isogeny trivializingthe Albanese morphism. Then

(1) φ′ factors though the minimal isogeny φ.(2) There exists a canonical Gal(φ′)-action on X.(3) The isogeny φ′ is minimal if and only if the Gal(φ′)-action on X is effective.

Proof. All of these can be directly deduced from the proof of Proposition 4.2. Recall Gal(Φ′) →Gal(φ′), (f, a) 7→ a is an isomorphism. Therefore, f = fa is uniquely determined by a, and thisdefines Gal(φ′) → Aut(X), a 7→ fa.

20 Y.-J. KIM

Now we can state the main result of this section. The ideas here were already contained in[Bea83a, Bea83b].

Proposition 4.5. Notation as in Proposition 4.2 and 4.4. Then Gal(φ′) acts on X by H2-trivialautomorphisms. That is, we have a canonical homomorphism

Gal(φ′) → Aut(X),

which is injective if and only if φ′ is minimal.

Proof. By Corollary 4.4, we may assume φ′ = φ is minimal and Gal(φ) ⊂ Aut(X). The content ofthe proposition is that it is further a subgroup of Aut(X).

Consider the diagram (4.1). Our first step is to equip T -actions on all the four spaces to makethe diagram T -equivariant. Equip a T -action on T by translation, and on X×T only on the secondfactor again by translation. The T -action on Alb(M) is by translation via the morphism φ: if a ∈ Tand z ∈ Alb(M) then we define a.z = z + φ(a).

To equip a T -action on M , we claim the T -action on X × T descends to M via Φ. The descentworks if the Gal(Φ)-action on X × T commutes with the T -action. Recall from the discussions inProposition 4.2 that Gal(Φ) acts on X × T by (f, a) where f is an automorphism of X and a is atranslation of T . Now let b ∈ T and (x, t) ∈ X × T . Then we have a sequence of identities

b.(

(f, a).(x, t))

= (f(x), t+ a+ b) = (f, a).(

b.(x, t))

.

This proves the T -action and Gal(Φ)-action commutes, yielding the descent T -action on M . Theconclusion is that Alb becomes automatically T -equivariant (and hence the diagram (4.1) becomesT -equivariant).

By definition, the stabilizer the T -action on Alb(M) is precisely kerφ = Gal(φ). Since theAlbanese map Alb : M → Alb(M) is T -equivariant, this induces a Gal(φ)-action on the fiber

Alb−1(0) = X . One easily shows this coincides with our previous Gal(φ)-action on X . Nownotice that any T -action on M is isotopic to the identity map because T is path connected. Inparticular, T acts on M trivially at the level of cohomology H∗(M,Q). The embedding X ⊂M isGal(φ)-equivariant, so we have a Gal(φ)-equivariant restriction homomorphism

H2(M,Q) → H2(X,Q).

Hence it suffices to prove this restriction homomorphism is surjective.Now the question became topological. Deform the complex structure of the hyper-Kahler mani-

fold X very generally so that H2(X,Q) becomes a simple Q-Hodge structure (we will have to losethe projectiveness of X). The complex structure ofM can be correspondingly chosen in a way thatthe finite covering map Φ : X × T → M becomes holomorphic. Therefore, the Hodge structuremorphism H2(M,Q) → H2(X,Q) is either 0 or surjective. We only need to rule out the formerpossibility.

To prove it is nonzero, consider any global holomorphic symplectic form σ on M . Pullingit back to X × T gives a global holomorphic symplectic form on X × T . But H2,0(X × T ) =H2,0(X) ⊕ H2,0(T ) by Kunneth. If σ was 0 in the H2,0(X)-component then this would mean σdoesn’t contain any 2-forms along the tangent direction of X , violating σ is a symplectic form.Hence σ|X cannot be 0. The claim follows.

Remark 4.6. The inclusion Gal(φ) ⊂ Aut(X) for the minimal isogeny φ in Proposition 4.5 may bea strict inclusion. For example, we will later see that when X if of Kumn-type then

Gal(φ) ∼= (Z/n+ 1)⊕4, Aut(X) ∼= Z/2⋉ (Z/n+ 1)⊕4.


5. The H2-trivial automorphisms and polarization scheme for generalized Kummer

varieties

The goal of this section is an explicit computation of the group Aut(X/B) and the polarizationscheme K for certain Lagrangian fibrations of Kumn-type hyper-Kahler manifolds. Since the groupAut(X/B) will be of interest for all known deformation types of hyper-Kahler manifolds, we statethe result in a more general form. The following is the first main theorem of this section.

Theorem 5.1. Let π : X → B be a Lagrangian fibration of a compact hyper-Kahler manifold.Then

Aut(X/B) ∼=

id if X is of K3[n] or OG10-type;

(Z/d1 ⊕ Z/d2)⊕2 if X is of Kumn-type; and

(Z/2)⊕4 if X is of OG6-type.

Here if X is of Kumn-type, we assume its polarization type is (1, · · · , 1, d1, d2).

Notice that the bigger group Aut(X) is already trivial for K3[n] and OG10-types (see [Bea83a]and [MW17]), so the theorem is clear in the first two cases. For Kumn and OG6-types, recall fromTheorem 2.3 that Aut(X/B) is deformation invariant on π. By [Wie18, §6.28], every Lagrangianfibration of a Kumn-type hyper-Kahler manifold is deformation equivalent to the moduli of sheavesconstruction, which will be recalled in Section 5.1. By [MR21], every Lagrangian fibration of anOG6-type hyper-Kahler manifold is deformation equivalent to each other. Therefore, Theorem 5.1follows from the following more concrete results.

Proposition 5.2. Let π : X → B be a Lagrangian fibration of a Kumn-type hyper-Kahler manifold,obtained by the moduli of sheaves construction from a triple (S, l, s) in Definition 5.7. Let (d1, d2)be the polarization type of the ample class l. Then

Aut(X/B) ∼= (Z/d1 ⊕ Z/d2)⊕2.

Proposition 5.3 (Mongardi–Wandel). Let π : X → B be a Lagrangian fibration of an OG6-typehyper-Kahler manifold, obtained by the moduli of sheaves construction. Then

Aut(X/B) ∼= (Z/2)⊕4.

We note that the latter computation for OG6-type was already done by Mongardi–Wandel in[MW17, §5], as an intermediate step for their computation of the larger group Aut(X) ∼= (Z/2)⊕8.Thus proving Proposition 5.2 will be enough to conclude Theorem 5.1. As mentioned before, theproof will be done by an explicit computation. In fact, the computation can be carried out furtherand calculates the polarization scheme K as well. This is the second main result of this section.

Proposition 5.4. Let π : X → B be a Lagrangian fibration of a Kumn-type hyper-Kahler manifold,obtained by the moduli of sheaves construction from a triple (S, l, s) in Definition 5.7. Then we havea closed immersion (in fact, an isomorphism) of group schemes

Aut(X/B) → K.

Contrary to the previous sections, all the discussions in this section will be algebraic. In partic-ular, algebraic Chern classes and Chow groups will be used. Given a coherent sheaf E on a smoothprojective variety S, we denote by

ci(E) ∈ H2i(S,Z), ci(E) ∈ CHi(S)

the i-th numerical (i.e., cohomological) and algebraic Chern classes of E, respectively.

