Lagrangian fibrations for IHS fourfolds

Abstract

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 higherdimensional cases.

Lagrangian fibrations

“…Singularities of the base. It is generally expected that the base manifold B is smooth, but at the moment this is only known for n ď 2, see [Ou19,BK18,HX19]. The expectation is corroborated by the following computations of invariants of the singularities of B.…”

On intersection cohomology and Lagrangian fibrations of irreducible symplectic varieties

“…2 By [36, Theorem 3 (3)], a general fiber of a Lagrangian fibration associated with any irreducible symplectic variety is an abelian variety. 3 The fact that Matsushita's example is irreducible symplectic is verified by [34,Proposition 2.5].…”

“…This was confirmed by Hwang [23] assuming B nonsingular, and remained wide open in general. See also [33,22,3] for recent progress.…”

“…An interesting phenomenon occurs in the singular case that the base B may fail to be a projective space. In [28, Theorem 1.9], Matsushita constructed an irreducible symplectic variety 3 whose Lagrangian base is a Fano variety with quotient singularities. Nevertheless, Theorem 0.1 (c) guarantees that B shares the same intersection cohomology as P n .…”

On the Chow ring of some Lagrangian fibrations

“…1 For background on Lagrangian fibrations, cf. the foundational [29] as well as the recent [7], [18] and the references given there.…”