Intersection numbers of extremal rays on holomorphic symplectic varieties

Understand

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.