Singularities of Bridgeland moduli spaces for K3 categories: an update

TL;DR

This paper investigates the local structure and singularities of Bridgeland moduli spaces for K3 categories, establishing their normality and irreducibility, and relating stability condition variations to GIT quotient changes.

Contribution

It provides a direct proof of formality, confirms normality and irreducibility of these moduli spaces, and links GIT variation with stability condition changes.

Findings

Proves formality of moduli spaces

Establishes normality and irreducibility of moduli spaces

Connects GIT quotient variation with stability condition changes

Abstract

This survey is a continuation of the study undertaken in \cite{AS18}. We examine the local structure of Bridgeland moduli spaces $M_\sigma(v,\D)$, where the relevant triangulated category $\D$ is either the bounded derived category $\D=\D^b(X)$ of a K3 surface $X$, or the Kuznetsov component $\D=\Ku(Y)\subset \D ^b(Y)$ of a smooth cubic fourfold $Y\subset \PP^5$. For these moduli spaces, building on \cite{Bmm19}, \cite{Bmm21} we give a direct proof of formality and, using their local isomorphism with quiver varieties, we establish their normality and their irreducibility, as long as $\sigma$ does not lie on a totally semistable wall. We then connect the variation of GIT quotients for quiver varieties with the changing of stability conditions on moduli spaces.