On Hyperk\"ahler manifolds of K3$^{[n]}$-type with large Picard number

TL;DR

This paper proves that hyperk"ahler manifolds of K3$^{[n]}$-type with Picard number at least 4 are moduli spaces of twisted sheaves on K3 surfaces, and describes cases with lower Picard numbers that are not birational to such spaces.

Contribution

It establishes a threshold Picard number for hyperk"ahler manifolds to be isomorphic to moduli spaces of twisted sheaves, and characterizes those below this threshold.

Findings

Hyperk"ahler manifolds with Picard number ≥ 4 are moduli spaces of twisted sheaves.

Explicit examples of manifolds with Picard number 3 not birational to such moduli spaces.

Provides a classification based on Picard number thresholds.

Abstract

Inspired by well-known examples of hyperk\"ahler manifolds, we show that any hyperk\"ahler manifold $X$ of K3$^{[n]}$-type with Picard number $\rho(X) \geq 4$ is always isomorphic to a moduli space of twisted stable sheaves on a K3 surface. Additionally, we provide explicit descriptions of hyperk\"ahler manifolds of K3$^{[n]}$-type with Picard ranks below this crucial value (e.g., $\rho(X)=3$) that are not birational to such moduli spaces.