Dual Lagrangian fibration of known HK]The dual Lagrangian fibration of known hyper-Kähler manifolds

Y.-J. Kim]Yoon-Joo Kim
Stony Brook University, Department of Mathematics, Stony Brook, NY 11794-3651

Given a Lagrangian fibration $\pi : X \to \mathbb{P}^n$ of a compact hyper-Kähler manifold of $\text{K3}^{[n]}$, $\text{Kum}_n$, OG10 or OG6-type, we construct a natural compactification of its dual torus fibration. Specifically, this compactification is given by a quotient of $X$ by $\text{Aut}^{\circ}(X / \mathbb{P}^n)$, the group of automorphisms acting trivially on the second cohomology and respecting the Lagrangian fibration. It is a compact hyper-Kähler orbifold with identical period mapping behavior as $X$.

§ INTRODUCTION

Mirror symmetry is a highly active research area in modern mathematics. Among various approaches to it, the Strominger–Yau–Zaslow (SYZ) conjecture predicts mirror symmetry can be realized by dualizing certain Lagrangian torus fibrations. More specifically, a compact Calabi–Yau manifold $X$ approaching a large complex structure limit point is expected to admit a special Lagrangian torus fibration, and its mirror dual space $\check X$ should be an appropriate compactification of the dual of this torus fibration. One case where this picture is better understood is for K3 surfaces; K3 surfaces are hyper-Kähler, and so admit a $2$-sphere family of complex structures. This allows one to hyper-Kähler rotate the complex structure and transform a special Lagrangian fibration into a holomorphic elliptic fibration. Since elliptic curves are self-dual, one concludes that K3 surfaces are topologically self-mirror dual. This becomes the starting point of mirror symmetry for K3 surfaces; the interested reader may consult <cit.>, <cit.> or <cit.> for more details.

A compact hyper-Kähler manifold is a higher dimensional generalization of a K3 surface. It is a simply connected compact Kähler manifold with a unique global holomorphic symplectic form. In <cit.>, mirror symmetry for K3 surfaces is generalized to certain compact hyper-Kähler manifolds (see <cit.> for a different approach). As for K3 surfaces, a hyper-Kähler rotation can transform a special Lagrangian fibration into a holomorphic Lagrangian fibration $\pi : X \to B$. Gross–Tosatti–Zhang studied such holomorphic Lagrangian fibrations $\pi$ under the following two conditions: (1) $\pi$ admits at least one section, and (2) all the torus fibers of $\pi$ are principally polarized. In this setting, they showed $X$ is topologically self-dual and described its mirror symmetry at the level of the period domain. In this paper, we are interested in the duality of Lagrangian fibered hyper-Kähler manifolds without the two conditions above. Our main result is the construction of a compactification of the dual torus fibration for all currently known examples of hyper-Kähler manifolds. However, we do not attempt to describe their mirror symmetry further; we hope to return to this topic in the future.

Let $X$ be a compact hyper-Kähler manifold of dimension $2n$. By a Lagrangian fibration of $X$ we mean a holomorphic surjective morphism $\pi : X \to B$ such that it has connected fibers and $B$ is a complex manifold with $0 < \dim B < 2n$. By <cit.> and <cit.>, the base $B$ is necessarily isomorphic to the projective space $\PP^n$. It is well-known that any smooth fiber of $\pi$ is a complex Lagrangian subtorus of $X$, so by restricting the Lagrangian fibration $\pi$ to its smooth locus $B_0 \subset B$ we get a torus fibration $\pi_0 : X_0 \to B_0$, a smooth proper family of complex tori.

We want to construct the dual torus fibration $\check \pi_0$, and ultimately its compactification $\check \pi$. A construction of the dual torus fibration as the relative Picard scheme was proposed in <cit.>. While this definition behaves well when $\pi_0$ admits a section, we would like to propose a slightly different approach to handle the case when $\pi_0$ has no section. Specifically, our starting point is the standard fact that all the torus fibers of $\pi_0$ are canonically polarized (e.g., Voisin's argument in <cit.>). That is, each torus fiber $F$ admits a natural isogeny $F \to \check F$ to its dual torus. Denoting the kernel of this isogeny by $G$, we obtain an isomorphism $\check F \cong F / G$. The idea is to make this discussion global over the entire base $B_0$. In <Ref>, we will attach a canonically polarized abelian scheme $P_0 \to B_0$ to $\pi_0$ so that $X_0$ becomes a $P_0$-torsor (this combines the results of Arinkin–Fedorov and van Geemen–Voisin). Let $K$ be the kernel of the canonical polarization $P_0 \to \check P_0$ over $B_0$. The group scheme $K$ then acts on both $P_0$ and $X_0$. After taking the associated quotients, we recover on the one hand the dual abelian scheme $\check P_0 \cong P_0 / K$, and on the other hand obtain a new space

\[ \check X_0 = X_0 / K, \qquad \check \pi_0 : \check X_0 \to B_0 .\]

By construction, $\check X_0$ is a $\check P_0$-torsor. Therefore, it is a smooth proper family of complex tori which are fiberwise dual to the original fibration $\pi_0$. This is our definition of the dual torus fibration $\check \pi_0$.

It is important to notice that $K$ is only a finite étale group scheme over $B_0$. One can think of this as the total space of a local system on $B_0$; 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 of this paper.

Let $\pi : X \to B$ be a Lagrangian fibration of a compact hyper-Kähler manifold. Assume $X$ is of $\text{K3}^{[n]}$, $\Kum_n$, OG10 or OG6-type. Then

* The group scheme $K$ is a constant group scheme with fiber

\[ \Aut^{\circ} (X/B) = \{ f \in \Aut(X) : \pi \circ f = \pi, \ \ f^* \mbox{ acts as the identity on } H^2 (X, \ZZ) \} .\]

* The quotient map

\[ \check X = X / \Aut^{\circ} (X/B), \qquad \check \pi : \check X \to B \]

compactifies the dual torus fibration $\check \pi_0$.

* $\check X$ is a compact hyper-Kähler orbifold and $\check \pi$ is a Lagrangian fibration. Moreover, $\check X$ has the same period mapping/deformation behavior as $X$.[(Note added in proof) While circulating the preliminary version of our manuscript, we have learned about the paper <cit.>. In Theorem 24 of the paper, an alternative construction for the dual Lagrangian fibration of certain $\Kum_n$-type hyper-Kähler manifolds is announced (without details).]

If $X$ is of $\text{K3}^{[n]}$ or OG10-type, then in fact the group $\Aut^{\circ} (X/B)$ is trivial and thus these hyper-Kähler manifolds are self-dual, as studied in <cit.>. On the other hand, if $X$ is of $\Kum_n$ or OG6-type, then $\Aut^{\circ}(X/B)$ is nontrivial and $\check X$ is not even homeomorphic to $X$. Still, one can interpret the third item as a certain self-duality at the level of period domains. Note that $\check X$ is a global quotient of $X$ by a specific group $\Aut^{\circ}(X/B)$ acting trivially on $H^2 (X, \ZZ)$. As a result, the second rational cohomology of $\check X$ and $X$ are canonically isometric as Beauville–Bogomolov quadratic spaces. The higher cohomology of $\check X$ may be strictly smaller than that of $X$ by <cit.>, but they are still tightly connected via their Looijenga–Lunts–Verbitsky (LLV) structures (see <cit.>, <cit.> and <cit.>). Finally, the singularities of $\check X$ are quotient singularities of high codimension ($\ge 4$), so they do not admit any symplectic resolutions. Thus, $\check X$ is truly a singular hyper-Kähler variety. For the reader's convenience, we briefly recall the notion of a singular hyper-Kähler variety and of a Lagrangian fibration for such a variety in <Ref>.

<cit.> and <cit.> introduced an explicit geometric construction of certain $4$-dimensional Lagrangian fibered hyper-Kähler orbifolds, and realized their dual Lagrangian fibrations using the same construction. It would be interesting to find a connection between their results and our perspective.

There are two key ingredients for our proof of <Ref>: the group $\Aut^{\circ}(X/B)$ and the notion of a polarization type. The definition of the group $\Aut^{\circ}(X/B)$ is inspired by the similar group $\Aut^{\circ}(X)$, which has already played an important role in the theory of hyper-Kähler manifolds. The two main properties of $\Aut^{\circ} (X)$ are its finiteness <cit.> and deformation invariance <cit.>. Furthermore, $\Aut^{\circ}(X)$ is computed for all known deformation types of hyper-Kähler manifolds (see <cit.>, <cit.> and <cit.>). We prove similar results for the group $\Aut^{\circ}(X/B)$: it is finite abelian (<Ref>) and deformation invariant (<Ref>). We also compute $\Aut^{\circ}(X/B)$ for all known deformation types in <Ref>. The idea of considering the polarization type of the fibers of $\pi_0$ has long been used, but only recently comprehensively studied in <cit.>. We relate the polarization type to the study of our group scheme $K$ in this work.

§.§ Structure of the paper

In <Ref>, we prove the group $\Aut^{\circ} (X/B)$ is deformation invariant on the Lagrangian fibration $\pi$. This is inspired by Hassett–Tschinkel's proof of deformation invariance of $\Aut^{\circ}(X)$ in <cit.>. In <Ref>, we start by attaching an abelian scheme $P_0$ to any Lagrangian fibration of a hyper-Kähler manifold: $P_0$ is the identity component of the relative automorphism scheme of $\pi$. There exists a unique primitive polarization $\lambda$ on $P_0$ so that we can define its kernel group scheme $K$. We then reinterpret the automorphisms in $\Aut^{\circ}(X/B)$ as certain global sections of $K$. This section also discusses the notion of a polarization type. In essence, the polarization type is the study of a single fiber of the group scheme $K$.

The goal of <Ref> is twofold. First, we compute the group $\Aut^{\circ} (X/B)$ for all currently known deformation types of hyper-Kähler manifolds. Second, we prove the equality $K = \Aut^{\circ}(X/B)$ in the special case of a certain construction of $\Kum_n$-type hyper-Kähler manifolds. The material here will be mostly concrete computations. Still, a systematic approach would be beneficial and reduce the amount of computation. We briefly discuss this approach in <Ref>. In <Ref>, we prove the main result of this article: there exists a natural compactification of the dual torus fibration for all currently known deformation types of hyper-Kähler manifolds. In <Ref>, we give an illustration of the geometry and cohomology of $\check X$ when $X$ is of $\Kum_2$-type.

We provide two appendices. <Ref> contains various definitions of singular hyper-Kähler varieties appearing in the literature. In <Ref>, we discuss certain special quotients of compact hyper-Kähler manifolds. The quotient $\check X = X / \Aut^{\circ}(X/B)$ is a special instance of this more general set-up.

§.§ Notation and conventions

In this paper, a hyper-Kähler manifold $X$ will be assumed compact, but not necessarily projective unless stated explicitly. When $X$ further admits a Lagrangian fibration $\pi : X \to B$, it is helpful to keep in mind that the following three conditions are equivalent:

* $X$ is projective;

* $\rho(X) \ge 2$; and

* $\pi$ admits at least one rational multisection.

Indeed, the first two conditions are equivalent by a direct application of Huybrechts's projectiveness criterion in <cit.>. The implication (3) $\Rightarrow$ (1) is shown in <cit.>, and the implication (1) $\Rightarrow$ (3) is a general scheme-theoretic fact that any smooth morphism between algebraic varieties admits an étale local section.

Any Lagrangian fibration $\pi : X \to B$ in this paper will always have base $B = \PP^n$ since we are assuming $B$ is smooth and $0 < \dim B < 2n$ (see <cit.> and <cit.>). It is conjectured that only the normalness of $B$ would imply $B \cong \PP^n$. 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 $\pi$ admits a section <cit.>, $X$ is of $\text{K3}^{[n]}$ or $\Kum_n$-type <cit.> <cit.>, or when $\dim X \le 4$ <cit.> <cit.>.

Assume $X$ has dimension $2n$. The Beauville–Bogomolov form and the Fujiki constant of $X$ are a unique primitive symmetric bilinear form $q : H^2 (X, \ZZ) \otimes H^2 (X, \ZZ) \to \ZZ$ and a positive rational number $c_X$ satisfying the Fujiki relation

\begin{equation} \label{eq:fujiki relation}
\int_X x^{2n} = c_X \cdot \frac{(2n)!}{2^n \cdot n!} \cdot q(x)^n \qquad\mbox{for}\quad x \in H^2 (X, \ZZ) .
\end{equation}

The Fujiki constant is computed for all currently known deformation types of hyper-Kähler manifolds: (1) $c_X = 1$ for $\text{K3}^{[n]}$ or OG10-type, and (2) $c_X = n+1$ for $\Kum_n$ or OG6-type (see <cit.> and <cit.>). In practice, we will mostly need a stronger version of the Fujiki relation, which follows from the polarization process

\[ \int_X x_1 \cdots x_{2n} = c_X \sum_{\sigma} q(x_{\sigma(1)}, x_{\sigma(2)}) \cdots q(x_{\sigma(2n-1)}, x_{\sigma(2n)}) \qquad\mbox{for}\quad x_i \in H^2 (X, \ZZ) .\]

Here $\sigma \in \mathfrak S_{2n}$ runs through all the $2n$-permutations but up to $2^n \cdot n!$ ambiguities inducing the same expression in the summation. The divisibility of $x \in H^2 (X, \ZZ)$ is defined to be the positive integer $\operatorname{div}(x) = \gcd \{ q(x,y) : y \in H^2 (X, \ZZ) \}$. To study the full cohomology $H^* (X, \QQ)$, we will use the notion of the LLV algebra $\mathfrak g$ and its action on the cohomology, which are introduced by Looijenga–Lunts <cit.> and Verbitsky <cit.>. For the concrete computations we will follow the representation theoretic notation used in <cit.>.

Throughout, group schemes will be used both in algebraic and analytic context. A group scheme is a morphism $G \to S$ equipped with an identity section $S \to G$, group law morphism $G \times_S G \to G$ and inverse $G \to G$ satisfying the usual axioms (either in the algebraic or analytic setting). In particular, by an abelian scheme $P \to S$ we mean an analytically proper connected commutative group scheme over $S$ with complex torus fibers. Any abelian scheme $P$ admits a dual abelian scheme $\check P$. A polarization of an abelian scheme $P$ is a finite étale homomorphism $\lambda : P \to \check P$ over $S$ such that for each fiber $F$, the restriction $\lambda_{|F} : F \to \check F$ is of the form $x \mapsto [t_x^* L \otimes L^{-1}]$ for an ample line bundle $L$ on $F$. Given a group scheme $G \to S$, an analytic torsor under $G$ (or analytic $G$-torsor) is a morphism $Y \to S$ equipped with a $G$-action, such that there exists an analytic covering $\tilde S = \bigsqcup_{\alpha} U_{\alpha} \to S$ where the base change $\tilde Y = Y \times_S \tilde S$ and $\tilde G = G \times_S \tilde S$ are $\tilde G$-equivariantly isomorphic over $\tilde S$. In the algebraic setting, one can use a different topology, e.g., étale topology to define an étale torsor. Our reference for the theory of abelian schemes is <cit.>, <cit.> and <cit.>. For the notion of torsors, see <cit.> or <cit.>.

§.§ Acknowledgment

I would like to thank my advisor Radu Laza for his invaluable comments and suggestions. Without his guidance, I wouldn't have completed this paper. I would also like to thank Mark Gross, Brendan Hassett, Klaus Hulek, Daniel Huybrechts and Qizheng Yin for some helpful 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 this project. This work was partially supported by NSF Grant DMS-1802128 (PI Laza).

§ DEFORMATION INVARIANCE OF THE $H^2$-TRIVIAL AUTOMORPHISMS

Let $X$ be a compact hyper-Kähler manifold. Consider the group of $H^2$-trivial automorphisms

\[ \Aut^{\circ} (X) = \ker \big( \Aut(X) \to \operatorname{O} (H^2 (X, \ZZ), q), \quad f \mapsto (f^*)^{-1} \big) .\]

Here $\Aut(X)$ is the group of biholomorphic automorphisms of $X$. Huybrechts <cit.> together with Hassett–Tschinkel <cit.> proved that $\Aut^{\circ} (X)$ is a finite group which is invariant under deformations of $X$.

Now let us further assume $X$ admits a Lagrangian fibration $\pi : X \to B$. Then we can restrict our attention to $H^2$-trivial automorphisms that respect the Lagrangian fibration

\begin{equation}
\Aut^{\circ} (X/B) = \Aut(X/B) \cap \Aut^{\circ}(X) .
\end{equation}

Since $\Aut^{\circ}(X)$ is finite, so is $\Aut^{\circ}(X/B)$. In fact, we can further prove $\Aut^{\circ}(X/B)$ is abelian: this will be showed later in <Ref>. Notice that $\Aut^{\circ}(X/B)$ not only depends on $X$ but also on the Lagrangian fibration $\pi : X \to B$. Hence, if $X$ admits two different Lagrangian fibrations then they can have different $\Aut^{\circ}(X/B)$. In <Ref>, we will compute $\Aut^{\circ} (X/B)$ for all currently known deformation types of hyper-Kähler manifolds $X$. In <Ref>, we will reinterpret $\Aut^{\circ} (X/B)$ as a subgroup of the group of global sections of the “polarization scheme” $K$. These results will lead us to conjecture that the polarization scheme $K$ coincides with the group $\Aut^{\circ} (X/B)$.

But before doing so, here we establish a more basic fact in this section; we prove $\Aut^{\circ} (X/B)$ is deformation invariant on $\pi$. To make this more precise, we first need to define the notion of a family of Lagrangian fibered hyper-Kähler manifolds.

A family of Lagrangian fibered compact hyper-Kähler manifolds is a commutative diagram

\[\begin{tikzcd}[row sep=tiny]
\mathcal X \arrow[rd, "\pi"] \arrow[dd, "p"] \\
& \mathcal B \arrow[ld, "q"] \\
\end{tikzcd}\]

with the following conditions.

