Twisted cotangent bundles of Hyperk\"ahler manifolds

TL;DR

This paper establishes a lower bound for the pseudoeffectivity of twisted cotangent bundles on Hyperk"ahler manifolds, with explicit bounds for Hilbert schemes of K3 surfaces, advancing understanding of their geometric properties.

Contribution

It provides a new lower bound criterion for the pseudoeffectivity of twisted cotangent bundles on Hyperk"ahler manifolds, including explicit bounds for specific deformation types.

Findings

Lower bound for pseudoeffectivity in terms of Beauville-Bogomolov form

Explicit bounds for Hilbert schemes of K3 surfaces

Analysis of the optimality of bounds

Abstract

Let $X$ be a Hyperk\"ahler manifold, and let $H$ be an ample divisor on $X$. We give a lower bound in terms of the Beauville-Bogomolov form $q(H)$ for the twisted cotangent bundle $\Omega_X \otimes H$ to be pseudoeffective. If $X$ is deformation equivalent to the Hilbert scheme of a K3 surface the lower bound can be written down explicitly and we study its optimality.