Abstracts Mathematics

Stability conditions on Kuznetsov components of Gushel-Mukai threefolds and Serre functor

by Ethan Robinett

This dissertation focuses on the construction of Serre-invariant Bridgeland stability conditions on Kuznetsov components of Gushel-Mukai threefolds. In particular, for a Gushel-Mukai threefold $X$ with Kuznetsov component $\textrm{Ku}(X)$ and Serre functor $S_{\textrm{Ku}(X)}$, we find a family of stability conditions $\sigma (s,q)$ on $\textrm{Ku}(X)$ such that $S_{\textrm{Ku}(X)} \cdot \sigma(s,q)= \sigma(s,q) \cdot \tilde{g}$ for some $\tilde{g}$ residing in the universal cover of $\textrm{GL}_2^+(\mathbb{R})$. This leads to an explicit construction of Bridgeland stability conditions on Kuznetsov components of special Gushel-Mukai fourfolds, which previously was not known.