q-Stability conditions on Calabi-Yau-X categories

Abstract: We introduce $q$-stability conditions $(\sigma,s)$ on Calabi-Yau-$\mathbb{X}$ categories $\mathcal{D}\mathbb{X}$, where $\sigma$ is a stability condition on $\mathcal{D}\mathbb{X}$ and $s$ a complex number. We prove the corresponding deformation theorem, that $\operatorname{QStab}s\mathcal{D}\mathbb{X}$ is a complex manifold of dimension $n$ for fixed $s$, where $n$ is the rank of the Grotendieck group of $\mathcal{D}\mathbb{X}$ over $\mathbb{Z}[q^{\pm 1}]$. When $s=N$ is an integer, we show that the $q$-stability conditions can be identified with the stability conditions on $\mathcal{D}_N$, provided the orbit category $\mathcal{D}_N=\mathcal{D}\mathbb{X}/[\mathbb{X}-N]$ is well defined. To attack the questions on existence and deformation along $s$ direction, we introduce the inducing method. Sufficient and necessary conditions are given, for a stability condition on an $\mathbb{X}$-baric heart (that is, an usual triangulated category) of $\mathcal{D}\mathbb{X}$ to induce $q$-stability conditions on $\mathcal{D}\mathbb{X}$. As a consequence, we show that the space $\operatorname{QStab}^\oplus\mathcal{D}_\mathbb{X}$ of (induced) open $q$-stability conditions is a complex manifold of dimension $n+1$. Our motivating examples for $\mathcal{D}_\mathbb{X}$ are coming from Calabi-Yau-$\mathbb{X}$ completions of dg algebras. In the case of smooth projective varieties, the $\mathbb{C}^*$-equivariant coherent sheaves on canonical bundles provide the Calabi-Yau-$\mathbb{X}$ categories. Another application is that we show the prefect derived categories can be realized as cluster-$\mathbb{X}$ categories for acyclic quivers.