Stability Conditions and Exceptional Objects in Triangulated Categories

TL;DR

This paper investigates the structure of stability conditions related to exceptional collections on projective varieties, establishing a connection with braid groups and demonstrating the topological properties of certain stability subspaces.

Contribution

It demonstrates a correspondence between loops in stability subspaces and braid group words, and proves the connectedness and simple connectivity of these spaces for specific cases.

Findings

The subspace $oldsymbol{ ext{Stab}}(X)$ associated to an exceptional collection is studied.

A correspondence between loops in the subspace and braid group elements is established.

The space $oldsymbol{ ext{Stab}}_{oldsymbol{ ext{E}}}$ for $oldsymbol{ ext{X}=oldsymbol{ ext{P}}^3}$ and a specific exceptional collection is shown to be a connected, simply connected 4-manifold.

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.