22 Y.-J. KIM

5.1. Moduli of coherent sheaves on an abelian variety. In this subsection, we recall theconstruction of Kumn-type hyper-Kahler manifolds obtained from certain moduli spaces of sheaveson abelian varieties. We will mostly follow [Yos01].

Let S be an abelian surface and l ∈ NS(S) an ample cohomology class with∫

S l2 = 2n+ 2. Fix

a nonzero class s ∈ H4(S,Z) so that we have a primitive Mukai vector

v = (0, l, s) ∈ H∗even(S,Z). (5.5)

Then the moduli space M of stable sheaves on S with Chern character v, with respect to a v-generic ample line bundle, becomes a smooth projective holomorphic symplectic variety of dimension〈v, v〉+2 = 2n+4. Denote by PiclS a connected component of the Picard scheme of S with numerical

first Chern class l. Yoshioka proved the Albanese variety of M is isomorphic to S × PiclS , so that

we can define the Albanese morphism Alb :M → S × PiclS .To be a little more precise, we first need to fix a reference coherent sheaf [E0] ∈ M . Let

[L0] ∈ PiclS be a closed point with c1(L0) = c1(E0), and consider it as the origin of the abelian

surface PiclS . The Albanese morphism is defined (on closed points) by

Alb : M → S × PiclS , [E] 7→(

c(E), c1(E))

, (5.6)

where we define c(E) = Σ(

c2(E) − c2(E0))

for the summation map Σ : CH2(S) → S(C). Themorphism becomes an etale trivial fiber bundle with a fiber X , a Kumn-type projective hyper-Kahler manifold.

Due to our choice of the Mukai vector v in (5.5), the above construction further comes with a

Lagrangian fibration. Consider a connected component B of the Chow variety of effective divisorson S with numerical first Chern class l. Le Potier [LP93] constructed a morphism

Supp :M → B, [E] 7→ [Fitt0E],

where Fitt0E is the Fitting support of a coherent sheaf E. Finally, consider the Poincare linebundle P on S×PiclS , the universal family of line bundles with the numerical Chern class l. Denote

by r : S × PiclS → PiclS the second projection. Then by Riemann–Roch, r∗P is a vector bundle ofrank n+ 1. Its projectivization is a Zariski locally trivial Pn-bundle

LB : B → PiclS , [C] 7→ [OS(C)].

Gathering all the morphisms together, one easily checks we have a commutative diagram

M

S × B

S × PiclS

Alb

(c, Supp)

id×LB

.

This is an isotrivial family of Lagrangian fibered hyper-Kahler manifolds in the sense of Defi-nition 2.2. For simplicity, we will take a fiber at (0, [L0]) of the family so that we have X =

Alb−1(0, [L0]) is a Kumn-type hyper-Kahler manifold and B = LB−1([L0]) = |L0| ∼= Pn is a com-plete linear system.

Definition 5.7. Let (S, l) be a degree n + 1 polarized abelian surface with a polarization type(d1, d2) and s ∈ H4(S,Z) any nonzero class with gcd(d1, s) = 1. Then the above constructionπ : X → B is called the moduli construction of Kumn-type associated to the triple (S, l, s). It is aLagrangian fibration of a projective hyper-Kahler manifold of Kumn-type to a projective space.


5.2. Describing the H2-trivial automorphisms of Kumn-type moduli constructions. Re-call that [BNWS11] and [HT13] proved that any Kumn-type hyper-Kahler manifold have the groupof H2-trivial automorphisms

Aut(X) ∼= Z/2⋉ (Z/n+ 1)⊕4.

The goal of this subsection is to explicitly describe such automorphisms for the moduli construction.Note that describing such automorphisms is about X itself but not about the Lagrangian fibrationπ : X → B. Hence, the Lagrangian fibration plays no role in this subsection.

Recall that we have fixed a reference line bundle [L0] ∈ PiclS . It is an ample line bundle on S.By the general theory of abelian varieties, there exists a dual ample line bundle L0 on the dualabelian variety S (defined only up to numerical equivalence. See, cf., [BL04, §14.4] for the notionof a dual polarization). The ample line bundles L0 and L0 induce polarization isogenies

ϕ : S → S, ϕ : S → S,

making their compositions the multiplication endomorphisms

[n+ 1] : Sϕ−→ S

ϕ−→ S, [n+ 1] : S

ϕ−→ S

ϕ−→ S. (5.8)

Since L0 has a polarization type (d1, d2), the dual line bundle L0 has a polarization type (d1, d2)as well. In particular, we have an isomorphism

ker ϕ ∼= (Z/d1 ⊕ Z/d2)⊕2. (5.9)

A closed point x on S defines a translation automorphism by x. Our notation for the translationautomorphism is

tx : S → S, y 7→ y + x.

A closed point ξ on S represents a numerically trivial line bundle on S. Considering ξ both as aclosed point on S and a line bundle on S can possibly lead to a confusion. Thus, we will write

Pξ : numerically trivial line bundle on S corresponding to ξ ∈ S.

With these notation in mind, we can explicitly realize the Aut(X)-action for the moduli of sheavesconstruction X .

Proposition 5.10. Let X be a Kumn-type moduli construction associated to a triple (S, l, s) inDefinition 5.7. Then

(1) We have an isomorphism

Aut(X) = ±1⋉ (x, ξ) ∈ S[n+ 1]× S[n+ 1] : ϕ(x) = 0, ϕ(ξ) = sx.

(2) With the above identification, the Aut(X)-action on X is defined by

(1, x, ξ).[E] = [t∗xE ⊗ Pξ], (−1, x, ξ).[E] = [t∗x([−1]∗E)⊗ Pξ],

where [−1] : S → S is the multiplication by −1 automorphism on S.

The rest of this subsection is devoted to the proof of Proposition 5.10. To start, we note thatYoshioka has already computed an explicit trivialization of Albanese morphism Alb :M → S×PiclS .

Yoshioka’s trivialization is obtained by the base change [n + 1] : S × PiclS → S × PiclS , which is adegree (n+ 1)8 isogeny. As we will see in a moment, this is not a minimal isogeny in the sense ofDefinition 4.3. Using the methods in Section 4, we first prove the morphism

φ : S × PiclS → S × PiclS , (y, [L]) 7→ (sy − ϕ(L ⊗ L−10 ), [L0 ⊗ Pϕ(y)]) (5.11)

is the minimal isogeny trivializing the Albanese morphism.

24 Y.-J. KIM

Proposition 5.12. The base change (5.11) is the minimal isogeny trivializing the Albanese mor-

phism Alb : M → S × PiclS in the sense of Definition 4.3.

Proof. Start from Yoshioka’s diagram [Yos01, §4.1] trivializing the Albanese morphism, which is acartesian diagram

X × (S × PiclS) M

S × PiclS S × PiclS

Φ′

pr2 Alb

[n+1]

. (5.13)

Here Φ′ : X × (S × PiclS) →M is defined to be a Galois etale morphism

Φ′([E], y, [L]) =[

t∗ϕ(L⊗L−1

0 )E ⊗ (L⊗ L−1

0 )⊗s ⊗ P−ϕ(y)

]

.

