斜体域中的交换族与 Beauville 纤维化的量子化

A remark on a question of Beauville about lagrangian fibrations

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.

Lagrangian fibrations on hyperkähler manifolds - On a question of Beauville

Let X be a compact hyperkähler manifold containing a complex torus L as a Lagrangian subvariety. Beauville posed the question whether X admits a Lagrangian fibration with fibre L. We show that this is indeed the case if X is not projective. If X is projective we find an almost holomorphic Lagrangian fibration with fibre L under additional assumptions on the pair (X, L), which can be formulated in topological or deformation-theoretic terms. Moreover, we show that for any such almost holomorphic Lagrangian fibration there exists a smooth good minimal model, i.e., a hyperkähler manifold birationa…

伴随丛与弱博韦尔-博戈莫洛夫分解的一个判别准则

A criterion for associated bundles and the weak Beauville--Bogomolov decomposition

本文给出纤维丛成为伴随纤维丛的判别准则，证明满足特定条件的满射纤维丛均为伴随纤维丛，进而推出丰反典范除子代数簇的弱博韦尔-博戈莫洛夫分解。

Lagrangian fibrations on generalized Kummer varieties

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.

Birational geometry of Beauville-Mukai systems III: asymptotic behavior

Suppose that a Hilbert scheme of points on a K3 surface S of Picard rank one admits a rational Lagrangian fibration. We show that if the degree of the surface is sufficiently large compared to the number of points, then the Hilbert scheme is the unique hyperkähler manifold in its birational class. In particular, the Hilbert scheme is a Lagrangian fibration itself, which we realize as coming from a (twisted) Beauville-Mukai system on a Fourier-Mukai partner of S. We also show that when the degree of the surface is small our method can be used to find all birational models of the Hilbert scheme.

On Lagrangian fibrations by Jacobians II

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.