相关论文: Multiple fibers of holomorphic Lagrangian fibratio…

Let $X \to S$ be a minimal abelian fibration of relative dimension $n$ over a curve. We classify all possible singular fibers $X_s$ having $(n-1)$-dimensional ``abelian variety parts''. This generalizes Kodaira's work on elliptic…

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,…

Let $\pi : X \to B$ be a Lagrangian fibration of a compact hyper-K\"ahler manifold. We prove that every codimension $1$ fiber of $\pi$ has multiplicity $1$.

Let $\pi:X\rightarrow\mathbb{P}^n$ 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…

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…

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…

We prove that every irreducible component of a fibre of a complex Lagrangian fibration is Lagrangian subvariety. Especially, complex Lagrangian fibations are equidimensional.

The Milnor fibre of any isolated hypersurface singularity contains many exact Lagrangian spheres: the vanishing cycles associated to a Morsification of the singularity. Moreover, for simple singularities, it is known that the only possible…

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 hyperk\"ahler Lagrangian…

For the sake of hyperk{\"a}hler SYZ conjecture, finding holomorphic Lagrangian fibrations becomes an important issue. Toric hyperk{\"a}hler manifolds are real dimension $4n$ non-compact hyperk{\"a}hler manifolds which are quaternion analog…

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…

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.

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…

This paper classifies Lagrangian fibrations over surfaces with compact total spaces up to fiberwise symplectomorphism identical on the base.

We give an algorithm to classify singular fibers of finite cyclic covering fibrations of a ruled surface by using singularity diagrams. As the first application, we classify all fibers of 3-cyclic covering fibrations of genus 4 of a ruled…

In this note, we reduce various conjectures in birational geometry, including Shokurov conjecture on singularities of the base of log Calabi-Yau fibrations of Fano type and boundedness conjecture for rationally connected Calabi-Yau…

Let $f:X@>>>\Bbb P^1$ be a fibered surface with fibers of genus g>1. If f is semistable and non isotrivial we prove that X of non negative Kodaira dimension implies that the number s of singular fibers is at least 5. Information about the…

We survey Lagrangian fibrations of holomorphic symplectic varieties, both compact and non-compact, whose fibres are Jacobians and Prym varieties.

A Kodaira fibration is a compact, complex surface admitting a holomorphic submersion onto a complex curve, such that the fibers have nonconstant moduli. We consider Kodaira fibrations X with nontrivial invariant rational cohomology in…

We define topological invariants of regular Lagrangian fibrations using the integral affine structure on the base space and we show that these coincide with the classes known in the literature. We also classify all symplectic types of…