* $p : \mathcal X \to S$ is a smooth proper family of compact hyper-Kähler manifolds of dimension $2n$ over a complex analytic space $S$.

* $q : \mathcal B \to S$ is the projectivization of a rank $n+1$ holomorphic vector bundle on $S$.

* For all $t \in S$, the fiber $\pi : X_t \to B_t$ is a Lagrangian fibration.

Note that the second condition ensures $\mathcal B$ is projective over $S$ and admits a relative ample line bundle $\mathcal O_{\mathcal B/S}(1)$. It is also possible to consider a weaker version of this definition which only assumes $\mathcal B \to S$ to be a $\PP^n$-bundle. The obstruction for a $\PP^n$-bundle to be the projectivization of a vector bundle lies in the analytic Brauer group $H^2 (S, \mathcal O_S^*)$. Thus, if $H^2 (S, \mathcal O_S^*) = 0$ (for example, when $S$ is a complex open ball) then the second axiom is in fact equivalent to the weaker one. Notice that the pullback $\mathcal H = \pi^* \mathcal O_{\mathcal B/S}(1)$ can be considered as a family of line bundles $H_t$ on $X_t$. Conversely, starting from the line bundle $\mathcal H$ we can recover $\mathcal B \cong \PP_S (p_* \mathcal H)$. Therefore, <Ref> is in some sense equivalent to the family of pairs $(X, H)$, where $H = \pi^* \mathcal O_B(1)$.

As usual, two Lagrangian fibrations $\pi : X \to B$ and $\pi' : X' \to B'$ are deformation equivalent if there exists a family of Lagrangian fibered compact hyper-Kähler manifolds $\mathcal X / \mathcal B / S$ over a connected union of $1$-dimensional open disks $S$, realizing them as two fibers at $t, t' \in S$. Matsushita in <cit.> proved such a deformation problem admits a local universal deformation.

We can now state the main theorem of this section.

The group $\Aut^{\circ}(X/B)$ is invariant under deformations of $\pi : X \to B$.

The rest of this section will be devoted to the proof of <Ref>. The sketch of the proof is as follows. First, we descend the $\Aut^{\circ}(X)$-action on $X$ to $B$ so that the Lagrangian fibration $\pi : X \to B$ becomes an equivariant morphism. This means we have a group homomorphism $\Aut^{\circ} (X) \to \Aut(B)$ whose kernel is precisely $\Aut^{\circ}(X/B)$. Descending such an action is a nontrivial problem (this is quite similar to the result of <cit.>), so we need to overcome this issue using the notion of $G$-linearizability of a line bundle. Next, we need to sheafify the discussions as we are interested in the deformation behavior of them. The result will follow from formal properties of the kernel of a sheaf homomorphism.

§.§ $G$-linearizability of a line bundle

Before we get into the proof of <Ref>, let us recall the notion of $G$-linearizability of a line bundle on a complex manifold. For simplicity we only consider finite group actions. Our references are <cit.>, <cit.> and <cit.>, but we need to take some additional care since these references only consider the algebraic setting.

Let $G$ be an arbitrary finite group and $\mathcal X$ be a complex manifold equipped with a holomorphic $G$-action. A $G$-linearized line bundle on $\mathcal X$ is a holomorphic line bundle $\mathcal L$ together with a collection of isomorphisms $\Phi_g : g^* \mathcal L \to \mathcal L$ for $g \in G$, satisfying the condition $\Phi_{gg'} = \Phi_{g'} \circ g'^* \Phi_g$ for $g, g' \in G$. A $G$-invariant line bundle on $\mathcal X$ is a holomorphic line bundle $\mathcal L$ such that $g^* \mathcal L \cong \mathcal L$ for all $g \in G$ (without any condition). We denote by $\Pic^G(\mathcal X)$ and $\Pic(\mathcal X)^G$ the groups of $G$-linearized line bundles and $G$-invariant line bundles on $\mathcal X$ up to isomorphisms, respectively. The second group is precisely the $G$-invariant subgroup of $\Pic(\mathcal X)$.

There is a forgetful homomorphism $\Pic^G (\mathcal X) \to \Pic (\mathcal X)^G$, which is neither injective nor surjective in general. To understand the obstruction to its surjectivity, one considers an exact sequence of abelian groups (<cit.> or <cit.>)

\[\begin{tikzcd}
\Pic^G(\mathcal X) \arrow[r] & \Pic(\mathcal X)^G \arrow[r] & H^2 (G, \Gamma)
\end{tikzcd}, \qquad \Gamma = H^0 (\mathcal X, \mathcal O_{\mathcal X}^*) .\]

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

Every $G$-invariant line bundle $\mathcal H$ on $\mathcal X$ is $G$-linerizable up to a suitable tensor power.

It is a general fact in the theory of group cohomology of finite groups that all the higher degree cohomologies $H^{\ge 1} (G, \Gamma)$ are $|G|$-torsion for any $G$-module $\Gamma$ (e.g., <cit.>). Hence by the previous lemma, the $|G|$-th tensor $\mathcal H^{\otimes |G|}$ vanishes in $H^2 (G, \Gamma)$ and hence comes from $\Pic^G (\mathcal X)$.

For us, the importance of the $G$-linearizability of a line bundle comes from the induced $G$-action on the higher direct images of a linearized line bundle. If $\mathcal L$ is a $G$-linearized line bundle on $\mathcal X$ and $p : \mathcal X \to S$ is a $G$-invariant holomorphic map, then we have a contravariant $G$-action on all the higher direct image sheaves

\[ g^* : R^k p_* \mathcal L \to R^k p_* \mathcal L, \qquad (g \circ g')^* = g'^* \circ g^* .\]

Now assume further $\mathcal L$ is globally generated over $S$ and $p_* \mathcal L$ is a vector bundle on $S$. Then we have a $G$-action on $\PP_S (p_* \mathcal L)$ making the holomorphic map $\mathcal X \to \PP_S (p_* \mathcal L)$ $G$-equivariant over $S$. See <cit.>.

§.§ The automorphism sheaves and deformation invariance of the $H^2$-trivial automorphisms

Suppose we have a smooth proper family of hyper-Kähler manifolds $p : \mathcal X \to S$. Let $U \subset S$ be an analytic open subset and denote by $p : \mathcal X_U = p^{-1} (U) \to U$ the restricted family over $U$. We define the sheaf of $H^2$-trivial automorphism groups $\SheafAut^{\circ}_{\mathcal X/S}$ on $S$ by

\[ \SheafAut^{\circ}_{\mathcal X/S} (U) = \{ f : \mathcal X_U \to \mathcal X_U : U \mbox{-automorphism such that } f^* : R^2 p_* \ZZ \to R^2 p_* \ZZ \mbox{ is the identity} \} .\]

By the work of Huybrechts and Hassett–Tschinkel, this sheaf is a local system of finite groups. We can consider it as a family of groups $\Aut^{\circ} (X_t)$ for $t \in S$. Similarly, given a family of Lagrangian fibered hyper-Kähler manifolds, we can define a family of groups $\Aut^{\circ}(X_t/B_t)$:

Given a family of Lagrangian fibered hyper-Kähler manifolds $p : \mathcal X \xrightarrow{\pi} \mathcal B \xrightarrow{q} S$, we define a sheaf of groups $\SheafAut^{\circ}_{\mathcal X/ \mathcal B/S}$ on $S$ by

\[ \SheafAut^{\circ}_{\mathcal X / \mathcal B/S} (U) = \{ f : \mathcal X_U \to \mathcal X_U : \mathcal B_U \mbox{-automorphism such that } f^* : R^2 p_* \ZZ \to R^2 p_* \ZZ \mbox{ is the identity} \} .\]

Equivalently, we may define $\SheafAut^{\circ}_{\mathcal X/\mathcal B/S} = q_* \SheafAut_{\mathcal X/\mathcal B} \cap \SheafAut^{\circ}_{\mathcal X/S}$.

As mentioned, the sheaf of $H^2$-trivial automorphisms $\SheafAut^{\circ}_{\mathcal X/S}$ is a local system. The sheaf $\SheafAut^{\circ}_{\mathcal X / \mathcal B / S}$ is a subsheaf of $\SheafAut^{\circ}_{\mathcal X/S}$, 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 $\SheafAut^{\circ}_{\mathcal X/S}$ becomes a constant sheaf, so we may consider it as an abstract finite group

\[ G = \Aut^{\circ} (X) \]

acting on $\mathcal X \to S$ fiberwise.

Consider the automorphism sheaf $\SheafAut_{\mathcal B/S}$ of the $\PP^n$-bundle $\mathcal B \to S$. It is the sheaf of analytic local sections of the $\PGL(n+1, \CC)$-group scheme $\Aut_{\mathcal B/S} \to S$. Our first step is to realize the sheaf $\SheafAut^{\circ}_{\mathcal X/\mathcal B/S}$ as the kernel of a certain homomorphism $\SheafAut^{\circ}_{\mathcal X/S} \to \SheafAut_{\mathcal B/S}$.

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

\begin{equation} \label{eq:homomorphism of automorphism sheaves}
G = \SheafAut^{\circ}_{\mathcal X/S} \to \SheafAut_{\mathcal B/S}
\end{equation}

whose kernel is $\SheafAut^{\circ}_{\mathcal X/\mathcal B/S}$.

Equivalently, the proposition states that there exists a $G$-action on $\mathcal B$ making $\pi : \mathcal X \to \mathcal B$ a $G$-equivariant morphism over $S$. To prove the proposition, we need to use the $G$-linearizability of line bundles in the previous subsection. The following lemma proves every line bundle on $\mathcal X$ is $G$-invariant.

$G$ acts trivially on $\Pic(\mathcal X)$.

We first claim $G$ acts trivially on $H^2 (\mathcal X, \ZZ)$. Apply the Leray spectral sequence

\[ E_2^{p,q} = H^p (S, R^q p_* \ZZ) \quad \Rightarrow \quad H^{p+q} (\mathcal X, \ZZ) .\]

Noticing that $R^0 p_* \ZZ = \ZZ$, $R^1 p_* \ZZ = 0$, and $S$ is an open ball, we obtain an isomorphism $H^2 (\mathcal X, \ZZ) \cong H^0 (S, R^2 p_*\ZZ)$. This isomorphism respects the $G$-action as the Leray spectral sequence is functorial. Now $G$ acts on $H^2 (X_t, \ZZ)$ trivially for any fiber $X_t$, so $G$ acts on $R^2p_*\ZZ$ trivially and the claim follows.

It is enough to prove the first Chern class map $\Pic (\mathcal X) \to H^2 (\mathcal X, \ZZ)$ is injective. This homomorphism is induced by the exponential sequence $0 \to \ZZ \to \mathcal O_{\mathcal X} \to \mathcal O_{\mathcal X}^* \to 0$, so it suffices to prove $H^1(\mathcal X, \mathcal O_{\mathcal X}) = 0$. Again, use the Leray spectral sequence

\[ E_2^{p, q} = H^p (S, R^q p_* \mathcal O_{\mathcal X}) \quad \Rightarrow \quad H^{p+q} (\mathcal X, \mathcal O_{\mathcal X}) .\]

This time, we have $R^0 p_* \mathcal O_{\mathcal X} = \mathcal O_S$ and $R^1 p_* \mathcal O_{\mathcal X} = 0$. This implies $H^1(\mathcal X, \mathcal O_{\mathcal X}) = 0$.

Consider the line bundle $\mathcal H = \pi^* \mathcal O_{\mathcal B/S}(1)$ on $\mathcal X$. Since $\Pic(\mathcal X)$ is $G$-invariant, we can apply <Ref> to $\mathcal H$ and conclude $\mathcal H^{\otimes m}$ is $G$-linearizable for some positive integer $m$. As a result, we have a $G$-equivariant morphism $\pi_m : \mathcal X \to \mathcal B_m$ where $\mathcal B_m = \PP_S \big( p_* \mathcal H^{\otimes m} \big)$ is the dual of the complete linear system associated to $\mathcal H^{\otimes m}$. Consider the diagram

\begin{equation} \label{diag:iitaka}
\begin{tikzcd}[row sep=tiny]
\mathcal X \arrow[dd, "p"] \arrow[rd, "\pi"] \arrow[rrd, bend left=20, "\pi_m"] \\
& \mathcal B \arrow[r, hookrightarrow] \arrow[ld, "q"] & \mathcal B_m \arrow[lld, bend left=20, "q_m"] \\
\end{tikzcd}.
\end{equation}

The $m$-th relative Veronese embedding $\mathcal B \hookrightarrow \mathcal B_m$ makes the diagram (<ref>) commute.

Notice that the morphism $\pi : \mathcal X \to \mathcal B$ associated to $\mathcal H$ has connected fibers. Hence, for any $t \in S$, the fiber $\pi : X_t \to B_t$ becomes the Iitaka fibration of the line bundle $H_t$ (e.g., <cit.>). This in particular implies that any morphism $\pi_m : X_t \to (B_m)_t$ associated to $H_t^{\otimes m}$ factors through the Iitaka fibration $\pi$. The factorization is precisely the Stein factorization of $\pi_m$ and $B_t \hookrightarrow (B_m)_t$ is precisely the $m$-th Veronese embedding. In other words, the $m$-th relative Veronese embedding makes the diagram commute.

Now the equivariance of $\pi_m$ with the diagram (<ref>) implies $\pi$ is equivariant. This completes the proof of <Ref>. As mentioned, the result is quite similar to <cit.> (both of them use the connectedness of fibers) but we do not know the precise relation between them. With <Ref> at hand, the proof of the main theorem is immediate.

Let $\mathcal X \to \mathcal B \to S$ be a family of Lagrangian fibered hyper-Kähler manifolds over an open ball $S$. The sheaves $\SheafAut^{\circ}_{\mathcal X/S}$ and $\SheafAut_{\mathcal B/S}$ are represented by the constant group schemes

\[ \Aut^{\circ}_{\mathcal X/S} \cong \bigsqcup_{f \in \Aut^{\circ}(X)} S , \qquad \Aut_{\mathcal B/S} \cong \PGL(n+1) \times S .\]

The homomorphism (<ref>) becomes a morphism $\alpha : \Aut^{\circ}_{\mathcal X/S} \to \Aut_{\mathcal B/\mathcal S}$. The desired sheaf $\SheafAut^{\circ}_{\mathcal X/\mathcal B/S}$ is representable by $\ker \alpha$ by <Ref>.

To prove $\ker \alpha$ is a constant subgroup scheme, it is enough to show the following: let $S'$ be a connected component of $\Aut^{\circ}_{\mathcal X/S}$. Consider the restriction of $\alpha$ followed by the projection

\[ \beta : S' \to \PGL(n+1) .\]

Then we claim that either $\beta(S') = \{ \id \}$ or $\beta(S') \not \ni \id$. Notice that the image $\beta(S')$ consists of $|G|$-torsion matrices in $\PGL(n+1)$. Since the set of $|G|$-torsion matrices is a disjoint union of $\PGL(n+1)$-adjoint orbits (classified by eigenvalues), the connected set $\beta(S')$ has to lie in a single orbit. The adjoint orbit containing the identity matrix is a singleton set $\{ \id \}$. Hence the claim follows.

§ ABELIAN SCHEMES ASSOCIATED TO LAGRANGIAN FIBRATIONS

The aim of this section is to associate a polarized abelian scheme to every Lagrangian fibered compact hyper-Kähler manifold, and to discuss consequences. The following is the first main theorem of this section.

Let $\pi : X \to B$ be a Lagrangian fibration of a compact hyper-Kähler manifold and $B_0 \subset B$ its smooth locus. Set $X_0 = \pi^{-1} (B_0)$ so that it becomes a smooth proper family of complex tori over $B_0$.

* There exists a unique projective abelian scheme $\nu : P_0 \to B_0$ making $\pi : X_0 \to B_0$ an analytic torsor under $\nu$.

* Moreover, the abelian scheme is simple and has a unique choice of a primitive polarization

\begin{equation} \label{eq:polarization}
\lambda : P_0 \to \check P_0 .
\end{equation}

Here $\check P_0 \to B_0$ is the dual abelian scheme of $P_0 \to B_0$.

The abelian scheme $\nu : P_0 \to B_0$ in <Ref> is called the abelian scheme associated to $\pi$.

Our statement is motivated by Arinkin–Fedorov's result in <cit.>, van Geemen–Voisin's argument in <cit.>, and Sawon's result in <cit.>. The theorem combines and slightly generalizes these results. Before discussing the applications of this theorem, let us first present some examples.

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-Kähler manifold of $\text{K3}^{[n]}$-type equipped with a Lagrangian fibration $\pi : X \to B$ (see, e.g., <cit.>). In this case, it is known that the torus fibration $\pi : X_0 \to B_0$ is isomorphic to a relative Jacobian $\Pic^d_{\mathcal C/B_0}$ associated to a certain universal family $\mathcal C/B_0$ of smooth curves on the K3 surface. Now $\Pic^d_{\mathcal C/B_0}$ is a torsor under the numerically trivial relative Jacobian $\Pic^0_{\mathcal C/B_0}$ <cit.>. By the uniqueness assertion of <Ref>, this is the associated abelian scheme $P_0$.

When $\pi : X \to B = \PP^1$ is an elliptic K3 surface, <Ref> is a weaker version of the relative Jacobian fibration construction of $\pi$ (e.g., <cit.>). In this case, one may even construct a semi-abelian scheme $P \to B$ over the entire base (Néron model) so that the smooth locus of $\pi$ becomes a torsor under $P$. Arinkin–Fedorov generalized this result to certain higher dimensional projective hyper-Kähler manifolds. A stronger version of <Ref> would potentially improve the arguments in this paper, but we will not discuss this further.

For higher dimensional compact hyper-Kähler manifolds, one may still consider the relative Picard scheme $\Pic^0_{X_0/B_0} \to B_0$. However, in general this is the dual of the abelian scheme $P_0$. This means we can consider $P_0$ as the “double Picard scheme” of the original $X_0$ <cit.>. However, for us it will be more useful to consider $P_0$ as the identity component of the relative automorphism scheme of $X_0/B_0$. We will show this in <Ref>.

