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

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 $X$.