Finiteness of isomorphic classes in the set of moduli schemes of sheaves on a surface

TL;DR

This paper proves that for certain complex surfaces with trivial canonical bundle, the set of moduli schemes of semistable sheaves with fixed Chern classes has only finitely many isomorphic classes as abstract schemes, when varying over all generic polarizations.

Contribution

It establishes the finiteness of isomorphic classes of moduli schemes of sheaves on Calabi-Yau surfaces with fixed Chern classes, varying over all generic polarizations.

Findings

Finiteness of isomorphic classes of moduli schemes on surfaces with $K_X \,\sim\, 0$

Finiteness holds for all $ ext{alpha}$-generic polarizations

Applicable to non-singular complex projective surfaces with trivial canonical bundle

Abstract

When a non-singular complex projective surface $X$ satisfies that $K_X\sim 0$, we shall show that there are only finitely many isomorphic classes as abstract schemes in the set of moduli scheme of $H$-semistable sheaves with fixed Chern classes $\alpha$ on $X$, where $H$ runs over the set of all $\alpha$-generic polarizations on $X$.