When $\pi$ admits at least one rational section, then the abelian scheme $P_0$ is in fact isomorphic to $X_0$. This is because the rational section must be defined over $B_0$ by <Ref>, so that $X_0$ becomes a trivial $P_0$-torsor. In some sense, <Ref> is thus a generalization of certain properties of $X_0$ to the case where $\pi$ does not have any rational section. For example, one can study the Mordell–Weil group of $\nu$, generalizing the study of the Mordell–Weil group of $\pi$.

One application of <Ref> is a more systematic study of the polarization type of the torus fibers arising in $\pi$. The study of the polarization type of the torus fibers goes back to at least <cit.>, 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 <cit.> were the first work to consider the polarization type as an invariant attached to a Lagrangian fibration and study them in great details for $\text{K3}^{[n]}$ and $\Kum_n$-type hyper-Kähler manifolds. Using <Ref>, we can given an alternative definition of the polarization type.

* The polarization scheme of $\pi$ is the kernel

\[ K = \ker \lambda \]

of the polarization (<ref>).

* The polarization type of $\pi$ is an $n$-tuple of positive integers $(d_1, \cdots, d_n)$ with $d_1 \mid \cdots \mid d_n$ such that the fibers of the polarization scheme are isomorphic to $(\ZZ/d_1 \oplus \cdots \oplus \ZZ/d_n)^{\oplus 2}$.

The polarization scheme $K$ is a finite étale commutative group scheme over $B_0$. Hence its fibers are all isomorphic and the polarization type is well-defined. The polarization type will be an important ingredient for our method. We devote a short subsection <ref> to collect its properties.

The second theme of this section is a relation between the group $\Aut^{\circ}(X/B)$ and the polarization scheme $K$. We will see in <Ref> that every automorphism $f \in \Aut^{\circ} (X/B)$ defines a global section of the abelian scheme $P_0 \to B_0$. If we consider $\Aut^{\circ}(X/B)$ as a constant group scheme over $B_0$, this means we have a closed immersion of group schemes

\begin{equation} \label{eq:aut defines global section of P}
\Aut^{\circ} (X/B) \hookrightarrow P_0 .
\end{equation}

We expect the image of this injective map will be precisely the polarization scheme $K \subset P_0$. This is a nontrivial claim; for example, this would imply the polarization scheme is a constant group scheme. We were not able to prove this claim in general, and a large part of this paper will be devoted to showing this for known deformation types of hyper-Kähler manifolds. The following proposition will become the first step to our approach.

Let $\pi : X \to B$ be a Lagrangian fibration of a compact hyper-Kähler manifold. Let $(d_1, \cdots, d_n)$ be its polarization type and $h \in H^2 (X, \ZZ)$ be the associated cohomology class of the line bundle $\pi^* \mathcal O_B(1)$. If either one of the following two conditions holds, then (<ref>) factors through the polarization scheme $K$ and yields a closed immersion

\begin{equation} \label{eq:aut defines global section of K}
\Aut^{\circ} (X/B) \hookrightarrow K .
\end{equation}

* The claim (<ref>) holds for a deformation of $\pi$.

* We have equalities $c_X = d_1 \cdots d_n$ and $\operatorname{div}(h) = 1$.

The first item says the desired property (<ref>) is deformation invariant on $\pi$. The second item proves (<ref>) directly in certain cases. We note that the condition $c_X = d_1 \cdots d_n$ is satisfied for all known deformation types of hyper-Kähler manifolds by <Ref>. However, the condition $\operatorname{div}(h) = 1$ is more serious and sometimes fails for $\text{K3}^{[n]}$ or $\Kum_n$-type hyper-Kähler manifolds.

Finally, we would like to mention a special consequence of the above discussion when $\pi$ admits a rational section. This may give the readers some more ideas on these objects. Recall from <Ref> that the existence of a rational section implies $X_0 \cong P_0$. Therefore, (<ref>) implies the existence of certain torsion rational sections of the Lagrangian fibration

\[ \Aut^{\circ}(X/B) \subset \MW(X/B) .\]

For example, let us consider the case when $X$ is of $\Kum_n$-type. We will prove in <Ref> that the order of $\Aut^{\circ}(X/B)$ is $(n+1)^2$. Thus any Lagrangian fibration of a $\Kum_n$-type hyper-Kähler manifold must have at least $(n+1)^2$ torsion rational sections (once it admits a single torsion rational section), and the dual hyper-Kähler orbifold $\check X$ is precisely the quotient of $X$ by these special torsion rational sections. To our knowledge, this phenomenon has not been observed before and it became one of our original motivations. See also <cit.> for some related ideas on the Mordell–Weil group and birational automorphisms defined by torsion rational sections.

§.§ Abelian scheme associated to a Lagrangian fibration

In this subsection, we present the proof of <Ref>. Note again that we are assuming neither $X$ is projective nor $\pi$ has a rational section.

Recall that every smooth closed fiber $F$ of $\pi$ is a complex torus (holomorphic Arnold–Liouville theorem). In fact, $F$ is necessarily an abelian variety as observed by Voisin <cit.>. It would be helpful for us to review this fact. The key idea is the following cohomological lemma, which has been discovered several times independently in <cit.> and recently generalized into higher degree cohomologies by Shen–Yin and Voisin <cit.>.

Let $F$ be any smooth fiber of $\pi$ and $h \in H^2 (X, \ZZ)$ the cohomology class of $\pi^* \mathcal O_B(1)$. Then the restriction map

\[ -_{|F} : H^2 (X, \ZZ) \to H^2 (F, \ZZ) \]

has $\ker (-_{|F}) = h^{\perp}$. Consequently, it has $\im (-_{|F}) \cong \ZZ$.

The image of the restriction map $-_{|F}$ is generated by an ample class of $F$. As a result, $F$ is an abelian variety.

Say $y$ is an integral generator of <Ref>. Choose any Kähler class $\omega \in H^2 (X, \RR)$ and consider its restriction $\omega_{|F}$, a Kähler class on $F$. It has to be a nonzero real multiple of $y$. This means, up to sign, $y$ has to be a Kähler class on $F$. Hence $y$ is an integral Kähler class, so it is ample.

We caution the reader to be aware that the ample generator $y$ of the restriction map need not be primitive (see <Ref>). One reasonable choice of a polarization on an abelian variety fiber $F$ is a unique primitive ample class in $H^2 (F, \ZZ)$ parallel to $y$. <Ref> is essentially a more global way to formulate this over the whole base $B_0$.

We divide the proof of <Ref> into three parts: (1) an explicit construction of the polarized abelian scheme $P_0$, (2) proving such a construction makes $X_0$ a torsor under $P_0$, and finally (3) its uniqueness. The uniqueness should be a more general fact about arbitrary torsors, at least in the algebraic case (see  Moret-Bailly's answer in <cit.>). The construction of $P_0$ works for any proper family of complex tori. The uniqueness of the polarization is the only part that needs the fact $X_0$ is obtained from a Lagrangian fibered hyper-Kähler manifold $X$.

The proof of the construction part closely follows <cit.>, but for completeness we reproduce their argument here.

Apply the global invariant cycle theorem (for proper maps between compact Kähler manifolds <cit.>) and <Ref> to obtain

\[ H^0 (B_0, R^2 \pi_* \QQ) = \im (H^2 (X, \QQ) \to H^2 (F, \QQ)) \cong \QQ .\]

Hence, there exists a unique homomorphism $(R^2 \pi_* \QQ)^{\vee} \to \QQ$ of local systems on $B_0$ up to scalar. This is a homomorphism of $\QQ$-VHS: fiberwise, <Ref> proves the image of $H^2 (X, \QQ) \to H^2 (F, \QQ)$ is an ample class. Restrict it to the morphism of $\ZZ$-VHS $(R^2 \pi_* \ZZ)^{\vee} \to \ZZ$. The morphism can be uniquely determined once we assume it to be primitive and represents an ample class on each fiber. Finally, use the fact that $\pi : X_0 \to B_0$ is a family of complex tori (abelian varieties) and obtain an isomorphism $R^2 \pi_* \ZZ = \wedge^2 R^1 \pi_* \ZZ$. The result is a primitive polarization

\begin{equation} \label{eq:polarization_vhs}
(R^1 \pi_* \ZZ)^{\vee} \otimes (R^1 \pi_* \ZZ)^{\vee} \to \ZZ .
\end{equation}

We have constructed a weight $-1$ $\ZZ$-VHS $(R^1 \pi_* \ZZ)^{\vee}$ equipped with a polarization (<ref>). Now use a formal equivalence of categories between polarized weight $-1$ $\ZZ$-VHS and that of polarized abelian schemes (e.g., <cit.> <cit.>). This constructs our desired abelian scheme $\nu : P_0 \to B_0$ with a unique primitive polarization $\lambda : P_0 \to \check P_0$ over $B_0$. To prove $P_0$ is simple, we may prove the corresponding VHS $R^1 \pi_* \QQ$ is simple. This is tacitly proved in <cit.> and later explicitly stated in <cit.>. The idea is that if $R^1 \pi_* \QQ$ splits as a direct sum $\mathcal V_1 \oplus \mathcal V_2$ of two VHS, then each of them has their own polarizations, forcing $h^0 (B_0, R^2 \pi_* \QQ) \ge h^0 (B_0, \wedge^2 \mathcal V_1) + h^0 (B_0, \wedge^2 \mathcal V_2) \ge 2$. We omit the details here.

Consider an analytic open covering $\{ B_i : i \in I \}$ of $B_0$ so that over each $B_i$, the restriction of the Lagrangian fibration $\pi : X_i \to B_i$ admits at least one holomorphic section $s_i : B_i \to X_i$. Considering $s_i$ as a zero section, $\pi : X_i \to B_i$ becomes an abelian scheme. Hence by the equivalence of abelian schemes and $(R^1\pi_*\ZZ)^{\vee}$, $\nu$ and $\pi$ are isomorphic over $B_i$ by $\phi_i : X_i \to P_i$ sending $s_i$ to the zero section of $P_i$.

Now use the isomorphism $\phi_i$ to transform the group law $+ : P_i \times_{B_i} P_i \to P_i$ into a $P_i$-action on $X_i$. That is, we define a group action morphism by

\[ \rho_i : P_i \times_{B_i} X_i \to X_i, \qquad (p_i, x_i) \mapsto \phi_i^{-1} (\phi_i (x_i) + p_i) .\]

We want to patch $\rho_i$ together to define a group action $\rho : P_0 \times_{B_0} X_0 \to X_0$ over the entire $B_0$. To do so, we need to check whether the definitions of $\rho_i$ and $\rho_j$ coincides over the intersection $B_{ij} = B_i \cap B_j$, i.e.,

\begin{equation} \label{eq:descent_condition}
\phi_i^{-1} (\phi_i (x_{ij}) + p_{ij}) = \phi_j^{-1} (\phi_j (x_{ij}) + p_{ij}) \qquad \mbox{for all}\quad (p_{ij}, x_{ij}) \in P_{ij} \times_{B_{ij}} X_{ij} .
\end{equation}

Over $B_{ij}$, one has a transition function $\phi_j \circ \phi_i^{-1} : P_{ij} \to X_{ij} \to P_{ij}$, an automorphism of $P_{ij}$. Recall that the isomorphisms $\phi_i$ and $\phi_j$ are constructed by choosing the zero sections $s_i$ and $s_j$, and the corresponding isomorphisms $\phi_i : X_{ij} \cong P_{ij}$ and $\phi_j : X_{ij} \cong P_{ij}$ are as abelian schemes. From it, we notice the automorphism $\phi_j \circ \phi_i^{-1} : P_{ij} \to P_{ij}$ is a translation automorphism. The translation is by $\phi_j \circ \phi_i^{-1}(0)$, the difference of the two zero sections. With this, we have a sequence of identities

\begin{align*}
\phi_j(x_{ij}) + p_{ij} = \phi_j \circ \phi_i^{-1} (\phi_i (x_{ij})) + p_{ij} &= \big( \phi_i (x_{ij}) + \phi_j \circ \phi_i^{-1} (0) \big) + p_{ij} \\
&= \big( \phi_i (x_{ij}) + p_{ij} \big) + \phi_j \circ \phi_i^{-1} (0) = \phi_j \circ \phi_i^{-1} \big( \phi_i (x_{ij}) + p_{ij} \big) .
\end{align*}

This proves (<ref>). Hence $\rho_i$ patches together and defines a morphism $\rho : P_0 \times_{B_0} X_0 \to X_0$. The group action axioms are all easily verified. Also, $X_0$ is clearly a $P_0$-torsor by construction.

Let $\nu : P_0 \to B_0$ be a (not necessarily projective) abelian scheme so that $\pi$ becomes a torsor under $\nu$. We claim $R^1 \nu_* \ZZ \cong R^1 \pi_* \ZZ$ as VHS over $B_0$. Consider the group scheme action map

\[\begin{tikzcd}
P_0 \times_{B_0} X_0 \arrow[r, "\rho"] \arrow[dr, "\mu"'] & X_0 \arrow[d, "\pi"] \\
& B_0
\end{tikzcd}.\]

From the diagram, we have a pullback morphism between the VHS $\rho^* : R^1 \pi_* \ZZ \to R^1 \mu_* \ZZ$. The latter VHS is isomorphic to the direct sum $R^1 \nu_* \ZZ \oplus R^1 \pi_* \ZZ$ by the decomposition theorem for smooth proper morphisms. Hence composing with the first projection, we obtain a morphism $R^1 \pi_* \ZZ \to R^1 \nu_* \ZZ$. Now over a small analytic open subset $U \subset B_0$, fix any holomorphic section of $\pi : X_U \to U$ so that we can identify $P_U$ and $X_U$. Hence $\rho$ becomes the addition operation of the abelian scheme $X_U \times_U X_U \to X_U$. With this description, the pullback morphism is fiberwise $\rho^* : H^1(F, \ZZ) \to H^1 (F, \ZZ) \oplus H^1 (F, \ZZ)$, $x \mapsto (x,x)$. Hence the morphism $R^1 \pi_* \ZZ \to R^1 \nu_* \ZZ$ is an isomorphism over $U$, and the claim follows.

* A posteriori, one has an interpretation of <Ref> in terms of <Ref>. The abelian scheme $\nu$ is projective, and $\nu$ and $\pi$ are fiberwise isomorphic. Hence the smooth fibers of $\pi$ are projective, even when the hyper-Kähler manifold $X$ is not.

* <Ref> is also related to Oguiso's result <cit.> in the following sense. Consider the generic fiber $P_L \to \Spec L$ of $\nu : P_0 \to B_0$. It is an abelian variety over $L$. Since $P_0$ has a unique polarization (up to scalar), $P_L$ has a unique polarization. Now ampleness is an open condition in $\NS(P_L)_{\RR}$, so the uniqueness of the polarization implies $\rho(P_L) = 1$.

* If we further assume $X$ is projective, then the discussion becomes algebraic and hence the smooth morphism $\pi : X_0 \to B_0$ admits étale local sections. Thus $\pi$ becomes an étale torsor under $\nu$.

The unique abelian scheme $P_0$ above should be considered as the identity component of the “automorphism scheme” of $\pi$ (see <cit.>). Unfortunately we do not know the existence of an automorphism group scheme in the non-projective setting, so here we need a care (when $X$ is projective, this is a byproduct of the existence of a Hilbert scheme <cit.>). Nevertheless, one may concentrate only on the translation automorphisms to avoid such a technical problem. Let us define a subsheaf $\underline{\Aut}^{tr}_{X_0/B_0} \subset \SheafAut_{X_0/B_0}$ of the automorphism sheaf by

\begin{equation}
\underline{\Aut}^{tr}_{X_0/B_0} (U) = \{ f : X_U \to X_U : U \mbox{-automorphism acting by translation on each fibers} \} .
\end{equation}

The abelian scheme $\nu : P_0 \to B_0$ represents $\underline{\Aut}^{tr}_{X_0/B_0}$.

This almost follows from the definition. Let us temporarily denote by $\underline{P_0}$ the sheaf of analytic local sections of $\nu$. Since $P_0$ acts on $X_0$ by fiberwise translation, we have a sheaf homomorphism $\underline{P_0} \to \underline{\Aut}^{tr}_{X_0/B_0}$. The homomorphism is injective because the $P_0$-action is effective. Now $X_0$ was in fact a torsor under $P_0$. Over a small analytic open subset $U \subset B_0$, $X_U \to U$ admits a section so it becomes an abelian scheme isomorphic to $P_U \to U$. Hence the set of sections $P_U(U) = X_U(U)$ consists of precisely the translation automorphisms of $X_U$. This proves the homomorphism $\underline{P_0} \to \underline{\Aut}^{tr}_{X_0/B_0}$ is surjective stalkwise. Thus it is an isomorphism.

§.§ Polarization type and divisibility of $\pi^* \mathcal O_B(1)$

The purpose of this subsection is to study two numerical invariants associated to a Lagrangian fibered hyper-Kähler manifold and study their relations: they are the polarization type of $\pi$ in <Ref> and the divisibility of the line bundle $\pi^* \mathcal O_B(1)$. Throughout, we will write $h \in H^2 (X, \ZZ)$ the first Chern class of $\pi^* \mathcal O_B(1)$.

The polarization type of $\pi$ is an $n$-tuple of positive integers $(d_1, \cdots, d_n)$ with $d_1 \mid \cdots \mid d_n$ such that the fiber of the polarization scheme $K$ is isomorphic to $(\ZZ/d_1 \oplus \cdots \oplus \ZZ/d_n)^{\oplus 2}$. Since we are assuming the polarization $\lambda : P_0 \to \check P_0$ is primitive, we always have $d_1 = 1$. The polarization type was already computed for all currently known deformation types of hyper-Kähler manifolds. The computations were based on its original definition in <cit.>. Therefore, to use the previous results we first need to show our definition is equivalent to the original one.

