Effective global generation on varieties with numerically trivial canonical class

TL;DR

This paper establishes effective global generation results for varieties with numerically trivial canonical bundles, utilizing properties of semihomogeneous bundles and extending to moduli spaces on abelian surfaces.

Contribution

It proves a Fujita-type theorem for such varieties and derives explicit global generation bounds for line bundles on Hilbert squares of abelian surfaces.

Findings

Global generation of line bundles on Hilbert squares for m ≥ 3

Effective bounds for global generation on varieties with trivial canonical class

Connections between semihomogeneous bundles and moduli space properties

Abstract

We prove a Fujita-type theorem for varieties with numerically trivial canonical bundle using properties of semihomogeneous bundles on abelian varieties. We combine our results with work of Riess on compact hyperk\"{a}hler manifolds and work of Mukai, Pareschi and Yoshioka to obtain effective global generation statements for certain moduli spaces of sheaves on abelian surfaces. Among these is the statment that if $\cL$ is an ample line bundle on the Hilbert square $S^{[2]}$ of an abelian surface $S,$ then $\cL^{\otimes m}$ is globally generated for $m \geq 3.$