Moving and ample cones of holomorphic symplectic fourfolds

TL;DR

This paper studies the geometric cones of holomorphic symplectic fourfolds, providing a numerical criterion for divisor ampleness based on recent minimal model program developments.

Contribution

It introduces a new numerical criterion for ampleness of divisors on certain fourfolds, advancing understanding of their geometric structure.

Findings

Established a numerical criterion for divisor ampleness

Analyzed the structure of ample and moving cones

Connected cone structure with minimal model program advances

Abstract

We analyze the ample and moving cones of holomorphic symplectic manifolds, in light of recent advances in the minimal model program. As an application, we establish a numerical criterion for ampleness of divisors on fourfolds deformation-equivalent to punctual Hilbert schemes of K3 surfaces.