The polarization type in <Ref> is equivalent to the definition in <cit.>.

Recall our definition of the polarization is constructed as the primitive morphism $(R^2\pi_*\ZZ)^{\vee} \to \ZZ$ of local systems, or equivalently a primitive morphism $\ZZ \to R^2\pi_* \ZZ$. Fix any smooth fiber $F$ of $\pi$. As a local system, $R^2\pi_*\ZZ$ is identified with a $\ZZ$-module $H^2(F,\ZZ)$ with a monodromy $\pi_1(B_0)$-action. In this setting, the primitive morphism $\ZZ \to R^2\pi_* \ZZ$ of local systems corresponds to a primitive homomorphism $\ZZ \to H^2 (F, \ZZ)$ of $\pi_1(B_0)$-modules. In other words, our definition of the polarization type is equivalent to the primitive polarization type of a single smooth fiber $F$ coming from the image of $H^2 (X, \QQ) \to H^2 (F, \QQ)$ by the global invariant cycle theorem. This was precisely how Wieneck defined the polarization type.

We can now use the previous results on computations of the polarization type of $\pi$. The following theorem collects all possible polarization types that can occur for known deformation types of hyper-Kähler manifolds. For $\text{K3}^{[n]}$ and $\Kum_n$-types the computations are done by <cit.>. For OG10 and OG6, the computations are contained in <cit.> and <cit.>, respectively.

Let $\pi : X \to B$ be a Lagrangian fibered compact hyper-Kähler manifold. Then the polarization type of $\pi$ is

\[\begin{cases}
(1,\cdots,1) \quad & \mbox{if } X \mbox{ is of } \text{K3}^{[n]} \mbox{-type;} \\
(1,1,1,1,1) \quad & \mbox{if } X \mbox{ is of OG10-type;} \\
(1,\cdots,1, d_1, d_2) \quad & \mbox{if } X \mbox{ is of } Kum_n \mbox{-type; and} \\
(1,2,2) \quad & \mbox{if } X \mbox{ is of OG6-type}.
\end{cases}\]

When $X$ is of $\Kum_n$-type, we set $d_1 = \operatorname{div}(h)$ in $H^2 (X, \ZZ)$ and $d_2 = \frac{n+1}{d_1}$.

It is also important for us that the polarization type is deformation invariant on $\pi$ (see <cit.>). We will later recover this result in <Ref>. Observe in <Ref> that we have an equality $c_X = d_1 \cdots d_n$ for all known deformation types of hyper-Kähler manifolds. In this sense, we expect the polarization type should be considered as a refinement of the Fujiki constant $c_X$. When such an equality holds, we can explain why the restriction homomorphism $H^2 (X, \ZZ) \to H^2 (F, \ZZ)$ may be non-primitive in certain cases.

Assume we have an equality $c_X = d_1 \cdots d_n$. Then the image of the restriction homomorphism $H^2 (X, \ZZ) \to H^2 (F, \ZZ)$ in <Ref> is generated by $a \theta$, where $a = \operatorname{div}(h)$ and $\theta$ is a primitive ample class representing the canonical polarization of $F$.

Choose a cohomology class $x \in H^2 (X, \ZZ)$ with $q(h, x) = a$. By <Ref>, the class $x_{|F} \in H^2 (F, \ZZ)$ must be a positive integer multiple of the primitive polarization class $\theta$. Set $x_{|F} = b \theta$. Now the claim directly follows from the Fujiki relation

\[ d_1 \cdots d_n = \frac{1}{n!} \int_F \theta^n = \frac{1}{n!} \int_X h^n \big( \tfrac{1}{b} x \big)^n = c_X \cdot q \big(h, \tfrac{1}{b} x \big)^n = c_X \left( \frac{a}{b} \right)^n . \qedhere\]

Though not used in this paper, the divisibility of $h$ is also related to the existence of a rational section of $\pi$. We end this subsection with the following observation.

Assume $c_X = d_1 \cdots d_n$ and $\pi$ admits at least one rational section. Then $\operatorname{div}(h) = 1$ or $2$.

If $\pi$ admits a rational section, then $X_0 \cong P_0$ becomes a projective abelian scheme (<Ref>). By the general theory of abelian schemes, twice a polarization is always associated to a line bundle (e.g., <cit.> or <cit.>). This means $2\theta \in H^2 (F, \ZZ)$ is contained in the image of $\Pic(X) \subset H^2 (X, \ZZ) \to H^2 (F, \ZZ)$. By <Ref>, this implies $\operatorname{div}(h) = 1$ or $2$.

If $X$ is of $\text{K3}^{[n]}$ or $\Kum_n$-type then its Lagrangian fibration $\pi : X \to B$ may have $\operatorname{div}(h) > 2$. In such cases, $\pi$ (and any of its deformation) would never admit any rational section and the notion of the $P_0$-torsor is necessary.

§.§ The polarization scheme and $H^2$-trivial automorphisms

The goal of this subsection is to prove <Ref>. In fact, we obtain the following more precise versions of it.

Let $\pi : X \to B$ and $\pi' : X' \to B'$ be two deformation equivalent Lagrangian fibrations of compact hyper-Kähler manifolds. Let $a$ be any positive integer. Then

\begin{equation} \label{eq:polarization scheme stronger}
\Aut^{\circ} (X/B) \hookrightarrow \ker (a \lambda : P_0 \to \check P_0)
\end{equation}

if and only if the same holds for $\pi'$.

Let $\pi : X \to B$ be a Lagrangian fibration of a compact hyper-Kähler manifold and $(d_1, \cdots, d_n)$ its polarization type. Assume we have an equality $c_X = d_1 \cdots d_n$. Then (<ref>) holds for $a = \operatorname{div}(h)$.

<Ref> is simply the case $a = 1$ for the above two propositions. The rough strategy of the proof of these propositions is as follows. First, we prove $\Aut^{\circ}(X/B)$ acts on $\pi : X_0 \to B_0$ fiberwise by translations. Since $P_0$ was the group scheme representing translation automorphisms of $\pi$ (<Ref>), this proves $\Aut^{\circ}(X/B) \hookrightarrow P_0$. <Ref> will follow from a simple observation about torsion sections (<Ref>). For <Ref>, the reason why we need $a = \operatorname{div}(h)$ is because the image of the restriction map $H^2 (X, \ZZ) \to H^2 (F, \ZZ)$ is precisely $a$ times the primitive polarization (<Ref>). Such a polarization on $F$ can be lifted to a line bundle on $X$, and the claim follows because $\Aut^{\circ}(X/B)$ acts trivially on $\Pic(X) \subset H^2 (X, \ZZ)$.

The actual realization of the above idea will be quite lengthy due to the technical difficulties arising in deformation arguments. The three main steps will be <Ref>, <ref> and <ref>.

Any rational section of $\nu : P_0 \to B_0$ can be uniquely extended to an honest section.

Assume $s : B_0 \dashrightarrow P_0$ is a rational section undefined at $b \in B_0$. Let $S \subset P_0$ be the closure of the image of $s$, so that we obtain a proper birational morphism $\nu_{|S} : S \to B_0$. Since $B_0$ is smooth and $s$ is undefined at $b$, the fiber $S_b = (\nu_{|S})^{-1}(b)$ is a uniruled variety (e.g., <cit.>). This means an abelian variety $\nu^{-1}(b)$ contains a uniruled variety $S_b$. Contradiction. See <cit.> for an alternative proof.

The same argument applies to $\pi$ and proves the following: any rational section of $\pi$ is necessarily defined over $B_0$.

Every $H^2$-trivial automorphism in $\Aut^{\circ} (X/B)$ defines a global section of $P_0 \to B_0$. That is, we have a closed immersion of group schemes

\[ \Aut^{\circ} (X/B) \hookrightarrow P_0 .\]

Recall from <Ref> that $P_0$ is the abelian scheme representing the translation automorphism sheaf $\underline{\Aut}^{tr}_{X_0/B_0}$. Hence our goal is to prove $\Aut^{\circ}(X/B)$ acts on $\pi : X_0 \to B_0$ by fiberwise translation automorphisms. Consider the quotient $\bar X = X / \Aut^{\circ}(X/B)$ with a commutative diagram

\[\begin{tikzcd}[row sep=tiny]
X \arrow[rd, "p"] \arrow[dd, "\pi"] \\
& \bar X \arrow[ld, "\bar \pi"] \\
\end{tikzcd}.\]

We first claim $p$ is étale on general fibers over $B$. Let $S \subset X$ be the ramified locus of $p$. It has codimension $\ge 2$ because $p$ is quasi-étale by <Ref>. Let $b \in B$ be a general point, so that the fibers $F = X_b$ and $\bar F = \bar X_b$ are both smooth. Observe the ramification locus of $p : F \to \bar F$ is precisely $S \cap F$, which is of codimension $\ge 2$ since $b$ is general. The purity of the branch locus theorem forces $p$ to be étale.

Now we have a finite étale quotient $p : F \to \bar F = F / \Aut^{\circ} (X/B)$ between smooth projective varieties. Its Galois group $\Aut^{\circ} (X/B)$ acts on $F$ by fixed point free automorphisms. Since $F$ is an abelian variety, this means $\Aut^{\circ} (X/B)$ acts on $F$ by translations. The conclusion is that on a general fiber of $\pi$, the group $\Aut^{\circ}(X/B)$ acts by translation. Finally, by <Ref> this means $\Aut^{\circ}(X/B)$ defines a rational section of $\nu : P_0 \to B_0$. By <Ref>, the rational section must be defined over the entire $B_0$ and becomes an honest section. Hence $\Aut^{\circ}(X/B)$ acts by translations over the entire $B_0$.

An immediate byproduct is that $\Aut^{\circ}(X/B)$ is abelian.

$\Aut^{\circ}(X/B)$ is a finite abelian group.

The next step is to understand the behavior of the polarization $\lambda$ under deformations of $\pi$. A related result is Wieneck's deformation invariance of the polarization type of $\pi$ <cit.>. Recall that the polarization scheme $K$ was defined to be the kernel of the polarization $\ker \lambda$. To deal with a more technical <Ref> and <ref>, we also need to consider the kernels $K_a = \ker (a\lambda)$ for each positive integer $a$:

\[\begin{tikzcd}
0 \arrow[r] & K_a \arrow[r] & P_0 \arrow[r, "a\lambda"] & \check P_0 \arrow[r] & 0
\end{tikzcd}.\]

Since the abelian scheme $P_0$ was associated to the VHS $(R^1\pi_*\ZZ)^{\vee}$, there is a VHS version of this sequence

\begin{equation} \label{eq:polarization_vhs_ses}
\begin{tikzcd}
0 \arrow[r] & (R^1 \pi_* \ZZ)^{\vee} \arrow[r, "a\lambda_*"] & R^1 \pi_* \ZZ \arrow[r] & \underline {K_a} \arrow[r] & 0
\end{tikzcd} .
\end{equation}

The cokernel $\underline {K_a}$ is a local system of finite abelian groups on $B_0$ and is related to $K_a$ as follows: $\underline {K_a}$ is a sheaf of analytic sections of the group scheme $K_a \to B_0$, and $K_a$ is the total space of the local system $\underline {K_a}$. Therefore, we can relate $K_a$ to either abelian schemes or variation of Hodge structures. This technical flexibility will be useful to describe deformation behaviors of $K_a$.

Let $p : \mathcal X \xrightarrow{\pi} \mathcal B \xrightarrow{q} \Delta$ be a family of Lagrangian fibered compact hyper-Kähler manifolds over a complex open disc $\Delta$. Set $\mathcal B_0 \subset \mathcal B$ the smooth locus of $\pi$. Then for each positive integer $a$, there exists a finite étale group scheme $\mathcal K_a$ over $\mathcal B_0$ parametrizing the group schemes $K_a$ over $(B_0)_t$ for all $t \in \Delta$.

Let $\mathcal X_0 = \pi^{-1} (\mathcal B_0)$ be the preimage of $\mathcal B_0$, so that the restriction $\pi_0 : \mathcal X_0 \to \mathcal B_0$ of $\pi$ is a smooth proper family of abelian varieties. Consider the local system $R^2 \pi_{0,*} \ZZ$ on $\mathcal B_0$. Our first claim is $H^0 (\mathcal B_0, R^2 \pi_{0,*} \ZZ) \cong \ZZ$. Denoting by $j : \mathcal B_0 \to \mathcal B$ an open immersion, it is enough to prove $H^0 (\Delta, q_* j_* R^2 \pi_{0, *} \ZZ) \cong \ZZ$. Notice that $q_* j_* R^2 \pi_{0, *} \ZZ$ is a constructible sheaf, because $R^2 \pi_{0, *} \ZZ$ is a local system, its pushforward by $j_*$ is a constructible sheaf on $\mathcal B$ (e.g., <cit.>), and again its pushforward by $q_*$ is a constructible sheaf on $\Delta$. For each $t \in S$, we have a Lagrangian fibered hyper-Kähler manifold $\pi : X_t \to B_t$ and we may apply our previous discussions

\[ H^0 ((B_0)_t, R^2 \pi_* \ZZ) \cong \ZZ .\]

This proves every fiber of $q_* j_* R^2 \pi_{0, *} \ZZ$ is isomorphic to $\ZZ$. In this setting, we will formally prove the sheaf has $\ZZ$ global sections in <Ref>. This proves the claim $H^0 (\mathcal B_0, R^2 \pi_{0,*} \ZZ) \cong \ZZ$.

We have a unique primitive morphism $(R^2 \pi_{0, *} \ZZ)^{\vee} \to \ZZ$ of local systems on $\mathcal B_0$. As $\pi_0$ is a family of abelian varieties, we have an isomorphism $R^2 \pi_{0, *} \ZZ = \wedge^2 R^1 \pi_{0, *} \ZZ$. This gives us a unique primitive morphism of local systems (in fact, a polarization of VHS by <Ref>) $(R^1\pi_{0, *}\ZZ)^{\vee} \otimes (R^1\pi_{0, *}\ZZ)^{\vee} \to \ZZ$ over $\mathcal B_0$. Consider the scalar multiple $a$ of it and induce a morphism $(R^1\pi_{0, *}\ZZ)^{\vee} \to R^1\pi_{0, *}\ZZ$ whose cokernel $\underline {\mathcal K_a}$ is a local system on $\mathcal B_0$, parametrizing the family of local systems $\underline {K_a}$ on each $(B_0)_t$ in (<ref>). The total space of $\underline {\mathcal K_a}$ gives our desired finite étale group scheme $\mathcal K_a \to \mathcal B_0$.

Let $F$ be a constructible sheaf on a complex open disc $\Delta$. If every fiber of $F$ is isomorphic to $\ZZ$, then we have $H^0 (\Delta, F) \cong \ZZ$.

Since $F$ is constructible, $F_{|U}$ is a local system on a completement $U$ of a finite set of points $t_1, \cdots, t_k \in \Delta$. Let $U_i \subset U$ be a small punctured disc around $t_i$. The restriction $F_{|U_i}$ is determined by the representation

\[ \rho_i : \ZZ \cong \pi_1(U_i) \to \Aut(\ZZ) = \{ \pm 1 \} .\]

We have only two possibilities $\rho_i (1) = \pm 1$ for each $i$. Suppose we have $\rho_i(1) = -1$ for some $i$. Consider the total space $f : \operatorname{Et}(F) \to \Delta$ of the entire constructible sheaf $F$ (espace étalé). The map $f$ is holomorphic and étale, i.e., a local isomorphism. The condition $\rho_i(1) = -1$ geometrically translates to the fact that $f^{-1} (U_i)$ consists of a single copy of $U_i$ (the zero section) and infinite number of two-sheeted coverings of the punctured disc $U_i$. By the very assumption, the preimage $f^{-1} (t_i) = \{ p_1, p_2, \cdots \}$ should be isomorphic to $\ZZ$. Since $f$ is a local isomorphism, there should be an open disc neighborhood of each $p_i \in \operatorname{Et}(F)$. Along the two-sheeted coverings of $U_i$ in $f^{-1} (U_i)$, this cannot happen. Therefore, the only possibility is that all $p_i$ are the non-Hausdorff points filling in the unique punctured disc component in $f^{-1} (U_i)$ (i.e., the zero section). Hence we obtain at least $\ZZ$ global sections around the zero section and we are done.

The remaining case is when $\rho_i(1) = 1$ for all $i$. This means $F_{|U}$ is a constant sheaf $\ZZ$. Be aware that this does not imply $F$ is a constant sheaf $\ZZ$ on $\Delta$. This can be again conveniently seen in the total space $f : \operatorname{Et}(F) \to \Delta$. Although $f$ is a local homeomorphism, it is not a covering space unless $\operatorname{Et}(F)$ is Hausdorff. Indeed, the fibers $f^{-1}(t_i)$ can consist of non-Hausdorff points in $\operatorname{Et}(F)$ and this gives us a classification of such a constructible sheaf $F$. In any case, there are always $\ZZ$ global sections.

<Ref> in particular recovers <cit.>.

The polarization type of $\pi$ is invariant under deformations of $\pi$.

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

Let $\mathcal P_0 \to \mathcal B_0$ be an abelian scheme over a complex manifold $\mathcal B_0$ and $a\lambda : \mathcal P_0 \to \check {\mathcal P}_0$ a polarization with $\mathcal K_a = \ker (a\lambda)$. Assume there exists a torsion section $f : \mathcal B_0 \to \mathcal P_0$. If $f(\mathcal B_0) \cap \mathcal K_a \neq \emptyset$ then $f(\mathcal B_0) \subset \mathcal K_a$.

The statement is topological and local on the base $\mathcal B_0$, so we may assume $\mathcal B_0$ is a complex open ball $S$ and $\mathcal P_0 \to \mathcal B_0$ is homeomorphic to the topological constant group scheme $(\RR/\ZZ)^{2n} \times S \to S$. In this setting, the kernel $\mathcal K_a$ is a constant subgroup scheme and the torsion section $f$ is a constant section. Hence $f(S) \cap \mathcal K_a \neq \emptyset$ if and only if $f(S) \subset \mathcal K_a$.

