Stability Conditions and Exceptional Objects in Triangulated Categories

Abstract

The goal of this paper is to study the subspace of stability condition $\Sigma_{\mathcal{E}}\subset \mathrm{Stab}(X)$ associated to an exceptional collection $\mathcal{E}$ on a projective variety $X$. Following Emanuele Macr\`{i}'s approach, we show a certain correspondence between the homotopy class of continuous loops in $\Sigma_{\mathcal{E}}$ and words of the braid group. In particular, we prove that in the case $X=\mathbb{P}^3$ and $\mathcal{E}=\{\mathcal{O},\mathcal{O}(1),\mathcal{O}(2),\mathcal{O}(3)\}$, the space $\Sigma_{\mathcal{E}}$ is a connected and simply connected 4-dimensional manifold.