Intersection numbers of extremal rays on holomorphic symplectic varieties

TL;DR

This paper develops a framework for understanding how extremal rays in holomorphic symplectic varieties interact under the Beauville-Bogomolov form, emphasizing the role of Lagrangian subspaces in controlling the cone of curves.

Contribution

It introduces a general framework linking extremal rays and Lagrangian subspaces, providing new insights into the geometry of holomorphic symplectic manifolds and their cone structures.

Findings

Extremal rays associated with Lagrangian projective subspaces influence the cone of curves.

Evidence supports conjectures relating extremal rays to geometric structures in specific examples.

Implications for Hilbert schemes of points on K3 surfaces and Kummer varieties are explored.

Abstract

We propose a general framework governing the intersection properties of extremal rays of irreducible holomorphic symplectic manifolds under the Beauville-Bogomolov form. Our main thesis is that extremal rays associated to Lagrangian projective subspaces control the behavior of the cone of curves. We explore implications of this philosophy for examples like Hilbert schemes of points on K3 surfaces and generalized Kummer varieties. We also collect evidence supporting our conjectures in specific cases.