Contractible flow of stability conditions via global dimension function

Abstract

We introduce an analytic method that uses the global dimension function $\mathrm{gldim}$ to produce contractible flows on the space $\mathrm{Stab}\mathcal{D}$ of stability conditions on a triangulated category $\mathcal{D}$. In the case when $\mathcal{D}=\mathcal{D}({\mathbf{S}}^{\lambda })$ is the topological Fukaya category of a graded surface ${\mathbf{S}}^{\lambda }$, we show that ${\mathrm{gldim}}^{-1}(0,y)$ contracts to ${\mathrm{gldim}}^{-1}(0,x)$ for any $1\le x\le y$, provided $(x,y)$ does not contain `critical' values $\{1+w_{\partial }/m_{\partial }\mid w_{\partial }\ge 0,\partial \in \partial {\mathbf{S}}^{\lambda }\}$, where the pair $(m_{\partial },w_{\partial })$ consists of the number $m_{\partial }$ of marked points and the winding number $w_{\partial }$ associated to a boundary component $\partial$ of ${\mathbf{S}}^{\lambda }$. One consequence is that the global dimension of $\mathcal{D}({\mathbf{S}}^{\lambda })$ must be one of these critical values. Besides, we remove the assumptions in Kikuta-Ouchi-Takahashi's classification result on triangulated categories with global dimension less than 1.

Many changes in Section 5 to correct some mistakes

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Cited by in corpus (1)

• Contractibility of space of stability conditions on the projective plane via global dimension function