Daisuke Matsushita
We prove that the higher direct images of the dualizing sheaf of a Lagrangian fibration between smooth projective manifolds are isomorphic to the cotangent bundles of base space. As a corollary, we obtain that every Hodge number of the base space of a fibre space of an irreducible symplectic manifold is the same to that of a projective space if the base space is smooth.
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We prove that the pull back of an ample line bundle by an almost holomorphic Lagrangian fibration is nef. As an application, we show birational semi rigidity of Lagrangian fibrations.
We prove that every irreducible component of a fibre of a complex Lagrangian fibration is Lagrangian subvariety. Especially, complex Lagrangian fibations are equidimensional.
We consider (holomorphic) Lagrangian fibrations X->P^n that satisfy some natural hypotheses. We prove that there are only finitely many such Lagrangian fibrations up to deformation.
Let $X$ be a hyperkähler variety admitting a Lagrangian fibration. Beauville's "splitting property" conjecture predicts that fibres of the Lagrangian fibration should have a particular behaviour in the Chow ring of $X$. We study this conjectural behaviour for two very classical examples of Lagrangian fibrations.
We prove that there are at most two possibilities for the base of a Lagrangian fibration from a complex projective irreducible symplectic fourfold.
A fibration is said to be isotrivial if all of its smooth fibres are isomorphic to a single fixed variety. We classify the elliptic K3 surfaces that are isotrivial, and use them to construct Lagrangian fibrations that are isotrivial. We then modify the construction to produce new examples of holomorphic symplectic orbifolds, that also admit isotrivial Lagrangian fibrations.
We classify singular fibres of a projective Lagrangian fibration over codimension one points. As an application, we obtain a canonical bundle formula for a projective Lagrangian fibration over a smooth manifold.
The base surface $B$ of a Lagrangian fibration $X\twoheadrightarrow B$ of a projective, irreducible symplectic fourfold $X$ is shown to be isomorphic to ${\mathbb P}^2$.
This note is a proof of the fact that a lagrangian torus on a hyperkaehler fourfold is always a fiber of an almost holomorphic lagrangian fibration.
In this paper we study the Lagrangian fibrations for projective irreducible symplectic fourfolds and exclude the case of non-smooth base. Our method could be extended to the higher-dimensional cases.
Isotrivial Lagrangian fibrations of compact hyper-Kähler manifolds
本文首次研究紧致超凯勒流形的等平凡拉格朗日纤维化，证明光滑纤维的同宗性质，提出A、B型二分性，给出A型纤维化的分类结果及一般性假设下的类型判定，拓展了K3曲面相关结论。
Let $π:X\rightarrow\mathbb{P}^n$ be a (holomorphic) Lagrangian fibration that is very general in the moduli space of Lagrangian fibrations. We conjecture that the singular fibres in codimension one must be semistable degenerations of abelian varieties. We prove a partial result towards this conjecture, and describe an example that provides further evidence.
A holomorphic Lagrangian fibration is stable if the characteristic cycles of the singular fibers are of type $I_m, 1 \leq m <\infty,$ or $A_{\infty}$. We will give a complete description of the local structure of a stable Lagrangian fibration when it is principally polarized. In particular, we give an explicit form of the period map of such a fibration and conversely, for a period map of the described type, we construct a principally polarized stable Lagrangian fibration with the given period map. This enables us to give a number of examples exhibiting interesting behavior of the characteristi…
Let $X\to¶^n$ be a $2n$-dimensional projective holomorphic symplectic manifold admitting a Lagrangian fibration over $¶^n$. Matsushita proved that the fibration can be deformed in a codimension one family in the moduli space $\mathrm{Def}(X)$ of deformations of $X$. We extend his result by proving that if the Lagrangian fibration admits a section, then there is a codimension two family of deformations which also preserve the section.
We review the theory of Lagrangian fibrations of hyperkähler manifolds as initiated by Matsushita. We also discuss more recent work of Shen-Yin and Harder-Li-Shen-Yin. Occasionally, we give alternative arguments and complement the discussion by additional observations.
Given a Lagrangian fibration $π: X \to \mathbb{P}^n$ of a compact hyper-Kähler manifold of $\text{K3}^{[n]}$, $\text{Kum}_n$, $\text{OG10}$ or $\text{OG6}$-type, we construct a natural compactification of its dual torus fibration. Specifically, this compactification is given by a quotient of $X$ by certain 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 $X$.