Consider a one-parameter family of Lagrangian fibered hyper-Kähler manifolds $\mathcal X \to \mathcal B \to \Delta$ over a complex disc $\Delta$. By <Ref>, there exists a notion of a family of abelian schemes $\mathcal P_0 \to \mathcal B_0$ and a family of finite étale group schemes $\mathcal K_a \subset \mathcal P_0$. <Ref> proves we have a closed immersion $\Aut^{\circ}(X/B) \hookrightarrow P_0$ for a single fiber. In fact, the argument applies to the entire family and produces $\Aut^{\circ}(X/B)$ global sections of $\mathcal P_0 \to \mathcal B_0$, or equivalently an embedding

\[ \Aut^{\circ}(X/B) \hookrightarrow \mathcal P_0 .\]

Since $\Aut^{\circ}(X/B)$ is finite, the global sections are torsion. Suppose we had $\Aut^{\circ}(X/B) \hookrightarrow K_a$ for the original Lagrangian fibration over $0 \in \Delta$. Then this forces $\Aut^{\circ}(X/B) \hookrightarrow \mathcal K_a$ over the entire $\Delta$ by <Ref>. Hence the claim follows.

Recall from <Ref> that the restriction map $H^2 (X, \ZZ) \to H^2 (F, \ZZ)$ has a rank $1$ image generated by the class $a\theta$, where $a = \operatorname{div}(h)$ and $\theta$ is the primitive ample class corresponding to our polarization $\lambda : F \to \check F$. The preimage of $a\theta \in H^2 (F, \ZZ)$ under this restriction homomorphism is precisely $S = \{ x \in H^2 (X, \ZZ) : q(x, h) = a \}$. By <Ref>, the claim is invariant under deformations of $\pi$. We may thus deform $\pi$ and assume $\Pic(X) \cap S \neq \emptyset$. In other words, we may assume the composition $\Pic(X) \subset H^2 (X, \ZZ) \to H^2 (F, \ZZ)$ is generated by $a\theta$.

The assertion $\Aut^{\circ}(X/B) \hookrightarrow K_a = \ker (a\lambda)$ is equivalent to $a\lambda (\Aut^{\circ}(X/B)) = 0$. The latter equality may be verified fiberwise, so we may concentrate on a single fiber $F = \nu^{-1}(b) = \pi^{-1}(b)$. Let $L$ be any line bundle on $X$ such that its image under $\Pic(X) \to H^2 (F, \ZZ)$ is $a\theta$. This means the polariztion $a\lambda$ can be described as

\[ a\lambda : F \to \check F, \qquad t_x \mapsto [t_x^* (L_{|F}) \otimes L_{|F}^{-1} ] .\]

If we assume $t_x = f_{|F}$ is from a global $H^2$-trivial automorphism $f \in \Aut^{\circ}(X/B)$, then we have a sequence of identities

\[ t_x^* (L_{|F}) = (f_{|F})^* (L_{|F}) = (f^* L)_{|F} \cong L_{|F} ,\]

where the last isomorphism follows from the fact $f$ acts on $\Pic(X) \subset H^2 (X, \ZZ)$ trivially. This proves $a\lambda$ sends $\Aut^{\circ}(X/B)$ to $0$ and the claim follows.

§ THE MINIMAL SPLIT COVERING AND $H^2$-TRIVIAL AUTOMORPHISMS

This section discusses an explicit construction of certain $H^2$-trivial automorphisms. This will be conveniently used in the next section when we describe the $\Aut^{\circ} (X)$-action explicitly for certain examples of $\Kum_n$-type hyper-Kähler manifolds. Recall that the group $\Aut^{\circ} (X)$ is computed for all known deformation types of hyper-Kähler manifolds; Beauville <cit.> for $\text{K3}^{[n]}$-types, Boissière–Nieper-Wißkirchen–Sarti <cit.> for $\Kum_n$-types, and Mongardi–Wandel <cit.> for OG10 and OG6-types. The strategy is to compute the group for a specific choice of a complex structure and then use Hassett–Tschinkel's deformation equivalence <cit.>. Unfortunately, this argument doesn't tell us how $\Aut^{\circ}(X)$ exactly acts on $X$ for the deformations. The goal of this section is to introduce <Ref> to partially resolve this problem.

Throughout the section, we stick to the following setting. Let $M$ be a projective holomorphic symplectic manifold, not necessarily irreducible. By Beauville–Bogomolov decomposition theorem, $M$ must admit a finite étale covering $X \times T \to M$, called a split covering, where $X$ is a finite product of projective hyper-Kähler manifolds and $T$ is an abelian variety. In fact, Beauville in <cit.> also considered minimal such a covering. A minimal split covering of $M$ is the smallest possible split covering of $M$, in the sense that every split covering factors through it. The minimal split covering of $M$ always exists and is unique up to a (non-unique) isomorphism. Moreover, it is a Galois covering. We refer to Beauville's original paper for more details about minimal split coverings.

Meanwhile, Kawamata <cit.> proved that if $M$ is a K-trivial smooth projective variety then its Albanese morphism $\Alb : M \to \Alb(M)$ has to be an étale fiber bundle. More concretely, there exists an isogeny $\phi : T \to \Alb(M)$ of abelian varieties such that the base change of $\Alb$ becomes a trivial fiber bundle over $T$. We obtain a cartesian diagram

\begin{equation} \label{diag:kawamata}
\begin{tikzcd}
X \times T \arrow[r, "\Phi"] \arrow[d, "\pr_2"] & M \arrow[d, "\Alb"] \\
T \arrow[r, "\phi"] & \Alb(M)
\end{tikzcd} ,
\end{equation}

where $X$ is a fiber of the Albanese morphism. In particular, one sees $\Phi: X \times T \to M$ becomes a split covering of $M$.

Combining the two discussions, we obtain:

Let $M$ be a projective holomorphic symplectic manifold and $\Alb : M \to \Alb(M)$ its Albanese morphism, an étale fiber bundle by Kawamata. Assume $X = \Alb^{-1}(0)$ is a projective hyper-Kähler manifold. Then there exists a unique isogeny $\phi : T \to \Alb(M)$ such that the morphism $\Phi$ in the fiber diagram (<ref>) becomes the minimal split covering of Beauville.

Use Kawamata's result to construct an isogeny $\phi' : T' \to \Alb(M)$ trivializing the Albanese map as in (<ref>). Since $\phi'$ is a finite Galois covering, $\Phi'$ is also a finite Galois covering with $\Gal(\Phi') \cong \Gal(\phi')$. The first lemma in <cit.> claims $\Aut (X \times T') = \Aut(X) \times \Aut(T')$. Hence the $\Gal(\Phi')$-action on $X \times T'$ is by $(f, a)$ where $f$ and $a$ are automorphisms on $X$ and $T$, respectively. The isomorphism $\Gal(\Phi') \to \Gal(\phi')$ is by the second projection $(f, a) \mapsto a$. Since $\Gal(\phi')$ is the kernel of the isogeny $\phi'$, the automorphisms $a$ must be translations of $T'$.

Now consider the homomorphism $\Gal(\Phi') \to \Aut(X)$ by $(f, a) \mapsto f$. Set $H$ by the kernel of it; it consists of elements of the form $(\id_X, a)$. Under the isomorphism $\Gal (\Phi') \cong \Gal(\phi')$, we can consider it as a subgroup of $\Gal(\phi')$, so there exists a Galois covering $T' \to T = T'/H$ corresponding to it. Let $\phi : T \to \Alb(M)$ be the morphism factorizing $\phi'$. We have a cartesian diagram

\[\begin{tikzcd}
X \times T' \arrow[r] \arrow[d, "\pr_2"] & X \times T \arrow[r, "\Phi"] \arrow[d, "\pr_2"] & M \arrow[d, "\Alb"] \\
T' \arrow[r] & T \arrow[r, "\phi"] & \Alb(M)
\end{tikzcd}.\]

By construction, $\Gal(\Phi)$ consists of automorphisms $(f, a)$ with no $(\id_X, a)$ (i.e., the $\Gal(\phi)$-action on $X$ is effective). But this means $\Phi$ is precisely Beauville's minimal split covering <cit.>. The uniqueness of $\phi$ follows from the uniqueness of the minimal split covering.

The proposition in particular proves that the minimal split covering can be always realized by an isogeny $\phi : T \to \Alb(M)$ and the base change (<ref>).

We call $\phi : T \to \Alb(M)$ in <Ref> the minimal isogeny trivializing the Albanese morphism $\Alb : M \to \Alb(M)$. It is unique up to a (non-unique) isomorphism.

In fact, the proof of <Ref> is saying more about an arbitrary isogny $\phi'$.

Notation as in <Ref>. Let $\phi' : T' \to \Alb(M)$ be any isogeny trivializing the Albanese morphism. Then

* $\phi'$ factors though the minimal isogeny $\phi$.

* There exists a canonical $\Gal(\phi')$-action on $X$.

* The isogeny $\phi'$ is minimal if and only if the $\Gal(\phi')$-action on $X$ is effective.

All of these can be directly deduced from the proof of <Ref>. Recall $\Gal(\Phi') \to \Gal(\phi')$, $(f,a) \mapsto a$ is an isomorphism. Therefore, $f = f_a$ is uniquely determined by $a$, and this defines $\Gal(\phi') \to \Aut(X)$, $a \mapsto f_a$.

Now we can state the main result of this section. The ideas here were already contained in <cit.>.

Notation as in <Ref> and <ref>. Then $\Gal(\phi')$ acts on $X$ by $H^2$-trivial automorphisms. That is, we have a canonical homomorphism

\[ \Gal(\phi') \to \Aut^{\circ} (X) ,\]

which is injective if and only if $\phi'$ is minimal.

By <Ref>, we may assume $\phi' = \phi$ is minimal and $\Gal(\phi) \subset \Aut(X)$. The content of the proposition is that it is further a subgroup of $\Aut^{\circ}(X)$.

Consider the diagram (<ref>). Our first step is to equip $T$-actions on all the four spaces to make the diagram $T$-equivariant. Equip a $T$-action on $T$ by translation, and on $X \times T$ only on the second factor again by translation. The $T$-action on $\Alb(M)$ is by translation via the morphism $\phi$: if $a \in T$ and $z \in \Alb(M)$ then we define $a.z = z + \phi(a)$.

To equip a $T$-action on $M$, we claim the $T$-action on $X \times T$ descends to $M$ via $\Phi$. The descent works if the $\Gal(\Phi)$-action on $X \times T$ commutes with the $T$-action. Recall from the discussions in <Ref> that $\Gal(\Phi)$ acts on $X \times T$ by $(f, a)$ where $f$ is an automorphism of $X$ and $a$ is a translation of $T$. Now let $b \in T$ and $(x, t) \in X \times T$. Then we have a sequence of identities

\[ b . \Big( (f, a) . (x, t) \Big) = (f(x), t + a + b) = (f, a) . \Big( b . (x,t) \Big) .\]

This proves the $T$-action and $\Gal(\Phi)$-action commutes, yielding the descent $T$-action on $M$. The conclusion is that $\Alb$ becomes automatically $T$-equivariant (and hence the diagram (<ref>) becomes $T$-equivariant).

By definition, the stabilizer the $T$-action on $\Alb(M)$ is precisely $\ker \phi = \Gal(\phi)$. Since the Albanese map $\Alb : M \to \Alb(M)$ is $T$-equivariant, this induces a $\Gal(\phi)$-action on the fiber $\Alb^{-1} (0) = X$. One easily shows this coincides with our previous $\Gal(\phi)$-action on $X$. Now notice that any $T$-action on $M$ is isotopic to the identity map because $T$ is path connected. In particular, $T$ acts on $M$ trivially at the level of cohomology $H^* (M, \QQ)$. The embedding $X \subset M$ is $\Gal(\phi)$-equivariant, so we have a $\Gal(\phi)$-equivariant restriction homomorphism

\[ H^2 (M, \QQ) \to H^2 (X, \QQ) .\]

Hence it suffices to prove this restriction homomorphism is surjective.

Now the question became topological. Deform the complex structure of the hyper-Kähler manifold $X$ very generally so that $H^2 (X, \QQ)$ becomes a simple $\QQ$-Hodge structure (we will have to lose the projectiveness of $X$). The complex structure of $M$ can be correspondingly chosen in a way that the finite covering map $\Phi : X \times T \to M$ becomes holomorphic. Therefore, the Hodge structure morphism $H^2 (M, \QQ) \to H^2 (X, \QQ)$ is either $0$ or surjective. We only need to rule out the former possibility.

To prove it is nonzero, consider any global holomorphic symplectic form $\sigma$ on $M$. Pulling it back to $X \times T$ gives a global holomorphic symplectic form on $X \times T$. But $H^{2,0} (X \times T) = H^{2,0} (X) \oplus H^{2,0} (T)$ by Künneth. If $\sigma$ was $0$ in the $H^{2,0}(X)$-component then this would mean $\sigma$ doesn't contain any $2$-forms along the tangent direction of $X$, violating $\sigma$ is a symplectic form. Hence $\sigma_{|X}$ cannot be $0$. The claim follows.

The inclusion $\Gal(\phi) \subset \Aut^{\circ}(X)$ for the minimal isogeny $\phi$ in <Ref> may be a strict inclusion. For example, we will later see that when $X$ if of $\Kum_n$-type then

\[ \Gal(\phi) \cong (\ZZ/n+1)^{\oplus 4}, \qquad \Aut^{\circ} (X) \cong \ZZ/2 \ltimes (\ZZ/n+1)^{\oplus 4} .\]

§ THE $H^2$-TRIVIAL AUTOMORPHISMS AND POLARIZATION SCHEME FOR GENERALIZED KUMMER VARIETIES

The goal of this section is an explicit computation of the group $\Aut^{\circ}(X/B)$ and the polarization scheme $K$ for certain Lagrangian fibrations of $\Kum_n$-type hyper-Kähler manifolds. Since the group $\Aut^{\circ}(X/B)$ will be of interest for all known deformation types of hyper-Kähler manifolds, we state the result in a more general form. The following is the first main theorem of this section.

Let $\pi : X \to B$ be a Lagrangian fibration of a compact hyper-Kähler manifold. Then

\begin{equation*}
\Aut^{\circ} (X/B) \cong \begin{cases}
\{ \id \} & \mbox{if } X \mbox{ is of K3}^{[n]} \mbox{ or OG10-type;} \\
(\ZZ/d_1 \oplus \ZZ/d_2)^{\oplus 2} & \mbox{if } X \mbox{ is of } \Kum_n \mbox{-type; and} \\
(\ZZ/2)^{\oplus 4} & \mbox{if } X \mbox{ is of OG6-type.}
\end{cases}
\end{equation*}

Here if $X$ is of $\Kum_n$-type, we assume its polarization type is $(1,\cdots,1,d_1,d_2)$.

Notice that the bigger group $\Aut^{\circ} (X)$ is already trivial for $\text{K3}^{[n]}$ and OG10-types (see <cit.> and <cit.>), so the theorem is clear in the first two cases. For $\Kum_n$ and OG6-types, recall from <Ref> that $\Aut^{\circ}(X/B)$ is deformation invariant on $\pi$. By <cit.>, every Lagrangian fibration of a $\Kum_n$-type hyper-Kähler manifold is deformation equivalent to the moduli of sheaves construction, which will be recalled in <Ref>. By <cit.>, every Lagrangian fibration of an OG6-type hyper-Kähler manifold is deformation equivalent to each other. Therefore, <Ref> follows from the following more concrete results.

Let $\pi : X \to B$ be a Lagrangian fibration of a $\Kum_n$-type hyper-Kähler manifold, obtained by the moduli of sheaves construction from a triple $(S, l, s)$ in <Ref>. Let $(d_1, d_2)$ be the polarization type of the ample class $l$. Then

\[ \Aut^{\circ}(X/B) \cong (\ZZ/d_1 \oplus \ZZ/d_2)^{\oplus 2} .\]

Let $\pi : X \to B$ be a Lagrangian fibration of an OG6-type hyper-Kähler manifold, obtained by the moduli of sheaves construction. Then

\[ \Aut^{\circ}(X/B) \cong (\ZZ/2)^{\oplus 4} .\]

We note that the latter computation for OG6-type was already done by Mongardi–Wandel in <cit.>, as an intermediate step for their computation of the larger group $\Aut^{\circ}(X) \cong (\ZZ/2)^{\oplus 8}$. Thus proving <Ref> will be enough to conclude <Ref>. As mentioned before, the proof will be done by an explicit computation. In fact, the computation can be carried out further and calculates the polarization scheme $K$ as well. This is the second main result of this section.

Let $\pi : X \to B$ be a Lagrangian fibration of a $\Kum_n$-type hyper-Kähler manifold, obtained by the moduli of sheaves construction from a triple $(S, l, s)$ in <Ref>. Then we have a closed immersion (in fact, an isomorphism) of group schemes

\[ \Aut^{\circ} (X/B) \hookrightarrow K .\]

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

\[ c_i (E) \in H^{2i} (S, \ZZ), \qquad \tilde c_i (E) \in \CH^i (S) \]

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

§.§ Moduli of coherent sheaves on an abelian variety

In this subsection, we recall the construction of $\Kum_n$-type hyper-Kähler manifolds obtained from certain moduli spaces of sheaves on abelian varieties. We will mostly follow <cit.>.

Let $S$ be an abelian surface and $l \in \NS(S)$ an ample cohomology class with $\int_S l^2 = 2n+2$. Fix a nonzero class $s \in H^4 (S, \ZZ)$ so that we have a primitive Mukai vector

\begin{equation} \label{eq:mukai_vector}
v = (0, l, s) \ \ \in H^*_{\operatorname{even}} (S, \ZZ) .
\end{equation}

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 $\langle v, v \rangle + 2 = 2n+4$. Denote by $\Pic^l_S$ 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 $S \times \Pic^l_S$, so that we can define the Albanese morphism $\Alb : M \to S \times \Pic^l_S$.