Note that our convention differs by sign to Yoshioka’s original paper, because Yoshioka’s dual linebundle L0 differs to ours by sign.

The Galois group of the base change [n+1] is the group of (n+1)-torsion points S[n+1]×S[n+1] ∼=(Z/n+ 1)⊕8. By Proposition 4.5, it acts on X × (S × PiclS) by translation on the second factor

(x, ξ).([E], y, [L]) = ([E], y + x, [L ⊗ Pξ]).

One computes the descent of this action to M via Φ′:

(x, ξ).[E] =[

t∗ϕ(ξ)E ⊗ Psξ−ϕ(x)

]

.

This is the S[n+1]× S[n+1]-action on X = Alb−1(0, [L0]) in Proposition 4.5. One sees this actionis not an effective action, and the kernel of the action is precisely

(x, ξ) ∈ S[n+ 1]× S[n+ 1] : ϕ(ξ) = 0, sξ − ϕ(x) = 0.

To kill the kernel and obtain an effective action, take a Galois quotient corresponding to thekernel (via the Galois correspondence). This is an isogeny ψ : S × PiclS → S × PiclS defined by

ψ(y, [L]) = (ϕ(L⊗ L−10 ), (L⊗ L−1

0 )⊗s ⊗ P−ϕ(y)).

One can check the morphism φ in (5.11) is precisely the isogeny making φ ψ = [n+ 1] (here oneneeds to use (5.8), but we omit the computation). The result is a factorization of (5.13) into theminimal isogeny

X × (S × PiclS) X × (S × PiclS) M

S × PiclS S × PiclS S × PiclS

Ψ

pr2

Φ

pr2 Alb

ψ φ

.

Here our new morphism Φ, Beauville’s minimal split covering ofM , turns out to have a better formthan the original Φ′:

Φ([E], y, [L]) =[

t∗yE ⊗ (L⊗ L−10 )]

. (5.14)

The claim follows.

Again thanks to Proposition 4.5, we have a canonical, effective and H2-trivial Gal(φ)-action onX . The Galois group Gal(φ) is captured by the kernel of φ, so we have

Gal(φ) = (x, ξ) ∈ S[n+ 1]× S[n+ 1] : ϕ(x) = 0, ϕ(ξ) = sx. (5.15)

This explains the isomorphism in Proposition 5.10. The Gal(φ)-action on the fiber X is obtained viathe description of Φ in (5.14). This explains how we obtained the group action in Proposition 5.10.


We can compute Gal(φ) more explicitly.

Lemma 5.16. Gal(φ) ∼= (Z/n+ 1)⊕4.

Proof. Let us compute the group (5.15) explicitly. The expression involves the abelian surfaces Sand its dual S, their (n + 1)-torsion points and their polarization isogenies ϕ and ϕ. Therefore,the expression is independent on the complex structure on S and the question is topological. Wemay fix polarization bases H1(S,Z) = Ze1, · · · , e4 and H1(S,Z) = Ze∗1, · · · , e

∗4 so that we can

identify S = (R/Z)e1, · · · , e4 and S = (R/Z)e∗1, · · · , e∗4. The polarization isogenies with respect

to them are

ϕ =

(

0 0 d1 00 0 0 d2

−d1 0 0 00 −d2 0 0

)

, ϕ =

(

0 0 −d2 00 0 0 −d1d2 0 0 00 d1 0 0

)

. (5.17)

Writing the coordinate (a1, · · · , a4) for S = (R/Z)4 and (b1, · · · , b4) for S = (R/Z)4, one explic-itly computes

Gal(φ) = (ai, bi)4i=1 ∈

(

1n+1Z/Z

)⊕8

: d1a1 = 0, d2b3 = sa1, · · · ∼= A⊕4,

where the abelian group A is defined by

A = (a, b) ∈ (Z/n+ 1)⊕2 : d1a = 0, d2b = sa.

Notice that gcd(d1, s) = 1 by the very assumption we had in Definition 5.7. Now A ∼= Z/n+ 1 bythe following simple computational lemma, and the desired isomorphism is proved.

Lemma 5.18. Let p, q, s be nonzero integers. Set m = pq and assume either gcd(p, s) = 1 orgcd(q, s) = 1. Then the abelian group

A = (a, b) ∈ (Z/m)⊕2 : pa = 0, qb = sa

is isomorphic to Z/m.

Proof. The group A is realized by the kernel of a homomorphism f : (Z/m)⊕2 → (Z/m)⊕2, f =(

p 0−s q

)

. Adjusting the bases of both the domain and codomain (i.e., performing elementary row

and column operations), the matrix can be transformed into(

1 00 pq

)

= ( 1 00 0 ). Here one needs the

assumption gcd(p, s) = 1 or gcd(q, s) = 1 to apply the Euclidean algorithm. The claim follows.

We have described Gal(φ) ∼= (Z/n+ 1)⊕4-action on X acting trivially on H2. Since Aut(X) ∼=Z/2 ⋉ (Z/n+ 1)⊕4, we still need an additional Z/2-part to describe. Fortunately, this is not hard

to guess. Construct an involution ι on X × (S × PiclS) by

ι([E], y, [L]) = ([[−1]∗E],−y, [L⊗20 ⊗ L−1]).

Because we are not relying on the general theory anymore, well-definedness needs to be checked inthis case (Chern class computation). The verification is easy and we omit the details.

The involution does not commute with the S× S-action on X× (S×PiclS), and this is the reasonwhy Z/2 should act on (Z/n+ 1)⊕4 nontrivially and leads to the semi-direct product. The actiondescends to M as a satisfying form

ι([E]) = [[−1]∗E],

and thus ι induces an action on X . It remains to prove ι acts on the second cohomology of X asthe identity. We have already proved in Proposition 4.5 that H2(M,Q) → H2(X,Q) is surjective.Hence we only need to prove ι acts on H2(M,Q) as the identity. This follows because ι is inducedfrom the automorphism [−1] on S, [−1] acts on H2(S,Q) trivially and finally the Hodge structure

26 Y.-J. KIM

H2(M,Q) is obtained by a tensor construction of H2(S,Q) by [B20]. This exhausts the entireAut(X)-action description on X and hence completes the proof of Proposition 5.10.

5.3. Automorphisms respecting the Lagrangian fibration. With Proposition 5.10 at hand,the proof of Proposition 5.2 becomes fairly straightforward. Any H2-trivial automorphism is of theform

f = (1, x, ξ) or (−1, x, ξ), x ∈ S[n+ 1], ξ ∈ S[n+ 1].