We show that the image of a dominant meromorphic map from an irreducible compact Calabi-Yau manifold $X$ whose general fiber is of dimension strictly between $0$ and $\dim X$ is rationally connected. Using this result, we construct for any hyper-Kähler manifold $X$ admitting a Lagrangian fibration a Lagrangian constant cycle subvariety $Σ_H$ in $X$ which depends on a divisor class $H$ whose restriction to some smooth Lagrangian fiber is ample. If $\dim X = 4$, we also show that up to a scalar multiple, the class of a zero-cycle supported on $Σ_H$ in $\mathrm{CH}_0(X)$ depend neither on $H$ nor…
A Hodge-theoretic proof of Hwang's theorem on base manifolds of Lagrangian fibrations
用Hodge理论证明Hwang定理：不可约全纯辛流形的拉格朗日纤维化若基光滑，则基为射影空间。
We show that the base manifold of a Lagrangian fibration on a hyperkähler manifold is isomorphic to complex projective space. This generalises a theorem of J.-M. Hwang to the Kähler case.
We investigate the existence of Lagrangian fibrations on the generalized Kummer varieties of Beauville. For a principally polarized abelian surface A of Picard number one we find the following: The Kummer variety K^n(A) is birationally equivalent to another irreducible symplectic variety admitting a Lagrangian fibration, if and only if n is a perfect square. And this is the case if and only if K^n(A) carries a divisor with vanishing Beauville-Bogomolov square.
Smoothness criterion for Fano orbifolds and bases of Lagrangian fibrations
该研究建立了具商奇点的法诺簇的光滑性准则，并将其应用于证明紧超凯勒流形的拉格朗日纤维化若基具商奇点则基为射影空间。
Let X be a projective hyperkähler manifold containing a Lagrangian subtorus L. We study intersections of deformations of L, defining a canonical almost holomorphic map called L-reduction, which is not birational if and only if X admits an almost holomorphic Lagrangian fibration with (strong) fibre L. In dimension four we prove that in the above situation there is always a holomorphic Lagrangian fibration with fibre L, thus answering a question of Beauville in this particular case.
In this paper, we show that there is a natural Lagrangian fibration structure on the map $Φ$ from the cotangent bundle of a del Pezzo surface $X$ of degree 4 to $\mathbb C^2$. Moreover, we describe explicitly all level surfaces of the above natural map $Φ$.
Let $\mathrm{Hilb}^gS$ be the Hilbert scheme of $g$ points on a K3 surface $S$. Suppose that $\mathrm{Pic}S\cong\Z C$ where $C$ is a smooth curve with $C^2=2(g-1)n^2$. We prove that $\mathrm{Hilb}^gS$ is a Lagrangian fibration.
A projective symplectic variety $\mathcal{P}$ of dimension 6, with only finite quotient singularities, $π(\mathcal{P})=0$ and $h^{(2,0)}(\mathcal{P}_{smooth})=1$, is described as a relative compactified Prym variety of a family of genus 4 curves with involution. It is a Lagrangian fibration associated to a K3 surface double cover of a generic cubic surface. It has no symplectic desingularization.
The monodromy of compact Lagrangian fibrations
研究紧致拉格朗日纤维化的monodromy表示，证明在周期映射通常浸没时表示是不可约的，在纤维为isotrivial时恢复Kim-Laza-Martin的结果，证明其纤维是椭圆曲线幂的isogeneous。
Let p: M -> B be a Lagrangian fibration on a hyperkahler manifold of maximal holonomy (also known as IHS), and H the generator of the Picard group of B. We prove that the pullback $p^*(H)$ is a primitive class on M.
We show that if $f:X\to B$ is a Lagrangian fibration from a compact connected Kähler hyperkähler manifold $X$ onto a projective normal variety $B$, then $f$ is locally projective. This answers a question raised by L. Kamenova and strengthens a former result (\cite{Ca05}, Proposition 2.1), according to which the smooth fibres of $f$ are projective. The proof below does not use the known additional results concerning fibres and base in this specific context).
Equivalences via twisted hyperholomorphic sheaves from transverse Lagrangian fibrations
该研究遵循Kapustka-Kapustka的思路，利用拉格朗日纤维化构造扭曲超全纯层，进而证明OG10型超凯勒流形的Lefschetz标准猜想与D-等价猜想。
We give a Kodaira-type classification of general singular fibers of a holomorphic Lagrangian fibration in Fujiki's class $\mathcal C$. Our approach is based on the study of the characteristic vector field of the discriminantal hypersurface, which naturally arises from the defining equation of the hypersurface via the symplectic form. As an application, we show that the characteristic foliation of the discriminantal hypersurface has algebraic leaves which are either rational curves or smooth elliptic curves.