To be a little more precise, we first need to fix a reference coherent sheaf $[E_0] \in M$. Let $[L_0] \in \Pic^l_S$ be a closed point with $\tilde c_1 (L_0) = \tilde c_1 (E_0)$, and consider it as the origin of the abelian surface $\Pic_S^l$. The Albanese morphism is defined (on closed points) by

\begin{equation} \label{eq:albanese morphism Kum}
\Alb : M \to S \times \Pic^l_S, \qquad [E] \mapsto \big( \mathfrak c(E), \ \tilde c_1(E) \big) ,
\end{equation}

where we define $\mathfrak c(E) = \Sigma \big( \tilde c_2 (E) - \tilde c_2(E_0) \big)$ for the summation map $\Sigma : \CH^2(S) \to S(\CC)$. The morphism becomes an étale trivial fiber bundle with a fiber $X$, a $\Kum_n$-type projective hyper-Kähler manifold.

Due to our choice of the Mukai vector $v$ in (<ref>), the above construction further comes with a Lagrangian fibration. Consider a connected component $\tilde B$ of the Chow variety of effective divisors on $S$ with numerical first Chern class $l$. Le Potier <cit.> constructed a morphism

\[ \Supp : M \to \tilde B, \qquad [E] \mapsto [\Fitt_0 E] ,\]

where $\Fitt_0 E$ is the Fitting support of a coherent sheaf $E$. Finally, consider the Poincaré line bundle $\mathcal P$ on $S \times \Pic^l_S$, the universal family of line bundles with the numerical Chern class $l$. Denote by $r : S \times \Pic^l_S \to \Pic^l_S$ the second projection. Then by Riemann–Roch, $r_* \mathcal P$ is a vector bundle of rank $n+1$. Its projectivization is a Zariski locally trivial $\PP^n$-bundle

\[ \operatorname{LB} : \tilde B \to \Pic^l_S, \qquad [C] \mapsto \mathcal [\mathcal O_S(C)] .\]

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

\[\begin{tikzcd}[sep=tiny]
M \arrow[dd, "\Alb"'] \arrow[rd, "{(\mathfrak c, \, \Supp)}"] \\
& S \times \tilde B \arrow[ld, "\id \times \operatorname{LB}"] \\
S \times \Pic^l_S
\end{tikzcd}.\]

This is an isotrivial family of Lagrangian fibered hyper-Kähler manifolds in the sense of <Ref>. For simplicity, we will take a fiber at $(0, [L_0])$ of the family so that we have $X = \Alb^{-1} (0, [L_0])$ is a $\Kum_n$-type hyper-Kähler manifold and $B = \operatorname{LB}^{-1}([L_0]) = |L_0| \cong \PP^n$ is a complete linear system.

Let $(S, l)$ be a degree $n+1$ polarized abelian surface with a polarization type $(d_1, d_2)$ and $s \in H^4 (S, \ZZ)$ any nonzero class with $\gcd(d_1, s) = 1$. Then the above construction $\pi : X \to B$ is called the moduli construction of $\Kum_n$-type associated to the triple $(S, l, s)$. It is a Lagrangian fibration of a projective hyper-Kähler manifold of $\Kum_n$-type to a projective space.

§.§ Describing the $H^2$-trivial automorphisms of $\Kum_n$-type moduli constructions

Recall that <cit.> and <cit.> proved that any $\Kum_n$-type hyper-Kähler manifold have the group of $H^2$-trivial automorphisms

\[ \Aut^{\circ}(X) \cong \ZZ/2 \ltimes (\ZZ/n+1)^{\oplus 4} .\]

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 fibration $\pi : X \to B$. Hence, the Lagrangian fibration plays no role in this subsection.

Recall that we have fixed a reference line bundle $[L_0] \in \Pic^l_S$. It is an ample line bundle on $S$. By the general theory of abelian varieties, there exists a dual ample line bundle $\check L_0$ on the dual abelian variety $\check S$ (defined only up to numerical equivalence. See, cf., <cit.> for the notion of a dual polarization). The ample line bundles $L_0$ and $\check L_0$ induce polarization isogenies

\[ \varphi : S \to \check S, \qquad \check \varphi : \check S \to S ,\]

making their compositions the multiplication endomorphisms

\begin{equation} \label{eq:isogeny composition}
[n+1] : S \xrightarrow{\varphi} \check S \xrightarrow{\check \varphi} S, \qquad [n+1] : \check S \xrightarrow{\check \varphi} S \xrightarrow{\varphi} \check S .
\end{equation}

Since $L_0$ has a polarization type $(d_1, d_2)$, the dual line bundle $\check L_0$ has a polarization type $(d_1, d_2)$ as well. In particular, we have an isomorphism

\begin{equation} \label{eq:kernel}
\ker \check \varphi \cong (\ZZ/d_1 \oplus \ZZ/d_2)^{\oplus 2} .
\end{equation}

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

\[ t_x : S \to S, \qquad y \mapsto y + x .\]

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

\[ P_{\xi} : \mbox{ numerically trivial line bundle on } S \mbox{ corresponding to } \xi \in \check S .\]

With these notation in mind, we can explicitly realize the $\Aut^{\circ}(X)$-action for the moduli of sheaves construction $X$.

Let $X$ be a $\Kum_n$-type moduli construction associated to a triple $(S, l, s)$ in <Ref>. Then

* We have an isomorphism

\[ \Aut^{\circ} (X) = \{ \pm 1 \} \ltimes \{ (x, \xi) \in S[n+1] \times \check S[n+1] : \ \varphi (x) = 0, \ \ \check \varphi (\xi) = sx \} .\]

* With the above identification, the $\Aut^{\circ}(X)$-action on $X$ is defined by

\[ (1, x, \xi).[E] = [t_x^* E \otimes P_{\xi}], \qquad (-1, x, \xi).[E] = [t_x^* ([-1]^* E) \otimes P_{\xi}] ,\]

where $[-1] : S \to S$ is the multiplication by $-1$ automorphism on $S$.

The rest of this subsection is devoted to the proof of <Ref>. To start, we note that Yoshioka has already computed an explicit trivialization of Albanese morphism $\Alb : M \to S \times \Pic^l_S$. Yoshioka's trivialization is obtained by the base change $[n+1] : S \times \Pic^l_S \to S \times \Pic^l_S$, which is a degree $(n+1)^8$ isogeny. As we will see in a moment, this is not a minimal isogeny in the sense of <Ref>. Using the methods in <Ref>, we first prove the morphism

\begin{equation} \label{eq:minimal isogeny}
\phi : S \times \Pic^l_S \to S \times \Pic^l_S, \qquad (y, [L]) \mapsto (sy - \check \varphi(L \otimes L_0^{-1}), \ [L_0 \otimes P_{\varphi(y)}])
\end{equation}

is the minimal isogeny trivializing the Albanese morphism.

The base change (<ref>) is the minimal isogeny trivializing the Albanese morphism $\Alb : M \to S \times \Pic^l_S$ in the sense of <Ref>.

Start from Yoshioka's diagram <cit.> trivializing the Albanese morphism, which is a cartesian diagram

\begin{equation} \label{diag:yoshioka}
\begin{tikzcd}
X \times (S \times \Pic^l_S) \arrow[r, "\Phi'"] \arrow[d, "\pr_2"] & M \arrow[d, "\Alb"] \\
S \times \Pic^l_S \arrow[r, "{[n+1]}"] & S \times \Pic^l_S
\end{tikzcd} .
\end{equation}

Here $\Phi' : X \times (S \times \Pic^l_S) \to M$ is defined to be a Galois étale morphism

\[ \Phi' ([E], y, [L]) = \Big[ t_{\check \varphi (L \otimes L_0^{-1})}^* E \otimes (L \otimes L_0^{-1})^{\otimes s} \otimes P_{-\varphi(y)} \Big] .\]

Note that our convention differs by sign to Yoshioka's original paper, because Yoshioka's dual line bundle $\check L_0$ differs to ours by sign.

The Galois group of the base change $[n+1]$ is the group of $(n+1)$-torsion points $S[n+1] \times \check S[n+1] \cong (\ZZ/n+1)^{\oplus 8}$. By <Ref>, it acts on $X \times (S \times \Pic^l_S)$ by translation on the second factor

\[ (x, \xi) . ([E], y, [L]) = ([E], y+x, [L \otimes P_{\xi}]) .\]

One computes the descent of this action to $M$ via $\Phi'$:

\[ (x,\xi).[E] = \Big[ t_{\check \varphi (\xi)}^* E \otimes P_{s \xi - \varphi(x)} \Big] .\]

This is the $S[n+1] \times \check S[n+1]$-action on $X = \Alb^{-1} (0, [L_0])$ in <Ref>. One sees this action is not an effective action, and the kernel of the action is precisely

\[ \{ (x, \xi) \in S[n+1] \times \check S[n+1] : \ \check \varphi(\xi) = 0, \ s\xi - \varphi(x) = 0 \} .\]

To kill the kernel and obtain an effective action, take a Galois quotient corresponding to the kernel (via the Galois correspondence). This is an isogeny $\psi : S \times \Pic^l_S \to S \times \Pic^l_S$ defined by

\[ \psi(y, [L]) = (\check \varphi (L \otimes L_0^{-1}), \ (L \otimes L_0^{-1})^{\otimes s} \otimes P_{-\varphi(y)}) .\]

One can check the morphism $\phi$ in (<ref>) is precisely the isogeny making $\phi \circ \psi = [n+1]$ (here one needs to use (<ref>), but we omit the computation). The result is a factorization of (<ref>) into the minimal isogeny

\[\begin{tikzcd}
X \times (S \times \Pic^l_S) \arrow[r, "\Psi"] \arrow[d, "\pr_2"] & X \times (S \times \Pic^l_S) \arrow[r, "\Phi"] \arrow[d, "\pr_2"] & M \arrow[d, "\Alb"] \\
S \times \Pic^l_S \arrow[r, "\psi"] & S \times \Pic^l_S \arrow[r, "\phi"] & S \times \Pic^l_S
\end{tikzcd}.\]

Here our new morphism $\Phi$, Beauville's minimal split covering of $M$, turns out to have a better form than the original $\Phi'$:

\begin{equation} \label{eq:Phi}
\Phi([E], y, [L]) = \Big[ t_y^* E \otimes (L \otimes L_0^{-1}) \Big] .
\end{equation}

The claim follows.

Again thanks to <Ref>, we have a canonical, effective and $H^2$-trivial $\Gal(\phi)$-action on $X$. The Galois group $\Gal(\phi)$ is captured by the kernel of $\phi$, so we have

\begin{equation} \label{eq:galois_group}
\Gal(\phi) = \{ (x, \xi) \in S[n+1] \times \check S[n+1] : \ \varphi (x) = 0, \ \check \varphi (\xi) = sx \} .
\end{equation}

This explains the isomorphism in <Ref>. The $\Gal(\phi)$-action on the fiber $X$ is obtained via the description of $\Phi$ in (<ref>). This explains how we obtained the group action in <Ref>.

We can compute $\Gal(\phi)$ more explicitly.

$\Gal(\phi) \cong (\ZZ/n+1)^{\oplus 4}$.

Let us compute the group (<ref>) explicitly. The expression involves the abelian surfaces $S$ and its dual $\check S$, their $(n+1)$-torsion points and their polarization isogenies $\varphi$ and $\check \varphi$. Therefore, the expression is independent on the complex structure on $S$ and the question is topological. We may fix polarization bases $H_1(S, \ZZ) = \ZZ \{ e_1, \cdots, e_4 \}$ and $H_1(\check S, \ZZ) = \ZZ \{ e_1^*, \cdots, e_4^* \}$ so that we can identify $S = (\RR/\ZZ) \{ e_1, \cdots, e_4 \}$ and $\check S = (\RR/\ZZ)\{ e_1^*, \cdots, e_4^* \}$. The polarization isogenies with respect to them are

\begin{equation} \label{eq:varphi matrix form}
\varphi = \begin{psmallmatrix}
0 & 0 & d_1 & 0 \\
0 & 0 & 0 & d_2 \\
-d_1 & 0 & 0 & 0 \\
0 & -d_2 & 0 & 0
\end{psmallmatrix}, \qquad \check \varphi = \begin{psmallmatrix}
0 & 0 & -d_2 & 0 \\
0 & 0 & 0 & -d_1 \\
d_2 & 0 & 0 & 0 \\
0 & d_1 & 0 & 0
\end{psmallmatrix} .
\end{equation}

Writing the coordinate $(a_1, \cdots, a_4)$ for $S = (\RR/\ZZ)^4$ and $(b_1, \cdots, b_4)$ for $\check S = (\RR/\ZZ)^4$, one explicitly computes

\[ \Gal(\phi) = \{ (a_i, b_i)_{i=1}^4 \in \Big( \tfrac{1}{n+1}\ZZ / \ZZ \Big)^{\oplus 8} : \ d_1 a_1 = 0, \ d_2 b_3 = sa_1, \ \cdots \} \cong A^{\oplus 4} ,\]

where the abelian group $A$ is defined by

\[ A = \{ (a, b) \in (\ZZ/n+1)^{\oplus 2} : \ d_1 a = 0, \ d_2 b = sa \} .\]

Notice that $\gcd(d_1, s) = 1$ by the very assumption we had in <Ref>. Now $A \cong \ZZ/n+1$ by the following simple computational lemma, and the desired isomorphism is proved.

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

\[ A = \{ (a,b) \in (\ZZ/m)^{\oplus 2} : \ pa = 0, \ qb = sa \} \]

is isomorphic to $\ZZ/m$.

The group $A$ is realized by the kernel of a homomorphism $f : (\ZZ/m)^{\oplus 2} \to (\ZZ/m)^{\oplus 2}$, $f = \begin{psmallmatrix} p & 0 \\ -s & q \end{psmallmatrix}$. Adjusting the bases of both the domain and codomain (i.e., performing elementary row and column operations), the matrix can be transformed into $\begin{psmallmatrix} 1 & 0 \\ 0 & pq \end{psmallmatrix} = \begin{psmallmatrix} 1 & 0 \\ 0 & 0 \end{psmallmatrix}$. 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 $\Gal(\phi) \cong (\ZZ/n+1)^{\oplus 4}$-action on $X$ acting trivially on $H^2$. Since $\Aut^{\circ}(X) \cong \ZZ/2 \ltimes (\ZZ/n+1)^{\oplus 4}$, we still need an additional $\ZZ/2$-part to describe. Fortunately, this is not hard to guess. Construct an involution $\iota$ on $X \times (S \times \Pic^l_S)$ by

\[ \iota ([E], y, [L]) = ([[-1]^*E], -y, [L_0^{\otimes 2} \otimes L^{-1}]) .\]

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

The involution does not commute with the $S \times \check S$-action on $X \times (S \times \Pic^l_S)$, and this is the reason why $\ZZ/2$ should act on $(\ZZ/n+1)^{\oplus 4}$ nontrivially and leads to the semi-direct product. The action descends to $M$ as a satisfying form

\[ \iota ([E]) = [[-1]^* E] ,\]

and thus $\iota$ induces an action on $X$. It remains to prove $\iota$ acts on the second cohomology of $X$ as the identity. We have already proved in <Ref> that $H^2 (M, \QQ) \to H^2 (X, \QQ)$ is surjective. Hence we only need to prove $\iota$ acts on $H^2 (M, \QQ)$ as the identity. This follows because $\iota$ is induced from the automorphism $[-1]$ on $S$, $[-1]$ acts on $H^2 (S, \QQ)$ trivially and finally the Hodge structure $H^2 (M, \QQ)$ is obtained by a tensor construction of $H^2 (S, \QQ)$ by <cit.>. This exhausts the entire $\Aut^{\circ} (X)$-action description on $X$ and hence completes the proof of <Ref>.

§.§ Automorphisms respecting the Lagrangian fibration

With <Ref> at hand, the proof of <Ref> becomes fairly straightforward. Any $H^2$-trivial automorphism is of the form

\[ f = (1, x, \xi) \quad\mbox{or}\quad (-1, x, \xi), \qquad x \in S[n+1], \quad \xi \in \check S[n+1] .\]

Let us first prove the automorphisms of the form $f = (-1, x, \xi)$ never respects $\pi$. The action is given by $f : [E] \mapsto [t_x^* ([-1]^* E) \otimes P_{\xi}]$. Recall $\pi : X \to B$ was by definition the (Fitting) support map $\Supp : [E] \mapsto [\Fitt_0 E]$. The support of $t_x^* ([-1]^* E) \otimes P_{\xi}$ is $[-1]^* \Supp E - x$. Therefore, if $f$ respects $\pi$ then we should get $\Supp E = [-1]^* \Supp E - x$ for all $[E] \in X$. In other words, we have $D = [-1]^* D - x$ for all $D \in |L_0|$. Fix any $\frac{1}{2}x \in S$ with $2 \cdot \left( \frac{1}{2} x \right) = x$. Then this condition is equivalent to saying that every divisor $D \in |t_{-\frac{1}{2}x}^* L_0|$ is a symmetric divisor. A simple way to show this is impossible is to choose an original reference line bundle $[L_0] \in \Pic^l_S$ so that none of $t^*_{\frac{1}{2} x} L_0$ becomes a symmetric line bundle for any $x \in S[n+1]$. Note that there are only $16$ symmetric line bundles in $\Pic^l_S$ (e.g., <cit.>).

Let us now consider the automorphisms of the form $f = (1, x, \xi)$. It acts on $X$ by $f : [E] \mapsto [t_x^*E \otimes P_{\xi}]$. Hence it respects $\pi$ if and only if $\Supp E = \Supp E - x$. This means $x = 0$, so we arrive

\[ \Aut^{\circ} (X/B) = \{ (1, 0, \xi) \in \{ \pm 1 \} \ltimes (S[n+1] \times \check S[n+1]) : \check \varphi(\xi) = 0 \} \cong \ker \check \varphi .\]

This is isomorphic to $(\ZZ/d_1 \oplus \ZZ/d_2)^{\oplus 2}$ by (<ref>). This completes the proof of <Ref>.

