Smoothness of the moduli space of complexes of coherent sheaves on an abelian or a projective K3 surface

TL;DR

This paper proves that the moduli space of certain complexes of coherent sheaves on abelian or K3 surfaces is smooth and symplectic, revealing geometric properties of these moduli spaces.

Contribution

It establishes the smoothness and symplectic structure of the moduli space of complexes with specific Ext and Hom conditions on abelian or K3 surfaces.

Findings

The moduli space is smooth.

The moduli space has a symplectic structure.

Results apply to complexes satisfying certain Ext and Hom conditions.

Abstract

For an abelian or a projective K3 surface $X$ over an algebraically closed field $k$, consider the moduli space $\splcpx_{X/k}\uet$ of the objects $E$ in $D^b(\mathrm{Coh}(X))$ satisfying $\Ext^{-1}_X(E,E)=0$ and $\Hom(E,E)\cong k$. Then we can prove that $\splcpx_{X/k}\uet$ is smooth and has a symplectic structure.