Let us first prove the automorphisms of the form f = (−1, x, ξ) never respects π. The action isgiven by f : [E] 7→ [t∗x([−1]∗E) ⊗ Pξ]. Recall π : X → B was by definition the (Fitting) supportmap Supp : [E] 7→ [Fitt0E]. The support of t∗x([−1]∗E) ⊗ Pξ is [−1]∗ SuppE − x. Therefore, iff respects π then we should get SuppE = [−1]∗ SuppE − x for all [E] ∈ X . In other words, wehave D = [−1]∗D − x for all D ∈ |L0|. Fix any 1

2x ∈ S with 2 ·(

12x)

= x. Then this condition isequivalent to saying that every divisor D ∈ |t∗

− 12xL0| is a symmetric divisor. A simple way to show

this is impossible is to choose an original reference line bundle [L0] ∈ PiclS so that none of t∗12xL0

becomes a symmetric line bundle for any x ∈ S[n+ 1]. Note that there are only 16 symmetric line

bundles in PiclS (e.g., [BL04, Lem 4.6.2]).Let us now consider the automorphisms of the form f = (1, x, ξ). It acts on X by f : [E] 7→

[t∗xE⊗Pξ]. Hence it respects π if and only if SuppE = SuppE−x. This means x = 0, so we arrive

Aut(X/B) = (1, 0, ξ) ∈ ±1⋉ (S[n+ 1]× S[n+ 1]) : ϕ(ξ) = 0 ∼= ker ϕ.

This is isomorphic to (Z/d1 ⊕ Z/d2)⊕2 by (5.9). This completes the proof of Proposition 5.2.

5.4. The polarization scheme of generalized Kummer varieties. This subsection will bedevoted to the proof of Proposition 5.4. Let us keep assume π : X → B is a Kumn-type moduliconstruction. The computations in this subsection are highly influenced by [Wie18, §6]. Recall from

§5.1 that we had a Fitting support morphism Supp : M → B over PiclS . Fix a point [L0] ∈ PiclSand consider the fibers of M and B over it. We obtain a morphism

Supp : Y → B,

where B = |L0| is a complete linear system and Y ⊂ M consists of torsion coherent sheaves E onS with ch(E) = v and c1(E) = c1(L0). The Kumn-type hyper-Kahler manifold X is obtained by afiber of the isotrivial fiber bundle c : Y → S.

Consider the universal family C → B of curves on S parametrizing effective divisors in B = |L0|.Since L0 is ample, by Bertini there exists a Zariski dense open subset B0 ⊂ B parametrizing thesmooth curves. The restriction of the universal family C0 → B0 becomes a smooth projective familyof curves. The following lemma is standard and we omit its proof.

Lemma 5.19. The morphism Supp : Y0 = Supp−1(B0) → B0 is isomorphic to the relative Picard

scheme of the universal family of curves PicdC0/B0→ B0 for d = s+ n+ 1.

The lemma in particular says Y0 → B0 is a torsor under the numerically trivial relative Picardscheme

J0 = Pic0C0/B0→ B0.

Since C0/B0 is a smooth projective family of curves, its relative Picard scheme J0 is a canonicallyprincipally polarized abelian scheme. As standard, we will call it a relative Jacobian of the familyand identify J0 = J0. Notice that we now have four different spaces over B0: P0, X0, J0 and Y0.The space X0 is a P0-torsor as usual, and we also have Y0 as a J0-torsor. Since P0 and J0 are


translation automorphism schemes of X0 ⊂ Y0, we have an inclusion P0 ⊂ J0. Our first goal is todescribe the quotient of this inclusion.

Proposition 5.20. There exists a short exact sequence of abelian schemes over B0:

0 P0 J0 S ×B0 0 . (5.21)

Proof. The universal family C0 → B0 is a subvariety of the product i : C0 → S×B0. This induces apullback morphism i∗ : S×B0 → J0 between their relative Picard schemes over B0. The morphismJ0 → S ×B0 can be constructed by the dual of it. Fiberwise, it is the morphism JC → S inducedby the universal property of the Albanese morphism applied to i : C → S.

We prove the kernel of the morphsim J0 → S × B0 is P0. The claim can be verified fiberwise.Fix a closed point [C] ∈ B0 corresponding to a smooth curve i : C → S. Over it, a closed pointof Y0 (resp., J0) is represented by a degree d line bundle L on C (resp., degree 0 line bundle Mon C). The J0-action on Y0 is given by [i∗M ].[i∗L] = [i∗(L ⊗M)]. Recall that X is a fiber of themorphism c : Y → S. Hence the abelian scheme P0 consists of translation automorphisms of J0invariant under the morphism c. Recall the definition of c in (5.6). A Riemann–Roch computationgives us

c([i∗(L⊗M)]) = c([i∗L])− Σi∗c1(M),

where c1(M) ∈ CH1(S), i∗ : CH1(S) → CH2(S) and Σ : CH2(S) → S(C) is a summation map.This proves the [i∗M ]-action on the fiber of Y0 is c-invariant if and only if Σi∗c1(M) = 0. The claimfollows by the following lemma, which is already proved in [Wie18, (6.8)].

Lemma 5.22. The morphism JC → S sends a closed point [M ] ∈ JC(C) to Σi∗c1(M) ∈ S(C).

The dual of (5.21) is automatically (e.g., [BL04, Prop 2.4.2]) a short exact sequence of abelianschemes

0 S ×B0 J0 P0 0 . (5.23)

In particular, P0 and S×B0 are both abealin subschemes of a bigger abelian scheme J0. The follow-ing proposition describes the polarization scheme K more explicitly for the moduli constructions.

Proposition 5.24. We have the following two additional descriptions of the polarization schemeK as B0-group schemes:

K = P0 ∩ (S ×B0) = ker(ϕ× id : S × B0 → S ×B0).

Proof. Fiberwise at a closed point [C] ∈ B0, the sequences (5.21) and (5.23) are short exact se-quences of abelian varieties

0 F JC S 0 , 0 S JC F 0 .

Here F = ν−1([C]) is a fiber of P0 and JC is the Jacobian of the curve C. The two abeliansubvarieties F and S of the principally polarized abelian variety JC are the so-called complementaryabelian subvarieties (see [Wie18, §6.4] and [BL04, §12.1]). In this case, we have an equality ([BL04,Cor 12.1.4])

ker(F → JC → F ) = F ∩ S = ker(S → JC → S).

We will soon prove in Lemma 5.25 that the composition S → JC → S is precisely the polarizationisogeny ϕ regardless of the choice of a closed point [C] ∈ B0. Given this, we obtain a sequence ofidentities of group schemes

ker(P0 → J0 → P0) = P0 ∩ (S ×B0) = ker(ϕ× id : S ×B0 → S ×B0).

28 Y.-J. KIM

From the last description and (5.9), this group scheme is a constant group scheme with fibers (Z/d1⊕Z/d2)

⊕2. The first description is describing the polarization scheme K; combine the uniqueness ofthe polarization in Theorem 3.1 and the computation of polarization types in Theorem 3.20. Theclaim follows.

Lemma 5.25. The composition S → JC → S is the polarization isogeny ϕ.

Proof. Denote by i : C → S the closed immersion. At the level of first homologies, the compositionS → JC → S becomes a Hodge structure homomorphism

H1(S,Z) = H1(S,Z)i∗−→ H1(C,Z)

i∗−→ H3(S,Z) = H1(S,Z).

Hence the composition is i∗i∗, which is the multiplication map by c1(OS(C)) ∈ H2(S,Z). Becausewe have chosen [C] in a complete linear system |L0|, it is a multiplication by c1(L0) = l.