§.§ The polarization scheme of generalized Kummer varieties

This subsection will be devoted to the proof of <Ref>. Let us keep assume $\pi : X \to B$ is a $\Kum_n$-type moduli construction. The computations in this subsection are highly influenced by <cit.>. Recall from <ref> that we had a Fitting support morphism $\Supp : M \to \tilde B$ over $\Pic^l_S$. Fix a point $[L_0] \in \Pic^l_S$ and consider the fibers of $M$ and $\tilde B$ over it. We obtain a morphism

\[ \Supp : Y \to B ,\]

where $B = |L_0|$ is a complete linear system and $Y \subset M$ consists of torsion coherent sheaves $E$ on $S$ with $\ch(E) = v$ and $\tilde c_1(E) = \tilde c_1(L_0)$. The $\Kum_n$-type hyper-Kähler manifold $X$ is obtained by a fiber of the isotrivial fiber bundle $\mathfrak c : Y \to S$.

Consider the universal family $\mathcal C \to B$ of curves on $S$ parametrizing effective divisors in $B = |L_0|$. Since $L_0$ is ample, by Bertini there exists a Zariski dense open subset $B_0 \subset B$ parametrizing the smooth curves. The restriction of the universal family $\mathcal C_0 \to B_0$ becomes a smooth projective family of curves. The following lemma is standard and we omit its proof.

The morphism $\Supp : Y_0 = \Supp^{-1}(B_0) \to B_0$ is isomorphic to the relative Picard scheme of the universal family of curves $\Pic^d_{\mathcal C_0/B_0} \to B_0$ for $d = s + n + 1$.

The lemma in particular says $Y_0 \to B_0$ is a torsor under the numerically trivial relative Picard scheme

\[ J_0 = \Pic^0_{\mathcal C_0/B_0} \to B_0 .\]

Since $\mathcal C_0 / B_0$ is a smooth projective family of curves, its relative Picard scheme $J_0$ is a canonically principally polarized abelian scheme. As standard, we will call it a relative Jacobian of the family and identify $\check J_0 = J_0$. Notice that we now have four different spaces over $B_0$: $P_0$, $X_0$, $J_0$ and $Y_0$. The space $X_0$ is a $P_0$-torsor as usual, and we also have $Y_0$ as a $J_0$-torsor. Since $P_0$ and $J_0$ are translation automorphism schemes of $X_0 \subset Y_0$, we have an inclusion $P_0 \subset J_0$. Our first goal is to describe the quotient of this inclusion.

There exists a short exact sequence of abelian schemes over $B_0$:

\begin{equation} \label{eq:ses to jacobian}
\begin{tikzcd}
0 \arrow[r] & P_0 \arrow[r] & J_0 \arrow[r] & S \times B_0 \arrow[r] & 0
\end{tikzcd}.
\end{equation}

The universal family $\mathcal C_0 \to B_0$ is a subvariety of the product $i : \mathcal C_0 \hookrightarrow S \times B_0$. This induces a pullback morphism $i^* : \check S \times B_0 \to J_0$ between their relative Picard schemes over $B_0$. The morphism $J_0 \to S \times B_0$ can be constructed by the dual of it. Fiberwise, it is the morphism $J_C \to S$ induced by the universal property of the Albanese morphism applied to $i : C \hookrightarrow S$.

We prove the kernel of the morphsim $J_0 \to S \times B_0$ is $P_0$. The claim can be verified fiberwise. Fix a closed point $[C] \in B_0$ corresponding to a smooth curve $i : C \hookrightarrow S$. Over it, a closed point of $Y_0$ (resp., $J_0$) is represented by a degree $d$ line bundle $L$ on $C$ (resp., degree $0$ line bundle $M$ on $C$). The $J_0$-action on $Y_0$ is given by $[i_* M] . [i_* L] = [i_* (L \otimes M)]$. Recall that $X$ is a fiber of the morphism $\mathfrak c : Y \to S$. Hence the abelian scheme $P_0$ consists of translation automorphisms of $J_0$ invariant under the morphism $\mathfrak c$. Recall the definition of $\mathfrak c$ in (<ref>). A Riemann–Roch computation gives us

\[ \mathfrak c ([i_* (L \otimes M)]) = \mathfrak c ([i_*L]) - \Sigma i_* \tilde c_1 (M) ,\]

where $\tilde c_1(M) \in \CH^1(S)$, $i_* : \CH^1 (S) \to \CH^2(S)$ and $\Sigma : \CH^2(S) \to S(\CC)$ is a summation map. This proves the $[i_* M]$-action on the fiber of $Y_0$ is $\mathfrak c$-invariant if and only if $\Sigma i_* \tilde c_1(M) = 0$. The claim follows by the following lemma, which is already proved in <cit.>.

The morphism $J_C \to S$ sends a closed point $[M] \in J_C(\CC)$ to $\Sigma i_* \tilde c_1(M) \in S(\CC)$.

The dual of (<ref>) is automatically (e.g., <cit.>) a short exact sequence of abelian schemes

\begin{equation} \label{eq:ses to jacobian dual}
\begin{tikzcd}
0 \arrow[r] & \check S \times B_0 \arrow[r] & J_0 \arrow[r] & \check P_0 \arrow[r] & 0
\end{tikzcd}.
\end{equation}

In particular, $P_0$ and $\check S \times B_0$ are both abealin subschemes of a bigger abelian scheme $J_0$. The following proposition describes the polarization scheme $K$ more explicitly for the moduli constructions.

We have the following two additional descriptions of the polarization scheme $K$ as $B_0$-group schemes:

\[ K = P_0 \cap (\check S \times B_0) = \ker (\check \varphi \times \id : \check S \times B_0 \to S \times B_0) .\]

Fiberwise at a closed point $[C] \in B_0$, the sequences (<ref>) and (<ref>) are short exact sequences of abelian varieties

\[\begin{tikzcd}
0 \arrow[r] & F \arrow[r] & J_C \arrow[r] & S \arrow[r] & 0
\end{tikzcd}, \qquad \begin{tikzcd}
0 \arrow[r] & \check S \arrow[r] & J_C \arrow[r] & \check F \arrow[r] & 0
\end{tikzcd}.\]

Here $F = \nu^{-1}([C])$ is a fiber of $P_0$ and $J_C$ is the Jacobian of the curve $C$. The two abelian subvarieties $F$ and $\check S$ of the principally polarized abelian variety $J_C$ are the so-called complementary abelian subvarieties (see <cit.> and <cit.>). In this case, we have an equality (<cit.>)

\[ \ker (F \to J_C \to \check F) = F \cap \check S = \ker (\check S \to J_C \to S) .\]

We will soon prove in <Ref> that the composition $\check S \to J_C \to S$ is precisely the polarization isogeny $\check \varphi$ regardless of the choice of a closed point $[C] \in B_0$. Given this, we obtain a sequence of identities of group schemes

\[ \ker (P_0 \to J_0 \to \check P_0) = P_0 \cap (\check S \times B_0) = \ker (\check \varphi \times \id : \check S \times B_0 \to S \times B_0) .\]

From the last description and (<ref>), this group scheme is a constant group scheme with fibers $(\ZZ/d_1 \oplus \ZZ/d_2)^{\oplus 2}$. The first description is describing the polarization scheme $K$; combine the uniqueness of the polarization in <Ref> and the computation of polarization types in <Ref>. The claim follows.

The composition $\check S \to J_C \to S$ is the polarization isogeny $\check \varphi$.

Denote by $i : C \hookrightarrow S$ the closed immersion. At the level of first homologies, the composition $\check S \to J_C \to S$ becomes a Hodge structure homomorphism

\[ H_1 (\check S, \ZZ) = H^1 (S, \ZZ) \xrightarrow{i^*} H^1 (C, \ZZ) \xrightarrow{i_*} H^3 (S, \ZZ) = H_1 (S, \ZZ) .\]

Hence the composition is $i_* \circ i^*$, which is the multiplication map by $c_1 (\mathcal O_S(C)) \in H^2 (S, \ZZ)$. Because we have chosen $[C]$ in a complete linear system $|L_0|$, it is a multiplication by $c_1 (L_0) = l$.

Therefore, the question reduces to the following claim: the dual polarization $\check \varphi : \check S \to S$ is given by $l \cup - : H^1(S, \ZZ) \to H^3(S, \ZZ)$. Again choose polarization bases $H_1(S, \ZZ) = \ZZ \{ e_1, \cdots, e_4 \}$ and $H_1(\check S, \ZZ) = H^1 (S, \ZZ) = \ZZ \{ e_1^*, \cdots, e_4^* \}$ as in <Ref>. The polarization isogenies $\varphi$ and $\check \varphi$ have the matrix forms (<ref>). The ample class $l$ is the skew-symmetric bilinear map $\varphi : H_1 (S, \ZZ) \otimes H_1 (S, \ZZ) \to \ZZ$ considered as an element of $H^2 (S, \ZZ)$. Hence it is $l = d_1 e_1^* \wedge e_3^* + d_2 e_2^* \wedge e_4^*$.

We can now explicitly compute the map $l \cup - : H^1 (S, \ZZ) \to H^3 (S, \ZZ)$:

\[\begin{aligned}
& e_1^* \mapsto d_2 e_1^* \wedge e_2^* \wedge e_4^* , \qquad && e_2^* \mapsto -d_1 e_1^* \wedge e_2^* \wedge e_3^*, \\
& e_3^* \mapsto -d_2 e_2^* \wedge e_3^* \wedge e_4^* , \qquad && e_4^* \mapsto d_1 e_1^* \wedge e_3^* \wedge e_4^* .
\end{aligned}\]

The Poincaré duality $H_1 (S, \ZZ) = H^3 (S, \ZZ)$ yields the basis of $H^3 (S, \ZZ)$:

\[ \{ e_2^* \wedge e_3^* \wedge e_4^*, \ \ -e_1^* \wedge e_3^* \wedge e_4^*, \ \ e_1^* \wedge e_2^* \wedge e_4^*, \ \ -e_1^* \wedge e_2^* \wedge e_3^* \} .\]

With respect to it, the matrix form of the multiplication coincides with precisely the matrix form of $\check \varphi$ above. (Compare this lemma with <cit.>.)

<Ref> already shows that the polarization scheme $K$ is a constant group scheme with fibers $(\ZZ/d_1 \oplus \ZZ/d_2)^{\oplus 2}$. We need to prove this coincides with $\Aut^{\circ} (X/B)$ considered as global sections of the abelian scheme $P_0$. Again by <Ref>, we have $K = P_0 \cap (\check S \times B_0)$. It is thus enough to show $\Aut^{\circ} (X/B) \subset \check S \times B_0$.

Recall from the description in <Ref> that $f = (1, 0, \xi) \in \Aut^{\circ}(X/B) \cong \ker \check \varphi$ acts on $Y$ by $[E] \mapsto [E \otimes P_{\xi}]$, where $P_{\xi}$ is the numerically trivial line bundle on $S$ represented by $\xi \in \ker \check \varphi \subset \check S$. On $Y_0$, closed points are of the form $[E] = [i_*L]$ where $L$ is a line bundle on a smooth curve $i : C \hookrightarrow S$. Hence $f$ acts on it by

\[ f . [i_*L] = [i_*L \otimes P_{\xi}] = [i_* (L \otimes i^* P_{\xi})] .\]

This means the global section of $J_0 \to B_0$ defined by $f$ represents a line bundle $[i^* P_{\xi}]$ over $[C] \in B_0$. The inclusion $\check S \times B_0 \subset J_0$ was by definition the pullback morphism of line bundles. Hence $f$ defines in fact a global section $\xi = [P_{\xi}] \in \check S$ of the constant group scheme $\check S \times B_0$.

§ THE DUAL LAGRANGIAN FIBRATION OF A COMPACT HYPER-KÄHLER MANIFOLD

Combining the previous results, we can prove the polarization scheme is precisely the group $\Aut^{\circ}(X/B)$ for known hyper-Kähler manifolds.

Let $\pi : X \to B$ be a Lagrangian fibration of a compact hyper-Kähler manifold of $\text{K3}^{[n]}$, $\Kum_n$, OG10 or OG6-type. Then the polarization scheme is a constant group scheme

\[ K = \Aut^{\circ} (X/B) .\]

When $X$ is of $\text{K3}^{[n]}$ or OG10-type, both the polarization scheme $K$ and the global sections defined by $\Aut^{\circ}(X/B)$ are the zero section of the abelian scheme $P_0$. Hence the equality is trivial. When $X$ is of OG6-type, lattice theory forces $\operatorname{div}(h) = 1$ as shown in <cit.>. The second item of <Ref> applies and we get an inclusion $\Aut^{\circ} (X/B) \hookrightarrow K$. Combining <Ref> and <ref>, the inclusion is forced to be an equality fiberwise. Hence we get the global equality $\Aut^{\circ}(X/B) = K$.

When $X$ is of $\Kum_n$-type, we have $a^2 \mid n+1$ for $a = \operatorname{div}(h)$ by <cit.>. 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 <Ref>. By <cit.>, any Lagrangian fibration of a $\Kum_n$-type hyper-Kähler manifold is deformation equivalent to a moduli construction in <Ref>. Thus the result follows by <Ref> with the fiberwise equality derived from <Ref> and <ref>.

A direct consequence of this theorem is a promised compactification of the dual torus fibration $\check \pi : \check X_0 \to B_0$.

Let $\pi : X \to B$ be a Lagrangian fibration of a compact hyper-Kähler manifold of $\text{K3}^{[n]}$, $\Kum_n$, OG10 or OG6-type. Then

\[ \check X = X / \Aut^{\circ}(X/B), \qquad \check \pi : \check X \to B \]

defines a compactification of the dual torus fibration $\check \pi : \check X_0 \to B_0$.

As explained in the introduction, we define the dual torus fibration by $\check X_0 = X_0 / K$. For known deformation types, $K = \Aut^{\circ}(X/B)$ is a constant group scheme by <Ref>. Therefore, the group scheme quotient $X_0 / K$ over $B_0$ becomes an honest quotient $X_0 / \Aut^{\circ} (X/B)$. Notice $\Aut^{\circ} (X/B)$ acts not only on $X_0$ but on the entire $X$. Therefore, we may consider the quotient $X/\Aut^{\circ}(X/B)$ and it gives a compactification of $\check X_0$.

When $X$ is of $\text{K3}^{[n]}$ or OG10-type, the group $\Aut^{\circ}(X/B)$ is trivial and hence $\check X = X$. When $X$ is of $\Kum_n$ or OG6-type, $\Aut^{\circ}(X/B)$ is nontrivial and hence $\check X$ cannot be homeomorphic to $X$ (their fundamental groups are different). We call the corresponding $\check X$ the dual Kummer variety and dual OG6, respectively.

Since the quotient is taken precisely by the $H^2$-trivial automorphisms $\Aut^{\circ} (X/B)$, the space $\check X$ has many satisfying properties. We provide an <ref> to collect properties of quotient spaces of $X$ by $H^2$-trivial automorphisms. Here we simply copy them for the special case of our dual hyper-Kähler $\check X$. For definitions of a primitive symplectic orbifold and irreducible symplectic variety used in the following proposition, see <Ref>.

Keep the notation from <Ref>. Then

* $\check X$ is a compact primitive symplectic orbifold and also an irreducible symplectic variety.

* The fundamental group and Fujiki constant of $\check X$ are $\pi_1(X) \cong \Aut^{\circ}(X/B)$ and $c_{\check X} = 1/c_X$.

* $H^2 (\check X, \QQ)$ and $H^2 (X, \QQ)$ are Hodge isomorphic and Beauville–Bogomolov isometric.

* The LLV algebras and Mumford–Tate algebras of $X$ and $\check X$ are isomorphic.

* The pullback $H^* (\check X, \QQ) \to H^* (X, \QQ)$ is an injective map of LLV structures.

* If $\mathcal X \to \Def(X)$ is the universal deformation of $X$, then $\mathcal X / \Aut^{\circ}(X/B) \to \Def(X)$ is the (locally trivial) universal deformation of $\check X$.

Everything is a direct consequence of <Ref> and <ref>. Only the first two items need further explanations. For the first item, it is enough to show $X^f$ has codimension $\ge 4$ for all $f \in \Aut^{\circ}(X/B)$. We will see later in <Ref> that the fixed loci of $H^2$-trivial automorphisms deform when $X$ deform. Hence we may prove this for any model in the deformation class on $X$. For OG6, the fixed loci are computed in <cit.>; they are either K3 surfaces or points. For $\Kum_n$, the fixed loci are computed in <cit.>, and similarly one can deduce their codimension is always $\ge 4$. For the second item, simply notice $\Aut^{\circ}(X/B)$ has order $c_X^2$ in all cases.

The proposition shows in particular the local deformation behavior and period domains of $X$ and $\check X$ are identical. Therefore, one can still apply the method in <cit.> at the level of period domains and obtain similar conclusions for all known deformation types of hyper-Kähler 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 $X' / \Aut^{\circ}(X/B)$ is still well-defined because we considered $\Aut^{\circ}(X/B)$ as an abstract subgroup of the group $\Aut^{\circ}(X)$. The local universal deformation space of the Lagrangian fibration $\pi : X \to B$ is a hyperplane $\Def(X, H) \subset \Def(X)$ (see <cit.>). If we choose a deformation of the Lagrangian fibration $[\pi' : X' \to B'] \in \Def(X, H)$ then we can say $\check \pi' : X' / \Aut^{\circ}(X/B) \to B'$ is the dual Lagrangian fibration of $\pi' : X' \to B'$.

