Justin Sawon
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.
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In this paper we give the Weierstrass equations and the generators of Mordell-Weil groups for Jacobian fibrations on the singular K3 surface of discriminant 3.
In this paper we classify all configurations of singular fibers of elliptic fibrations on the double cover of P^2 ramified along six lines in general position.
In this paper we give the Weierstrass equations for Jacobian fibrations on the K3 surface that is the minimal resolution of the double covering of projective plane ramified along generic six lines.
For a general cubic fourfold, it was observed by Donagi and Markman that the relative intermediate Jacobian fibration associated to the family of its hyperplane sections carries a natural holomorphic symplectic form making the fibration Lagrangian. In this paper, we obtain a smooth projective compactification of the intermediate Jacobian fibration giving a ten-dimensional compact hyper-Kähler manifold, which we then show to be deformation equivalent to the exceptional example of O'Grady. This proves a conjecture by Markushevich.
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。
The starting point of this note is our recent paper with Laza and Saccà on the construction of deformations of O'Grady's $10$-dimensional manifolds as compactifications of intermediate Jacobian fibrations associated to cubic fourfolds. The note provides a complement to that paper consisting in the analogous construction in the twisted case, leading to isogenous but presumably not isomorphic or birational hyper-Kähler manifolds.
Jacobian elliptic fibrations on K3s with a non-symplectic automorphism of order 3
本文依据3阶非辛自同构对纤维的作用，结合Kneser–Nishiyama方法分类相应K3曲面上的Jacobian椭圆纤维化，并给出可约纤维类型与σ不变Weierstrass方程。
We describe all the elliptic fibrations with section on the Kummer surface X of the Jacobian of a very general curve C of genus 2 over an algebraically closed field of characteristic 0, modulo the automorphism group of X and the symmetric group on the Weierstrass points of C. In particular, we compute elliptic parameters and Weierstrass equations for the 25 different fibrations and analyze the reducible fibers and Mordell-Weil lattices. This answers completely a question posed by Kuwata and Shioda in 2008.
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.
We prove that if $X$ and $S$ are smooth varieties and $f\colon X\to S$ is an elliptic fibration with singular fibers curves of types I$_N$ with $N\geq 1$, II, III and IV, then the relative Jacobian $\hat{f}\colon \bar{M}_{X/S}\to S$ of $f$, defined as the relative moduli space of semistable pure dimension one sheaves of rank 1 and degree 0 on the fibers of $f$, is an elliptic fibration such that all its fibers are irreducible. This extends known results when fibers are integral or of type I$_2$.
Sections of Jacobian fibrations over lines
本文研究光滑曲面线性系中直线对应的通用雅可比截面的上同调类，针对K3曲面的通用紧化雅可比，修正了排印错误与猜想7，探讨结果与文献步骤的协调问题。
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。
Jacobian elliptic fibrations on the generalized Inose quartic of Picard rank sixteen
本文研究 Picard 格包含 $H \oplus E_7(-1) \oplus E_7(-1)$ 的复代数 K3 曲面族，证明其一般成员恰有四个互不等价的 Jacobian 椭圆纤维化，并构造了显式线束。
On Néron-Severi lattices of Jacobian elliptic K3 surfaces
本文分类了有限自同构群 K3 曲面及无限自同构群且 Néron-Severi 格为 2-初等情形下具有有限 Mordell-Weil 群的 Jacobian 椭圆纤维化，并计算了相应格论重数。
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.
The defect of a cubic threefold and applications to intermediate Jacobian fibrations
本文研究具有孤立奇点的三次三流形的缺陷不变量，利用过奇点的直线曲线拓扑数据计算缺陷，刻画混合霍奇结构，并证明非Q-阶乘性等价于包含平面或三次直纹面，进而联系到三次四流形的中间雅可比纤维化。
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.
Twists of intermediate Jacobian fibrations
本文通过引入解析相对雅可比层，研究OG10超凯勒拉格朗日量的截面、Tate–Shafarevich扭曲与周期，并用解析Brauer群计算扭曲群，证明三次四维流形的本原Hodge格与OG10流形第二上同调的特殊子格等距。
Birational geometry of the intermediate Jacobian fibration of a cubic fourfold
该研究证明了任意光滑三次四fold的中等雅可比纤维化具有超凯勒紧化，并计算了其双有理几何中的锥，同时确定了Mordell-Weil群与代数上同调的同构关系。
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。
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.
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。
This paper proposes a new geometric construction of Enriques surfaces. Its starting point are K3 surfaces with jacobian elliptic fibration which arise from rational elliptic surfaces by a quadratic base change. The Enriques surfaces obtained in this way are characterised by elliptic fibrations with a rational curve as bisection which splits into two sections on the covering K3 surface. The construction has applications to the study of Enriques surfaces with specific automorphisms. It also allows us to answer a question of Beauville about Enriques surfaces whose Brauer groups show an exceptiona…
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…
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$\mathbb{P}^2$。
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。
Counting elliptic fibrations on K3 surfaces
本文研究任意复射影 K3 曲面上模自同构的 Jacobian 椭圆纤维化计数问题，给出求解方法并以显式示例加以说明。
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.
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