Therefore, the question reduces to the following claim: the dual polarization ϕ : S → S is givenby l ∪ − : H1(S,Z) → H3(S,Z). Again choose polarization bases H1(S,Z) = Ze1, · · · , e4 andH1(S,Z) = H1(S,Z) = Ze∗1, · · · , e

∗4 as in Lemma 5.16. The polarization isogenies ϕ and ϕ have

the matrix forms (5.17). The ample class l is the skew-symmetric bilinear map ϕ : H1(S,Z) ⊗H1(S,Z) → Z considered as an element of H2(S,Z). Hence it is l = d1e

∗1 ∧ e

∗3 + d2e

∗2 ∧ e

∗4.

We can now explicitly compute the map l ∪− : H1(S,Z) → H3(S,Z):

e∗1 7→ d2e∗1 ∧ e

∗2 ∧ e

∗4, e∗2 7→ −d1e

∗1 ∧ e

∗2 ∧ e

∗3,

e∗3 7→ −d2e∗2 ∧ e

∗3 ∧ e

∗4, e∗4 7→ d1e

∗1 ∧ e

∗3 ∧ e

∗4.

The Poincare duality H1(S,Z) = H3(S,Z) yields the basis of H3(S,Z):

e∗2 ∧ e∗3 ∧ e

∗4, −e∗1 ∧ e

∗3 ∧ e

∗4, e∗1 ∧ e

∗2 ∧ e

∗4, −e∗1 ∧ e

∗2 ∧ e

∗3.

With respect to it, the matrix form of the multiplication coincides with precisely the matrix formof ϕ above. (Compare this lemma with [Wie18, Lem 6.14].)

Proof of Proposition 5.4. Proposition 5.24 already shows that the polarization scheme K is a con-stant group scheme with fibers (Z/d1 ⊕Z/d2)

⊕2. We need to prove this coincides with Aut(X/B)considered as global sections of the abelian scheme P0. Again by Proposition 5.24, we haveK = P0 ∩ (S ×B0). It is thus enough to show Aut(X/B) ⊂ S ×B0.

Recall from the description in Proposition 5.10 that f = (1, 0, ξ) ∈ Aut(X/B) ∼= ker ϕ actson Y by [E] 7→ [E ⊗ Pξ], where Pξ is the numerically trivial line bundle on S represented by

ξ ∈ ker ϕ ⊂ S. On Y0, closed points are of the form [E] = [i∗L] where L is a line bundle on asmooth curve i : C → S. Hence f acts on it by

f.[i∗L] = [i∗L⊗ Pξ] = [i∗(L⊗ i∗Pξ)].

This means the global section of J0 → B0 defined by f represents a line bundle [i∗Pξ] over [C] ∈ B0.

The inclusion S×B0 ⊂ J0 was by definition the pullback morphism of line bundles. Hence f definesin fact a global section ξ = [Pξ] ∈ S of the constant group scheme S ×B0.

6. The dual Lagrangian fibration of a compact hyper-Kahler manifold

Combining the previous results, we can prove the polarization scheme is precisely the groupAut(X/B) for known hyper-Kahler manifolds.


Theorem 6.1. Let π : X → B be a Lagrangian fibration of a compact hyper-Kahler manifold of

K3[n], Kumn, OG10 or OG6-type. Then the polarization scheme is a constant group scheme

K = Aut(X/B).

Proof. When X is of K3[n] or OG10-type, both the polarization scheme K and the global sectionsdefined by Aut(X/B) are the zero section of the abelian scheme P0. Hence the equality is trivial.When X is of OG6-type, lattice theory forces div(h) = 1 as shown in [MR21, Lem 7.1]. Thesecond item of Proposition 3.10 applies and we get an inclusion Aut(X/B) → K. CombiningTheorem 3.20 and 5.1, the inclusion is forced to be an equality fiberwise. Hence we get the globalequality Aut(X/B) = K.

When X is of Kumn-type, we have a2 | n+ 1 for a = div(h) by [Wie18]. If we have a = 1 (e.g.,when n+1 is square-free) then we can apply the same strategy we used for OG6 and conclude. If a >1 then we use the first item of Proposition 3.10. By [Wie18], any Lagrangian fibration of a Kumn-type hyper-Kahler manifold is deformation equivalent to a moduli construction in Definition 5.7.Thus the result follows by Proposition 5.4 with the fiberwise equality derived from Theorem 3.20and 5.1.

A direct consequence of this theorem is a promised compactification of the dual torus fibrationπ : X0 → B0.

Theorem 6.2. Let π : X → B be a Lagrangian fibration of a compact hyper-Kahler manifold of

K3[n], Kumn, OG10 or OG6-type. Then

X = X/Aut(X/B), π : X → B

defines a compactification of the dual torus fibration π : X0 → B0.

Proof. As explained in the introduction, we define the dual torus fibration by X0 = X0/K. Forknown deformation types, K = Aut(X/B) is a constant group scheme by Theorem 6.1. Therefore,the group scheme quotient X0/K over B0 becomes an honest quotient X0/Aut

(X/B). NoticeAut(X/B) acts not only on X0 but on the entire X . Therefore, we may consider the quotientX/Aut(X/B) and it gives a compactification of X0.

When X is of K3[n] or OG10-type, the group Aut(X/B) is trivial and hence X = X . WhenX is of Kumn or OG6-type, Aut(X/B) is nontrivial and hence X cannot be homeomorphic to X(their fundamental groups are different). We call the corresponding X the dual Kummer varietyand dual OG6, respectively.

Since the quotient is taken precisely by the H2-trivial automorphisms Aut(X/B), the spaceX has many satisfying properties. We provide an appendix B to collect properties of quotientspaces of X by H2-trivial automorphisms. Here we simply copy them for the special case of ourdual hyper-Kahler X. For definitions of a primitive symplectic orbifold and irreducible symplecticvariety used in the following proposition, see Appendix A.

Proposition 6.3. Keep the notation from Theorem 6.2. Then

(1) X is a compact primitive symplectic orbifold and also an irreducible symplectic variety.(2) The fundamental group and Fujiki constant of X are π1(X) ∼= Aut(X/B) and cX = 1/cX .

(3) H2(X,Q) and H2(X,Q) are Hodge isomorphic and Beauville–Bogomolov isometric.(4) The LLV algebras and Mumford–Tate algebras of X and X are isomorphic.(5) The pullback H∗(X,Q) → H∗(X,Q) is an injective map of LLV structures.

30 Y.-J. KIM

(6) If X → Def(X) is the universal deformation of X, then X/Aut(X/B) → Def(X) is the(locally trivial) universal deformation of X.

Proof. Everything is a direct consequence of Proposition B.2 and B.3. Only the first two itemsneed further explanations. For the first item, it is enough to show Xf has codimension ≥ 4 for allf ∈ Aut(X/B). We will see later in Lemma 7.5 that the fixed loci of H2-trivial automorphismsdeform when X deform. Hence we may prove this for any model in the deformation class on X . ForOG6, the fixed loci are computed in [MW17, §6]; they are either K3 surfaces or points. For Kumn,the fixed loci are computed in [Ogu20, Lem 3.5], and similarly one can deduce their codimension isalways ≥ 4. For the second item, simply notice Aut(X/B) has order c2X in all cases.