§ EXAMPLE: THE DUAL KUMMER FOURFOLDS

To illustrate the geometry of dual Lagrangian fibrations more concretely, we now focus on the simplest nontrivial case of <Ref>: when $X$ is of $\Kum_2$-type. Throughout, we let $X$ to be a $\Kum_2$-type hyper-Kähler fourfold and $\pi : X \to B = \PP^2$ its Lagrangian fibration. We will use all the results in previous sections without mentioning them explicitly. We write for simplicity

\[ \tilde G = \Aut^{\circ} (X), \qquad G = \Aut^{\circ} (X/B) .\]

There exist isomorphisms $\tilde G \cong \ZZ/2 \ltimes (\ZZ/3)^{\oplus 4}$ and $G \cong (\ZZ/3)^{\oplus 2}$. Again for simplicity, we call $f \in \tilde G$ a translation if $f$ does not contain a $\ZZ/2$-part, and call $f$ an involution if $f$ has a nontrivial $\ZZ/2$-part. If $f \neq \id$ is a translation (resp., involution) then it has order $3$ (resp., order $2$). There are precisely $81$ translations and $81$ involutions. The subgroup $G \subset \tilde G$ consists of $9$ translations respecting the Lagrangian fibration. We define the dual Kummer fourfold by $\check \pi : \check X = X/G \to B$. The main result of this section will be <Ref>. It collects some more precise geometric and cohomological descriptions of $\check X$. We believe the method in this section can be similarly applied to the OG6-type and higher dimensional $\Kum_n$-types.

We will use the notion of the LLV structure to describe the cohomology of $\check X$. To do so, we first need to review the LLV structure of the generalized Kummer fourfolds (following <cit.>). Recall that the Beauville–Bogomolov quadratic space of $X$ (and hence $\check X$) is isomorphic to $(H^2 (X, \QQ), q) \cong U^{\oplus 3} \oplus \langle -6 \rangle$ where $U$ denotes the hyperbolic plane $\begin{psmallmatrix} 0 & 1 \\ 1 & 0 \end{psmallmatrix}$. For simplicity, we denote $\bar V = H^2 (X, \QQ)$ and its Mukai completion by

\[ (V, \tilde q) = (\bar V, q) \oplus U \quad \big( \cong U^{\oplus 4} \oplus \langle -6 \rangle \big) .\]

Set $\mathfrak g \cong \so (V, \tilde q)$ and $\bar {\mathfrak g} \cong \so (\bar V, q)$ the LLV algebra and reduced LLV algebra of $X$ (and $\check X$). It is a split semisimple $\QQ$-Lie algebra. Associated to any dominant weight $\mu$ of $\mathfrak g$, there exists an irreducible $\mathfrak g$-module $V_{\mu}$ over $\QQ$. The LLV structure of $X$ is explicitly computed in <cit.>; we have an isomorphism of $\mathfrak g$-modules

\[ H^* (X, \QQ) \cong V_{(2)} \oplus 80\QQ \oplus V_{(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2})} .\]

It is sometimes convenient to consider the reduced LLV structure on the fixed degree cohomologies $H^k (X, \QQ)$. The LLV structure restricts to the reduced LLV structure on the middle cohomology: there exists an isomorphism of $\bar {\mathfrak g}$-modules

\begin{equation} \label{eq:reduced LLV structure of Kum2}
H^4 (X, \QQ) \cong \bar V_{(2)} \oplus 81\QQ .
\end{equation}

Let us now state the main result of this section.

Let $\pi : X \to B$ be a Lagrangian fibration of a $\Kum_2$-type hyper-Kähler fourfold and $\check \pi : \check X \to B$ its dual fibration. Then (in addition to <Ref>)

* $\check X$ has precisely $18$ isolated cyclic quotient singularities of type $\frac{1}{3} (1,1,2,2)$.

* $\check X$ (and any of its deformation) contains $9$ smooth K3 surfaces. Each of them passes through $2$ singularities of $\check X$. The image of each K3 surface is a line in $B = \PP^2$.

* The LLV decomposition of the cohomology of $\check X$ is

\[ H^* (\check X, \QQ) \cong V_{(2)} \oplus 8\QQ \oplus V_{(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2})} .\]

The rest of the section will be devoted to the proof of <Ref>. The following proposition claims any Lagrangian fibered $\Kum_2$-type hyper-Kähler manifolds are deformation equivalent. The idea originates from the results of Markman (e.g., <cit.>).

Any Lagrangian fibration of a $\Kum_2$-type hyper-Kähler fourfold $\pi : X \to B$ is deformation equivalent to each other.

The polarization type of $\pi$ is $(1,3)$ by <Ref>. Setting $h \in H^2 (X, \ZZ)$ to be an associated cohomology class of $\pi^* \mathcal O_B(1)$, its divisibility $\operatorname{div}(h)$ is $1$ by <cit.>. We can now imitate the method of <cit.>. The lattice theory result in <cit.> forces any two primitive isotropic elements $h, h'$ with divisibility $1$ in $H^2 (X, \ZZ)$ are monodromy equivalent. We can imitate the proof of <cit.> (or the proof of <cit.>) and show that any pairs $(X, H)$ with a primitive isotropic $H$ with divisibility $1$ are deformation equivalent. This proves the claim.

Thanks to this proposition, we can often specialize our discussion to a single model. Our explicit model for Lagrangian fibered $\Kum_2$-type hyper-Kähler manifolds is the following example presented in <cit.>. Let $E$ and $E'$ be elliptic curves and $S = E' \times E$ be an abelian surface. Consider a commutative diagram

\begin{equation} \label{diag:Kum2}
\begin{tikzcd}[sep=tiny]
S^{[3]} \arrow[rd, "{(\Sigma \circ \pr_1, \, \pr_2)}"] \arrow[dd, "\Alb"'] \\
& E' \times E^{(3)} \arrow[ld, "\id \times \Sigma"] \\
E' \times E
\end{tikzcd},
\end{equation}

where $\pr_1 : S^{[3]} \to (E')^{(3)}$ and $\pr_2 : S^{[3]} \to E^{(3)}$ are the coordinate projections and $\Sigma$ are the summation maps. By the discussion we had in <Ref>, this is an isotrivial family of Lagrangian fibered hyper-Kähler manifolds of $\Kum_2$-type. The advantage of this construction to the moduli construction in <Ref> is that this gives us an honest generalized Kummer fourfold, and thus we can use the computational results in <cit.> and <cit.>.

Let us also recall some known facts about the fixed loci of $H^2$-trivial automorphisms.

Let $X$ be a compact hyper-Kähler manifold and $H \subset \Aut^{\circ} (X)$ any subgroup. If $X'$ is deformation equivalent to $X$ then $(X')^H$ is deformation equivalent to $X^H$.

Let $p : \mathcal X \to \Def(X)$ be a universal deformation of $X$. Since $H$ acts fiberwise on $p$, the morphism $\mathcal X^H \to \Def(X)$ gives a family of fixed loci $(X_t)^H$. Because $H$ is a finite group acting on a complex manifold $\mathcal X$, its fixed locus $\mathcal X^H$ is a complex manifold. Similarly, each $(X_t)^H$ is a complex (symplectic) manifold. Hence $\mathcal X^H \to \Def(X)$ is a smooth proper family and the claim follows.

Let $X$ be a $\Kum_2$-type hyper-Kähler manifold and $f \in G$ its $H^2$-trivial automorphism.

* If $f$ is an involution then its fixed locus $X^f$ is a disjoint union of a single point and a K3 surface.

* If $f \neq \id$ is a translation then its fixed locus $X^f$ consists of $27$ points.

When $X$ is an honest generalized Kummer fourfold the statements were proved in <cit.> and <cit.>, even in greater generality. Now if we deform $X$ then the fixed locus $X^f$ also deforms by <Ref>.

Following <cit.>, any $\Kum_2$-type hyper-Kähler manifold must always contain $81$ K3 surfaces obtained by the fixed loci of $81$ involutions (this was first observed in <cit.>). The $81$ K3 surfaces are related by $81$ translations, and represent the $81$ trivial reduced LLV classes in (<ref>). With these backgrounds, we can begin the proof of <Ref>.

Realize $\check X$ by a double quotient $\check X = (X/f)/f'$ where $f, f'$ are two translations generating $G \cong (\ZZ/3)^{\oplus 2}$. Since $X^f$ consists of $27$ points, the first quotient $X/f$ has precisely $27$ $\ZZ/3$-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 $9$ singularities in $\check X$. This proves there are precisely $18$ $\ZZ/3$-quotient singularities.

Any symplectic $\ZZ/3$-quotient singularity must be of type $\frac{1}{3}(1,1,2,2)$. By <cit.>, the $\ZZ/3$-action is locally biholomorphic to a symplectic linear action on $\CC^4$ around $0$. Its eigenvalues are either $1$, $\zeta$ and $\zeta^2$ where $\zeta$ is the third primitive root of unity. In this case, $1$ cannot arise because the fixed locus of the action should be the origin. Hence there are five possibilities of the linear action up to conjugate, and one easily checks $\operatorname{diag}(\zeta, \zeta, \zeta^2, \zeta^2)$ is the only symplectic linear map among them (for some symplectic form).

For the second item, notice that the $9$ K3 surfaces appearing in $\check X$ is a shadow of $81$ K3 surfaces in $X$ mentioned above. For the remaining statements, we deform the Lagrangian fibration (<Ref>) and assume we are in the construction (<ref>). In this case, the $81$ K3 surfaces are explicitly described in <cit.>. One can easily check the image of each K3 surface under $\pi$ can be considered as a sublinear system in $\PP^2$, so it is a line.

Finally, for the last item, recall $H^* (\check X, \QQ) = H^* (X, \QQ)^G$. The translations in $G$ act trivially on $H^2 (X, \QQ)$ by definition and trivially on $H^3 (X, \QQ)$ by the computations in <cit.>. Hence we only need to prove $H^4 (X, \QQ)^G \cong \bar V_{(2)} \oplus 9\QQ$. The Verbitsky component is preserved by $G$, so $\bar V_{(2)}$ is $G$-invariant. Again recall the $81$ trivial reduced LLV classes in $H^4 (X, \QQ)$ were represented by $81$ K3 surfaces which are bound by $81$ translation automorphisms. Since only $9$ of them survives in $\check X$, this proves the claim.

§ VARIOUS NOTIONS OF SINGULAR HYPER-KÄHLER VARIETIES

Many of the important properties of compact hyper-Kähler manifolds have been generalized to singular settings. However, there seem to be many natural frameworks to define the notion of singular hyper-Kähler varieties. To make our discussion less ambiguous, here we collect some definitions of singular hyper-Kähler varieties and compare them. Our main references are <cit.>, <cit.> and <cit.>.

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 $\Omega_X^{[k]} = (\Omega_X^k)^{\vee\vee}$, or equivalently $\Omega_X^{[k]} = j_* \Omega_{X_{\operatorname{reg}}}^k$ where $j : X_{\operatorname{reg}} \hookrightarrow X$ is the smooth locus of $X$. A quasi-étale morphism is a morphism étale outside of a codimension $\ge 2$ closed subvariety.

Let $X$ be a compact normal Kähler space and $\sigma \in H^0 (X, \Omega_X^{[2]})$ a reflexive $2$-form.

* $(X, \sigma)$ is called a symplectic variety if $X$ has rational singularities and $\sigma$ is nondegenerate on $X_{\operatorname{reg}}$.

* $X$ is called a primitive symplectic variety if

\[ H^0 (X, \Omega_X^{[1]}) = 0, \qquad H^0 (X, \Omega_X^{[2]}) = \CC \sigma, \]

and $(X, \sigma)$ is a symplectic variety.

* $X$ is called an irreducible symplectic variety if it is a primitive symplectic variety with the following condition: for any finite quasi-étale cover $f : X' \to X$, we have

\[ H^0 (X', \Omega_{X'}^{[2k+1]}) = 0, \qquad H^0 (X', \Omega_{X'}^{[2k]}) = \CC \cdot f^* \sigma^{[k]} \qquad \mbox{for} \quad k \ge 0 .\]

* $X$ is called a Namikawa symplectic variety if it is a $\QQ$-factorial and terminal primitive symplectic variety.

* $X$ is called a primitive symplectic orbifold if it is Namikawa symplectic with only finite quotient singularities.

We have a series of implications

\[\begin{tikzcd}
\mbox{primitive symplecitc orbifold} \arrow[r, Rightarrow] & \mbox{Namikawa symplectic} \arrow[d, Rightarrow] \\
\mbox{irreducible symplectic} \arrow[r, Rightarrow] & \mbox{primitive symplectic} \arrow[r, Rightarrow] & \mbox{symplectic.}
\end{tikzcd}\]

Eventually, the dual hyper-Kähler variety $\check X$ in <Ref> will be both a primitive symplectic orbifold and an irreducible symplectic variety (<Ref>). Hence all of the discussions here apply.

Many of the interesting properties of compact hyper-Kähler manifolds generalize to their singular analogues, especially to primitive symplectic varieties. We highlight some of their properties that will be useful to our discussion. Let $X$ be a primitive symplectic variety of dimension $2n$.

* The normailzation of the singular locus $X_{\operatorname{sing}}$ is again symplectic <cit.>. In particular, $X_{\operatorname{sing}}$ is always even dimensional.

* There exist a notion of the Beauville–Bogomolov form and Fujiki constant of $X$, so that the Fujiki relation (<ref>) holds <cit.> <cit.>.

* $X$ is Namikawa symplectic if and only if it is $\QQ$-factorial and $\codim X_{\operatorname{sing}} \ge 4$ <cit.> <cit.>.

* $X$ is smooth if and only if it is a compact hyper-Kähler manifold <cit.>.

* Every morphism $\pi : X \to B$ with connected fibers to a normal base $B$ (with $0 < \dim B < 2n)$ is a Lagrangian fibration <cit.>. That is, all the irreducible components of the fibers of $\pi$ are Lagrangian subvarieties of $X$.

* The Hodge structure $H^2 (X, \ZZ)$ is pure <cit.> <cit.>. If $X$ is a primitive symplectic orbifold, then the full cohomology $H^* (X, \QQ)$ is a pure Hodge structure.

* There exists a universal locally trivial deformation $\mathcal X \to \Def^{lt}(X)$ over a smooth complex germ $\Def^{lt}(X)$ of dimension $h^{1,1}(X)$ <cit.>. If $X$ is Namikawa symplectic, then any deformation is automatically locally trivial <cit.>.

* The local Torelli theorem holds for $\Def^{lt}(X)$. In fact, global Torelli theorem holds in a suitable form <cit.>.

We will use these facts in <Ref> and <Ref>, without mentioning them explicitly.

§ QUOTIENT OF A HYPER-KÄHLER MANIFOLD BY $H^2$-TRIVIAL AUTOMORPHISMS

Let $X$ be a compact hyper-Kähler manifold and $\Aut^{\circ} (X)$ the finite group of $H^2$-trivial automorphisms. Throughout the appendix, we always let

\[ G \subset \Aut^{\circ} (X) \]

to be any subgroup and write

\begin{eqnarray} \label{eq:quotient}
p : X \to \bar X = X/G .
\end{eqnarray}

The goal of this appendix is to gather basic geometric and cohomological properties of the quotient $\bar X$. Note that Lagrangian fibrations play no role in this appendix. The main results are <Ref> and <ref>.

Consider the quotient (<ref>) of a compact hyper-Kähler manifold $X$.

* The morphism $p$ is a finite quasi-étale symplectic quotient.

* $\bar X$ is a $\QQ$-factorial irreducible symplectic variety. If we further assume $\codim X^f > 2$ for all $f \in G$, then it is also a primitive symplectic orbifold.

* $\pi_1 (\bar X) \cong G$. In particular, $\bar X$ is never smooth unless $G$ is trivial.

* If $\mathcal X \to \Def(X)$ is the universal deformation of $X$ then the quotient $\mathcal X/G \to \Def(X)$ becomes the universal locally trivial deformation of $\bar X$.

The quotient $\bar X = X / G$ being an irreducible symplectic variety, its behavior is intimately related to its (second) cohomology. To talk about the precise cohomological behavior of $\bar X$, we first need to fix the Beauville–Bogomolov form; the Beauville–Bogomolov form is a priori only defined up to scalar. We define a symmetric bilinear form $q_{\bar X} : H^2 (\bar X, \ZZ) \otimes H^2 (\bar X, \ZZ) \to \ZZ$ by

\[ q_{\bar X} (x, y) = q_X (p^* x, p^* y) \qquad\mbox{for}\quad x, y \in H^2 (\bar X, \ZZ) .\]

Be aware that $q_{\bar X}$ may be a non-primitive bilinear form with this definition.

Notation as above.

* $q_{\bar X}$ is a Beauville–Bogomolov form of $\bar X$. The Fujiki constant of $\bar X$ is $c_{\bar X} = c_X/|G|$.

* The pullback

\[ p^* : H^2(\bar X, \ZZ)/(\mbox{torsion}) \to H^2 (X, \ZZ) \]

is an injective Hodge structure homomorphism and a Beauville–Bogomolov isometry. It is an isomorphism over $\QQ$.

* The LLV algebra of $X$ and $\bar X$ are canonically isomorphic. Denoting them by $\mathfrak g$, the pullback

\[ p^* : H^* (\bar X, \QQ) \to H^* (X, \QQ) \]

is an injective $\mathfrak g$-module homomorphism.

* For all $k$, the special Mumford–Tate algebra of $H^k (\bar X, \QQ)$ is isomorphic to that of $H^2 (X, \QQ)$. As a consequence, any $\mathfrak g$-module decomposition of $H^* (\bar X, \QQ)$ is a pure Hodge structure decomposition.

Again note that the subgroup $G \subset \Aut^{\circ} (X)$ was taken arbitrary. Hence we have a family of irreducible symplectic varieties corresponding to each subgroups of $\Aut^{\circ} (X)$. That is, we get a Galois correspondence between the subgroups $G \subset \Aut^{\circ} (X)$ and the symplectic quotients $\bar X = X/G$ with the same rational Beauville–Bogomolov forms. In particular, their deformation behaviors are all identical.

The rest of this appendix is devoted to the proof of <Ref> and <ref>. Most of the proofs will be straightforward so we will be brief.