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 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 investigate the stability of fibers of coisotropic fibrations on holomorphic symplectic manifolds and generalize Voisin's result on Lagrangian subvarieties to this framework. We present applications to the moduli space of holomorphic symplectic manifolds which are deformations equivalent to Hilbert schemes of points on a $K3$ surface or to generalized Kummer manifolds.
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.
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.
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 $π$.
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 ${\mathbb P}^2$.
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.
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…
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…
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.
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.
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.
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.
In this paper we analyse the birational geometry of O'Grady ten dimensional manifolds, giving a characterisation of Kaehler classes and lagrangian fibrations. Moreover, we study symplectic compactifications of intermediate jacobian fibrations of smooth cubic fourfolds.
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.
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).
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.
We describe certain Lagrangian fibrations by Prym varieties on holomorphic symplectic varieties that deform to compactifications of Sp-Hitchin systems.
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.
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
Columbia University（哥伦比亚大学）；Korea Institute for Advanced Study（韩国高等研究院）；National Center for Theoretical Sciences, National Taiwan University（台湾大学理论科学研究中心）；University of Sheffield（谢菲尔德大学）
We calculate curvature tensors of metrics on the total spaces of holomorphic fibrations. Our main tool is a theory of Chern connections and curvature forms for possibly degenerate Hermitian forms on holomorphic vector bundles. We prove a version of the Codazzi-Griffiths equations for curvatures of sub- and quotient bundles in that setting and apply them to the study of honest Hermitian metrics on fibrations. We apply this to calculate the curvature of a metric on Grassmannian bundles and prove they have positive holomorphic sectional curvature if the base does.
A hypercomplex manifold is a manifold equipped with a triple of complex structures satisfying the quaternionic relations. A holomorphic Lagrangian variety on a hypercomplex manifold with trivial canonical bundle is a holomorphic subvariety which is calibrated by a form associated with the holomorphic volume form; this notion is a generalization of the usual holomorphic Lagrangian subvarieties known in hyperkaehler geometry. An HKT (hyperkaehler with torsion) metric on a hypercomplex manifold is a metric determined by a local potential, in a similar way to the Kaehler metric. We prove that a ba…
A holomorphic Lagrangian fibration on a holomorphically symplectic manifold is a holomorphic map with Lagrangian fibers. It is known that a given compact manifold admits only finitely many holomorphic symplectic structures, up to deformation. We prove that a given compact manifold with $b_2 \geq 7$ admits only finitely many deformation types of holomorphic Lagrangian fibrations. We also prove that all known hyperkahler manifolds are never Kobayashi hyperbolic.
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 $ϕ:X\rightarrow B$ 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.
We study holomorphic symplectic manifolds which are fibred by abelian varieties. This structure is a higher dimensional analogue of an elliptic fibration on a K3 surface. We investigate when a holomorphic symplectic manifold is fibred in this way, and are led to several natural conjectures. We then study the geometry of these fibrations. The expectation is that this point of view will prove useful in understanding holomorphic symplectic manifolds, and possibly lead to a classification.
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.
We use twisted Fourier-Mukai transforms to study the relation between an abelian fibration on a holomorphic symplectic manifold and its dual fibration. Our reasoning leads to an equivalence between the derived category of coherent sheaves on one space and the derived category of twisted sheaves on the other space.
The main result of this note essentially is that if the base and fibers of a compact fibration carry Hermitian metrics of positive holomorphic sectional curvature, then so does the total space of the fibration. The proof is based on the use of a warped product metric as in the work by Cheung in case of negative holomorphic sectional curvature, but differs in certain key aspects, e.g., in that it does not use the subadditivity property for holomorphic sectional curvature due to Grauert-Reckziegel and Wu.
In this paper, we pose several conjectures on structures and images of maximal rationally connected fibrations of smooth projective varieties admitting semi-positive holomorphic sectional curvature. Toward these conjectures, we prove that the canonical bundle of images of such fibrations is not big. Our proof gives a generalization of Yang's solution using RC positivity for Yau's conjecture. As an application, we show that any compact Kähler surface with semi-positive holomorphic sectional curvature is rationally connected, or a complex torus, or a ruled surface over an elliptic curve.