The proposition shows in particular the local deformation behavior and period domains of Xand X are identical. Therefore, one can still apply the method in [GTZ13, §2] at the level of perioddomains and obtain similar conclusions for all known deformation types of hyper-Kahler manifolds.One subtlety here is that the quotient construction works for any deformation X ′ of X , even if X ′

does not admit any Lagrangian fibration; the quotient X ′/Aut(X/B) is still well-defined becausewe considered Aut(X/B) as an abstract subgroup of the group Aut(X). The local universaldeformation space of the Lagrangian fibration π : X → B is a hyperplane Def(X,H) ⊂ Def(X)(see [Mat16]). If we choose a deformation of the Lagrangian fibration [π′ : X ′ → B′] ∈ Def(X,H)then we can say π′ : X ′/Aut(X/B) → B′ is the dual Lagrangian fibration of π′ : X ′ → B′.

7. Example: the dual Kummer fourfolds

To illustrate the geometry of dual Lagrangian fibrations more concretely, we now focus on thesimplest nontrivial case of Theorem 6.2: when X is of Kum2-type. Throughout, we let X to be aKum2-type hyper-Kahler fourfold and π : X → B = P2 its Lagrangian fibration. We will use allthe results in previous sections without mentioning them explicitly. We write for simplicity

G = Aut(X), G = Aut(X/B).

There exist isomorphisms G ∼= Z/2 ⋉ (Z/3)⊕4 and G ∼= (Z/3)⊕2. Again for simplicity, we call

f ∈ G a translation if f does not contain a Z/2-part, and call f an involution if f has a nontrivialZ/2-part. If f 6= id is a translation (resp., involution) then it has order 3 (resp., order 2). There

are precisely 81 translations and 81 involutions. The subgroup G ⊂ G consists of 9 translationsrespecting the Lagrangian fibration. We define the dual Kummer fourfold by π : X = X/G → B.The main result of this section will be Proposition 7.2. It collects some more precise geometric andcohomological descriptions of X. We believe the method in this section can be similarly applied tothe OG6-type and higher dimensional Kumn-types.

We will use the notion of the LLV structure to describe the cohomology of X . To do so, we firstneed to review the LLV structure of the generalized Kummer fourfolds (following [GKLR]). Recallthat the Beauville–Bogomolov quadratic space of X (and hence X) is isomorphic to (H2(X,Q), q) ∼=U⊕3 ⊕ 〈−6〉 where U denotes the hyperbolic plane ( 0 1

1 0 ). For simplicity, we denote V = H2(X,Q)and its Mukai completion by

(V, q) = (V , q)⊕ U(

∼= U⊕4 ⊕ 〈−6〉)

.

Set g ∼= so(V, q) and g ∼= so(V , q) the LLV algebra and reduced LLV algebra of X (and X). Itis a split semisimple Q-Lie algebra. Associated to any dominant weight µ of g, there exists an


irreducible g-module Vµ over Q. The LLV structure of X is explicitly computed in [LL97, (4.7)];we have an isomorphism of g-modules

H∗(X,Q) ∼= V(2) ⊕ 80Q⊕ V( 12, 12, 12, 12).

It is sometimes convenient to consider the reduced LLV structure on the fixed degree cohomologiesHk(X,Q). The LLV structure restricts to the reduced LLV structure on the middle cohomology:there exists an isomorphism of g-modules

H4(X,Q) ∼= V(2) ⊕ 81Q. (7.1)

Let us now state the main result of this section.

Proposition 7.2. Let π : X → B be a Lagrangian fibration of a Kum2-type hyper-Kahler fourfoldand π : X → B its dual fibration. Then (in addition to Proposition 6.3)

(1) X has precisely 18 isolated cyclic quotient singularities of type 13 (1, 1, 2, 2).

(2) X (and any of its deformation) contains 9 smooth K3 surfaces. Each of them passes through2 singularities of X. The image of each K3 surface is a line in B = P2.

(3) The LLV decomposition of the cohomology of X is

H∗(X,Q) ∼= V(2) ⊕ 8Q⊕ V( 12, 12, 12, 12).

The rest of the section will be devoted to the proof of Proposition 7.2. The following propositionclaims any Lagrangian fibered Kum2-type hyper-Kahler manifolds are deformation equivalent. Theidea originates from the results of Markman (e.g., [Mar14, Prop 1.7]).

Lemma 7.3. Any Lagrangian fibration of a Kum2-type hyper-Kahler fourfold π : X → B is defor-mation equivalent to each other.

Proof. The polarization type of π is (1, 3) by Theorem 3.20. Setting h ∈ H2(X,Z) to be anassociated cohomology class of π∗OB(1), its divisibility div(h) is 1 by [Wie18, Thm 1.1]. We cannow imitate the method of [MR21, §7]. The lattice theory result in [MR21, Lem 2.6] forces anytwo primitive isotropic elements h, h′ with divisibility 1 in H2(X,Z) are monodromy equivalent.We can imitate the proof of [MR21, Thm 7.2] (or the proof of [Mar14, Prop 1.7]) and show thatany pairs (X,H) with a primitive isotropic H with divisibility 1 are deformation equivalent. Thisproves the claim.

Thanks to this proposition, we can often specialize our discussion to a single model. Our explicitmodel for Lagrangian fibered Kum2-type hyper-Kahler manifolds is the following example presentedin [Mat15, §2]. Let E and E′ be elliptic curves and S = E′ × E be an abelian surface. Consider acommutative diagram

S[3]

E′ × E(3)

E′ × E

(Σpr1, pr2)

Alb

id×Σ

, (7.4)

where pr1 : S[3] → (E′)(3) and pr2 : S[3] → E(3) are the coordinate projections and Σ are thesummation maps. By the discussion we had in Section 5, this is an isotrivial family of Lagrangianfibered hyper-Kahler manifolds of Kum2-type. The advantage of this construction to the moduliconstruction in Section 5 is that this gives us an honest generalized Kummer fourfold, and thus wecan use the computational results in [HT13] and [Ogu20].

Let us also recall some known facts about the fixed loci of H2-trivial automorphisms.

32 Y.-J. KIM

Lemma 7.5. Let X be a compact hyper-Kahler manifold and H ⊂ Aut(X) any subgroup. If X ′

is deformation equivalent to X then (X ′)H is deformation equivalent to XH .

Proof. Let p : X → Def(X) be a universal deformation of X . Since H acts fiberwise on p, themorphism XH → Def(X) gives a family of fixed loci (Xt)

H . Because H is a finite group acting on acomplex manifold X , its fixed locus XH is a complex manifold. Similarly, each (Xt)

H is a complex(symplectic) manifold. Hence XH → Def(X) is a smooth proper family and the claim follows.

Lemma 7.6. Let X be a Kum2-type hyper-Kahler manifold and f ∈ G its H2-trivial automorphism.

(1) If f is an involution then its fixed locus Xf is a disjoint union of a single point and a K3surface.

(2) If f 6= id is a translation then its fixed locus Xf consists of 27 points.

Proof. When X is an honest generalized Kummer fourfold the statements were proved in [Ogu20,Lem 3.5] and [HT13, Thm 4.4], even in greater generality. Now if we deform X then the fixed locusXf also deforms by Lemma 7.5.

