Irreducible symplectic varieties with a large second Betti number

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.

28 pages. Comments are welcome! v5: A footnote is added since Kollár pointed out that our example is not smooth (see footnote 2). v4: To appear in Crelle's Journal