A rational Lagrangian fibration f on an irreducible symplecitc variety V is a rational map which is birationally equivalent to a regular surjective morphism with Lagrangian fibers. By analogy with K3 surfaces, it is natural to expect that a rational Lagrangian fibration exists if and only if V has a divisor D with Bogomolov--Beauville square 0. This conjecture is proved in the case when V is the punctual Hilbert scheme of a generic algebraic K3 surface S. The construction of f uses a twisted Fourier--Mukai transform which induces a birational isomorphism of V with a certain moduli space of twi…
Let Y->P^n be a flat family of reduced Gorenstein curves, such that the compactified relative Jacobian X=\bar{J}^d(Y/P^n) is a Lagrangian fibration. We prove that X is a Beauville-Mukai integrable system if n=3, 4, or 5, and the curves are irreducible and non-hyperelliptic. We also prove that X is a Beauville-Mukai system if n=3, d is odd, and the curves are canonically positive 2-connected hyperelliptic curves.
Meromorphic Group Actions and the Support Theorem for Lagrangian Fibrations
研究全纯辛复空间上拉格朗日纤维化，构造亚纯群的亚纯作用，结合霍奇理论方法证明相关支撑定理及紧凯勒空间同调自由性定理。
Hyper-Kähler varieties: Lagrangian fibrations, atomic sheaves, and categories
本文综述了紧致超Kähler流形理论的最新发展，从拉格朗日纤维化、稳定sheaves模空间和导出范畴的角度探讨核心方法与贡献。
We classify singular fibres over general points of the discriminant locus of projective complex Lagrangian fibrations on 4-dimensional holomorphic symplectic manifolds. The singular fibre F is the following either one: F is isomorphic to the product of an elliptic curve and a Kodaira singular fibre up to finite unramified covering or F is a normal crossing variety consisting of several copies of a minimal elliptic ruled surface of which the dual graph is Dynkin diagram of type A_n, $\tilde{A_n}$, $\tilde{D_n}$ or D_n.
Multiplicity of codimension 1 fibers in Lagrangian fibrations
研究了紧凑超凯赫流形的拉格朗日纤维化中1维纤维的乘积性质，证明了每个1维纤维的乘积为1。
Let $X\to¶^n$ be an irreducible holomorphic symplectic manifold of dimension $2n$ fibred over $¶^n$. Matsushita proved that the generic fibre is a holomorphic Lagrangian abelian variety. In this article we study the discriminant locus $Δ\subset¶^n$ parametrizing singular fibres. Our main result is a formula for the degree of $Δ$, leading to bounds on the degree when $X$ is a four-fold.
Obstructions for codimension one multiple fibers of Lagrangian and Calabi--Yau fibrations
该研究证明了两类纤维化不存在余维一多重纤维的结论，为拉格朗日纤维化结构、相关猜想及卡拉比-丘纤维化研究提供了关键进展，仅存在一个奇维K-平凡簇的例外情形。
Lagrangian Fibrations onto Varieties with Isolated Quotient Singularities
本文证明了具有孤立商奇点的变分在拉格朗日纤维化下必为光滑，特别地，超Kähler四维叠的拉格朗日纤维化目标表面必为$\mathbb{P}^2$。
One constructs lagrangian fibrations on the flag variety $F^3$ and proves that the fibrations are special.
Lagrangian fibrations on hyperkahler manifolds have no multiple fibers in codimension one
本文证明紧超凯勒流形上的拉格朗日纤维化在余维一中无多重纤维，并给出其等价条件为超平面除子原像基本类的原始性。
We study the moduli space of logarithmic connections of rank 2 on the Riemann sphere minus n points with fixed spectral data. There are two natural Lagrangian maps: one towards apparent singularities of the associated fuchsian scalar equation, and another one towards moduli of parabolic bundles. We show that these are transversal and dual to each other. In case n=5, we recover the beautiful geometry of Del Pezzo surfaces of degree 4.
Let $Y$ be a smooth hypersurface in a projective irreducible holomorphic symplectic manifold $X$ of dimension $2n$. The characteristic foliation $F$ is the kernel of the symplectic form restricted to $Y$. Assume that $X$ is equipped with a Lagrangian fibration $π:X\to B$ and $Y=π^{-1}D$, where $D$ is a hypersurface in $B$. It is easy to see that the leaves of $F$ are contained in the fibers of $π$. We prove that a very general leaf is Zariski dense in a fiber of $π$.
Relative and absolute Lefschetz standard conjectures for some Lagrangian fibrations
本文证明了某些拉格朗日纤维化满足Lefschetz标准猜想，并探讨了其在相对设置中的应用及与经典情况的关系。
Let $ϕ:X\rightarrow B$ be a Lagrangian fibration on a projective irreducible hyper-Kähler manifold of dimension $\leq8$. Let $M\in {\rm Pic}\,X$ be a line bundle whose restriction to the general fiber $X_b$ of $ϕ$ is topologically trivial. We prove that if the fibration has maximal variation or is isotrivial, the set of points $b$ such that the restriction $M_{\mid X_b}$ is torsion is dense in $B$. We give an application to the Chow ring of $X$. We prove a similar result for elliptic fibrations which gives a toy model for the argument.