Following [HT13, Thm 4.4], any Kum2-type hyper-Kahler manifold must always contain 81 K3surfaces obtained by the fixed loci of 81 involutions (this was first observed in [KV98, §6]). The81 K3 surfaces are related by 81 translations, and represent the 81 trivial reduced LLV classes in(7.1). With these backgrounds, we can begin the proof of Proposition 7.2.

Proof of Proposition 7.2. Realize X by a double quotient X = (X/f)/f ′ where f, f ′ are two trans-lations generating G ∼= (Z/3)⊕2. Since Xf consists of 27 points, the first quotient X/f has precisely27 Z/3-quotient singularities. Taking the second quotient, we introduce additional 9 singularities

from the fixed locus (X/f)f′

and at the same time, identify 27 singularities together into 9 singu-larities in X. This proves there are precisely 18 Z/3-quotient singularities.

Any symplectic Z/3-quotient singularity must be of type 13 (1, 1, 2, 2). By [Pri67, Prop 6], the

Z/3-action is locally biholomorphic to a symplectic linear action on C4 around 0. Its eigenvaluesare either 1, ζ and ζ2 where ζ is the third primitive root of unity. In this case, 1 cannot arisebecause the fixed locus of the action should be the origin. Hence there are five possibilities of thelinear action up to conjugate, and one easily checks diag(ζ, ζ, ζ2, ζ2) is the only symplectic linearmap among them (for some symplectic form).

For the second item, notice that the 9 K3 surfaces appearing in X is a shadow of 81 K3 surfaces inX mentioned above. For the remaining statements, we deform the Lagrangian fibration (Lemma 7.3)and assume we are in the construction (7.4). In this case, the 81 K3 surfaces are explicitly describedin [HT13, Thm 4.4]. One can easily check the image of each K3 surface under π can be consideredas a sublinear system in P2, so it is a line.

Finally, for the last item, recall H∗(X,Q) = H∗(X,Q)G. The translations in G act trivially onH2(X,Q) by definition and trivially on H3(X,Q) by the computations in [Ogu20, §3]. Hence weonly need to prove H4(X,Q)G ∼= V(2) ⊕ 9Q. The Verbitsky component is preserved by G, so V(2)is G-invariant. Again recall the 81 trivial reduced LLV classes in H4(X,Q) were represented by 81K3 surfaces which are bound by 81 translation automorphisms. Since only 9 of them survives inX, this proves the claim.

Appendix A. Various notions of singular hyper-Kahler varieties

Many of the important properties of compact hyper-Kahler manifolds have been generalizedto singular settings. However, there seem to be many natural frameworks to define the notion


of singular hyper-Kahler varieties. To make our discussion less ambiguous, here we collect somedefinitions of singular hyper-Kahler varieties and compare them. Our main references are [BL18],[Sch20a] and [Men20].

If X is a normal complex space, then its sheaf of reflexive k-forms is defined to be the reflexive

closure of the sheaf of k-forms Ω[k]X = (ΩkX)∨∨, or equivalently Ω

[k]X = j∗Ω

kXreg

where j : Xreg → X

is the smooth locus of X . A quasi-etale morphism is a morphism etale outside of a codimension≥ 2 closed subvariety.

Definition A.1 ([BL18, Def 3.1], [Sch20a, Def 1], [Men20, Def 3.1]). Let X be a compact normal

Kahler space and σ ∈ H0(X,Ω[2]X ) a reflexive 2-form.

(1) (X, σ) is called a symplectic variety if X has rational singularities and σ is nondegenerateon Xreg.

(2) X is called a primitive symplectic variety if

H0(X,Ω[1]X ) = 0, H0(X,Ω

[2]X ) = Cσ,

and (X, σ) is a symplectic variety.(3) X is called an irreducible symplectic variety if it is a primitive symplectic variety with the

following condition: for any finite quasi-etale cover f : X ′ → X , we have

H0(X ′,Ω[2k+1]X′ ) = 0, H0(X ′,Ω

[2k]X′ ) = C · f∗σ[k] for k ≥ 0.

(4) X is called a Namikawa symplectic variety if it is a Q-factorial and terminal primitivesymplectic variety.

(5) X is called a primitive symplectic orbifold if it is Namikawa symplectic with only finitequotient singularities.

We have a series of implications

primitive symplecitc orbifold Namikawa symplectic

irreducible symplectic primitive symplectic symplectic.

Eventually, the dual hyper-Kahler variety X in Theorem 1.1 will be both a primitive symplecticorbifold and an irreducible symplectic variety (Proposition 6.3). Hence all of the discussions hereapply.

Many of the interesting properties of compact hyper-Kahler manifolds generalize to their singularanalogues, especially to primitive symplectic varieties. We highlight some of their properties thatwill be useful to our discussion. Let X be a primitive symplectic variety of dimension 2n.

• The normailzation of the singular locus Xsing is again symplectic [Kal06]. In particular,Xsing is always even dimensional.

• There exist a notion of the Beauville–Bogomolov form and Fujiki constant of X , so thatthe Fujiki relation (1.3) holds [Sch20a, Thm 2] [BL18, Prop 5.20].

• X is Namikawa symplectic if and only if it is Q-factorial and codimXsing ≥ 4 [Nam01][BL18, Thm 3.4].

• X is smooth if and only if it is a compact hyper-Kahler manifold [Sch20b].• Every morphism π : X → B with connected fibers to a normal base B (with 0 < dimB <2n) is a Lagrangian fibration [Sch20a, Thm 3]. That is, all the irreducible components ofthe fibers of π are Lagrangian subvarieties of X .

34 Y.-J. KIM

• The Hodge structure H2(X,Z) is pure [Sch20a, Thm 8] [BL18, Cor 3.5]. If X is a primitivesymplectic orbifold, then the full cohomology H∗(X,Q) is a pure Hodge structure.

• There exists a universal locally trivial deformation X → Def lt(X) over a smooth complex

germ Def lt(X) of dimension h1,1(X) [BL18, Thm 4.7]. If X is Namikawa symplectic, thenany deformation is automatically locally trivial [Nam06].

• The local Torelli theorem holds for Def lt(X). In fact, global Torelli theorem holds in asuitable form [BL18].

We will use these facts in Section 6 and Appendix A, without mentioning them explicitly.

Appendix B. Quotient of a hyper-Kahler manifold by H2-trivial automorphisms

Let X be a compact hyper-Kahler manifold and Aut(X) the finite group of H2-trivial automor-phisms. Throughout the appendix, we always let

G ⊂ Aut(X)

to be any subgroup and write

p : X → X = X/G. (B.1)

The goal of this appendix is to gather basic geometric and cohomological properties of the quo-tient X. Note that Lagrangian fibrations play no role in this appendix. The main results areProposition B.2 and B.3.

Proposition B.2. Consider the quotient (B.1) of a compact hyper-Kahler manifold X.

(1) The morphism p is a finite quasi-etale symplectic quotient.(2) X is a Q-factorial irreducible symplectic variety. If we further assume codimXf > 2 for

all f ∈ G, then it is also a primitive symplectic orbifold.(3) π1(X) ∼= G. In particular, X is never smooth unless G is trivial.(4) If X → Def(X) is the universal deformation of X then the quotient X/G→ Def(X) becomes

the universal locally trivial deformation of X.

