Irreducible symplectic varieties with a large second Betti number

TL;DR

This paper establishes the existence of certain irreducible symplectic varieties with large second Betti numbers, specifically in the context of Lagrangian fibrations related to cubic fivefolds, expanding the classification of such varieties.

Contribution

It proves a general existence result for irreducible symplectic compactifications of non-compact Lagrangian fibrations and constructs new examples with high second Betti numbers.

Findings

Existence of irreducible symplectic compactifications with second Betti number ≥ 24.

Construction of a new deformation type of irreducible symplectic varieties.

The relative Jacobian of certain cubic fivefolds admits a Lagrangian fibration.

Abstract

We prove a general result on the existence of irreducible symplectic compactifications of non-compact Lagrangian fibrations. As an application, we show that the relative Jacobian fibration of cubic fivefolds containing a fixed cubic fourfold can be compactified by a $\mathbb{Q}$-factorial terminal irreducible symplectic variety with the second Betti number at least 24, and admits a Lagrangian fibration whose base is a weighted projective space. In particular, it belongs to a new deformation type of irreducible symplectic varieties.