On the discriminant locus of a Lagrangian fibration

Abstract

Let X → P n be an irreducible holomorphic symplectic manifold of dimension 2n fibred over P n . Matsushita proved that the generic fibre is a holomorphic Lagrangian abelian variety. In this article we study the discriminant locus ∆ ⊂ P n parametrizing singular fibres. Our main result is a formula for the degree of ∆, leading to bounds on the degree when X is a four-fold.

A finiteness theorem for Lagrangian fibrations

“…The precise meanings of "rank-one semi-stable degeneration" and "maximal variation" will be made clear in Section 2, and these hypotheses are discussed further in Subsections 4.1 to 4.3. Some restrictions on the types of polarizations that can occur in dimension 2n = 4 were previously found by the author [28]. The idea of the proof is as follows.…”

Critical Loci for Higgs Bundles

“…1. One is to follow [22] where the author works in the compact case, using exact sequences of sheaves. In that case the base is P m rather than C m which gives a non-zero contribution if m > 1.…”

Lagrangian fibrations by Prym varieties

“…30 and 18 for the Beauville-Mukai and Debarre systems, respectively). For Lagrangian fibrations on smooth holomorphic symplectic man-ifolds, this can be related to topological invariants of the total space X (see [49]), but it is not clear whether the same relation holds for orbifolds.…”