Stability conditions and exceptional objects in triangulated categories

The goal of this paper is to study the subspace of stability condition $\mathcal{E} \subset \operatorname{Stab}(X)$ associated to an exceptional collection $\mathcal{E}$ on a projective variety $X$. Following Macrì’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 $\mathcal{E} = \lbrace \mathcal{O},\mathcal{O}(1), \mathcal{O}(2), \mathcal{O}(3) \rbrace$, the space $\Sigma_\mathcal{E}$ is a connected and simply connected $4$-dimensional complex manifold.

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Received 8 October 2018

Accepted 20 January 2019

Published 14 December 2020