The quotient X = X/G being an irreducible symplectic variety, its behavior is intimately relatedto its (second) cohomology. To talk about the precise cohomological behavior of X , we first needto fix the Beauville–Bogomolov form; the Beauville–Bogomolov form is a priori only defined up toscalar. We define a symmetric bilinear form qX : H2(X,Z)⊗H2(X,Z) → Z by

qX(x, y) = qX(p∗x, p∗y) for x, y ∈ H2(X,Z).

Be aware that qX may be a non-primitive bilinear form with this definition.

Proposition B.3. Notation as above.

(1) qX is a Beauville–Bogomolov form of X. The Fujiki constant of X is cX = cX/|G|.(2) The pullback

p∗ : H2(X,Z)/(torsion) → H2(X,Z)

is an injective Hodge structure homomorphism and a Beauville–Bogomolov isometry. It isan isomorphism over Q.

(3) The LLV algebra of X and X are canonically isomorphic. Denoting them by g, the pullback

p∗ : H∗(X,Q) → H∗(X,Q)

is an injective g-module homomorphism.


(4) For all k, the special Mumford–Tate algebra of Hk(X,Q) is isomorphic to that of H2(X,Q).As a consequence, any g-module decomposition of H∗(X,Q) is a pure Hodge structuredecomposition.

Again note that the subgroup G ⊂ Aut(X) was taken arbitrary. Hence we have a familyof irreducible symplectic varieties corresponding to each subgroups of Aut(X). That is, we get aGalois correspondence between the subgroupsG ⊂ Aut(X) and the symplectic quotients X = X/Gwith the same rational Beauville–Bogomolov forms. In particular, their deformation behaviors areall identical.

The rest of this appendix is devoted to the proof of Proposition B.2 and B.3. Most of the proofswill be straightforward so we will be brief.

Proof of Proposition B.2: Part 1. Let us present the proof of the theorem without the second item.The group G acts trivially on H2(X,Z), so it acts symplectically on X . Hence p is a symplecticquotient. The ramified locus of p is contained in the union of the fixed loci

⋃

f∈GXf , which is of

codimension ≥ 2. This means p is quasi-etale.The fundamental group is invariant under codimension ≥ 2 modifications. Hence X = X/G has

the fundamental group G. If X becomes smooth then it a smooth irreducible symplectic variety.By [Sch20b], X becomes a hyper-Kahler manifold. In particular, X is simply connected and henceG becomes trivial.

The last item follows directly from [Fuj83, Thm 3.5, Lem 3.10]. Since G acts on X holomor-phically and trivially on H2(X,Z), X → Def(X) equipped with a G-action is the universal defor-mation of the pair (X,G). Once we have a universal deformation of the pair (X,G), the quotientX/G→ Def(X) is the locally trivial universal family of X/G.

Lemma B.4. Let (X, σ) be a compact symplectic variety and f : X ′ → X a finite quasi-etalemorphism. Then (X ′, f∗σ) is a compact symplectic variety.

Proof. By [KM98, Prop 5.20] or [GKP16, Rmk 3.4], X ′ is Gorenstein and canonical. Therefore, it

has rational singularities by [KM98, Cor 5.24]. Now f∗σ ∈ H0(X ′,Ω[2]X′) is a symplectic form in

codimension 1 as f is etale in codimension 1. The claim follows.

Proof of Proposition B.2: Part 2. We prove the second item here. As a finite quotient of a smoothvariety X , the space X is certainly Q-factorial and has quotient singularities. If the additionalassumption on the fixed loci holds, then codimXsing ≥ 4 and it becomes Namikawa symplectic.

To prove X is irreducible symplectic, we follow the argument of Matsushita [Mat15, Lem 2.2].Let f : Y → X be an arbitrary finite quasi-etale morphism. Consider the diagram

Y X

Y X

g

q p

f

,

where Y is the normalization of the fiber product X ×X Y . We claim g and q are finite quasi-etale. The finiteness is clear, so we concentrate on their quasi-etaleness. Notice that the quasi-etaleproperty is stable under base change, so we need to prove the normalization in this case is quasi-etale. But notice that X is smooth and f is quasi-etale, so that X ×X Y is smooth in codimension1. Hence the normalization of it is in fact isomorphism in codimension 1. This proves g and q arequasi-etale.

NowX is smooth, Y is normal, and g : Y → X is finite quasi-etale. By the Zariski–Nagata puritytheorem of the branch locus (e.g., [Sta, Tag 0BMB]), this forces g to be etale. The hyper-Kahler

36 Y.-J. KIM

manifold X is simply connected, so this means Y must be a disjoint union of several isomorphiccopies of X . Let us fix a connected component Y0 of Y . It is a hyper-Kahler manifold isomorphicto X .

Consider the morphism q restricted to the connected component q : Y0 → Y . It is a finitequasi-etale morphism. Note that the target Y is canonical (Lemma B.4), so [GKKP11, Thm 4.3]

guarantees the existence of a reflexive pullback q∗ : H0(Y ,Ω[k]

Y) → H0(Y0,Ω

[k]Y0). Since q is quasi-

etale, this morphism is injective. But recall that Y0 ∼= X is a hyper-Kahler manifold, so this forcesY to satisfy the dimension condition of the definition of irreducible symplectic varieties. This provesX is an irreducible symplectic variety.

Proof of Proposition B.3. The following sequence of identities proves qX is the Beauville–Bogomolovform with the Fujiki constant cX = cX/|G|:

∫

X

x2n =1

|G|

∫

X

(p∗x)2n =cX|G|

qX(p∗x)n =cX|G|

qX(x)n.

Since X is a compact Kahler orbifold, its rational singular cohomology admits a well-behaved pureHodge structure (e.g., [PS08, §2.5]) and p∗ : H∗(X,Q) → H∗(X,Q) is an injective Hodge structurehomomorphism with the image H∗(X,Q)G. In particular, p∗ is an isomorphism in degree 2.

To prove H∗(X,Q) = H∗(X,Q)G is closed under the g-action, it is enough to prove the G-actionand g-action on H∗(X,Q) commutes. Recall that the LLV structure is diffeomorphism invariant.In other words, if f : X1 → X2 is a diffeomorphism between two compact hyper-Kahler manifoldsthen we have

f∗(Lx(ξ)) = Lf∗x(f∗ξ), f∗(Λx(ξ)) = Λf∗x(f

∗ξ)

for any x ∈ H2(X2,Q) and ξ ∈ H∗(X2,Q). Here Lx and Λx are Lefschetz and inverse Lefschetzoperators associated to x. If we set X1 = X2 = X and f ∈ G to be an H2-trivial automorphismthen this means f∗ commutes with the operators Lx and Λx. That is, G commutes with g.

To obtain the results about the Mumford–Tate algebras one imitates the method used in [GKLR,§2] and deduces f ∈ g for f a Weil operator on the cohomology H∗(X,Q) (which is the restrictionof Weil operator on H∗(X,Q)). This proves all the special Mumford–Tate algebra of Hk(X,Q) arethe same and even same for that of H2(X,Q).

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