On the existence of (H,A)-stable sheaves on K3 or abelian surfaces

TL;DR

This paper proves the existence of certain stable sheaves on K3 and abelian surfaces, leading to new symplectic structures on moduli spaces of sheaves, with implications for algebraic geometry.

Contribution

It establishes the existence of (H,A)-stable sheaves with primitive invariants on K3 or abelian surfaces, advancing understanding of moduli space structures.

Findings

Existence of (H,A)-stable sheaves with primitive invariants.

Construction of singular symplectic terminalisations of moduli spaces.

Application to Gieseker semistable sheaves on surfaces.

Abstract

We give an existence result on (H,A)-stable sheaves on a K3 or abelian surface X with primitive triple of invariants (rank,first Chern class,Euler characteristics) in the integral cohomology lattice. Such a result yields the existence of singular projective Q-factorial symplectic terminalisations of certain moduli spaces of sheaves on X that are Gieseker semistable with respect to a nongeneral ample divisor.