Let X be an irreducible symplectic manifold and Def(X) the Kuranishi space. Assume that X admits a Lagrangian fibration. We prove that X can be deformed preserving a Lagrangian fibration. More precisely, there exists a smooth hypersurface H of Def(X) such that the restriction family over H admits a family of Lagrangian fibrations over H.
Compactifying Lagrangian fibrations
提出紧化几何起源的拟射影拉格朗日纤维化的框架，包含紧化判据，并证明在局部截面条件下 Albanese 纤维化及其 torsor 存在全纯辛紧化。
Lagrangian fibrations on Nikulin-type orbifolds
本文研究Nikulin型orbifold上的拉格朗日纤维化，证明其满足SYZ猜想。
The generic fiber of a Lagrangian fibration on an irreducible holomorphic symplectic manifold is an abelian variety. Associate a polarization type to such Lagrangian fibrations coming from polarizations on a generic fiber. We prove that this polarization type is constant in families of Lagrangian fibrations. Further, we determine the polarization type of K3^[n]-type fibrations and conjecture that the polarization type should only depend on the deformation type of the total space.
We prove finiteness of hyperkaehler Lagrangian fibrations in any fixed dimension with fixed Fujiki constant and discriminant of the Beauville-Bogomolov-Fujiki lattice, up to deformation. We also prove finiteness of hyperkähler Lagrangian fibrations with an ample line bundle of a given degree on the general fiber of the fibration.
We identify the perverse filtration of a Lagrangian fibration with the monodromy weight filtration of a maximally unipotent degeneration of compact hyper-Kähler manifolds.
We establish a compact analog of the P = W conjecture. For a holomorphic symplectic variety with a Lagrangian fibration, we show that the perverse numbers associated with the fibration match perfectly with the Hodge numbers of the total space. This builds a new connection between the topology of Lagrangian fibrations and the Hodge theory of hyper-Kähler manifolds. We present two applications of our result, one on the topology of the base and fibers of a Lagrangian fibration, the other on the refined Gopakumar-Vafa invariants of a K3 surface. Furthermore, we show that the perverse filtration as…
Let Y->P^n be a flat family of integral Gorenstein curves, such that the compactified relative Jacobian X=\bar{J}^d(Y/P^n) is a Lagrangian fibration. We prove that the degree of the discriminant locus Delta in P^n is at least 4n+2, and we prove that X is a Beauville-Mukai integrable system if the degree of Delta is greater than 4n+20.
Finiteness of stable Lagrangian fibrations
该研究综述 hyperkähler Lagrangian 纤维化形变类的有限性，并证明给定判别除子下稳定 Lagrangian 纤维化的有限性。
Compactified Jacobians of Extended ADE Curves and Lagrangian Fibrations
本文提出扩展 ADE 曲线概念，描述其紧化雅可比簇分支与相交图的关系，并给出 Beauville-Mukai 型拉格朗日纤维化一般奇异纤维的显式描述。
We give a construction of Lagrangian torus fibrations with controlled discriminant locus on certain affine varieties. In particular, we apply our construction in the following ways. We find a Lagrangian torus fibration on the 3-fold negative vertex whose discriminant locus has codimension 2; this provides a local model for finding torus fibrations on compact Calabi-Yau 3-folds with codimension 2 discriminant locus. Then, we find a Lagrangian torus fibration on a neighbourhood of the one-dimensional stratum of a simple normal crossing divisor (satisfying certain conditions) such that the base o…
We survey Lagrangian fibrations of holomorphic symplectic varieties, both compact and non-compact, whose fibres are Jacobians and Prym varieties.
We determine all possible multiplicities of general singular fibers of a holomorphic Lagrangian fibration, under the assumption that all components of the fibers are of Fujiki class. The multiplicities are at most 6 and the possible values are intricately related to the Kodaira type of the characteristic cycle.
Symmetric tensors on the intersection of two quadrics and Lagrangian fibration
本文证明两个二次曲面光滑交 X 上的对称张量代数为 n 元多项式代数，并构造余切丛到 C^n 的 Lagrangian 纤维化，刻画其一般纤维及 n=3 时与 Hitchin 纤维化的关系。
The Intermediate Jacobian fibration of a cubic fourfold containing a plane and fibrations in Prym varieties
本文描述了含平面的三次四fold的中间雅可比纤维化，将其视为K3曲面上挠层模空间的Lagrangian子纤维化，并提出了Prym簇中Lagrangian纤维化的一般构造。
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