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 ${\mathrm{QStab}}_s{\mathcal{D}}_{\mathbb{X}}$ 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 ${\mathrm{QStab}}^{\oplus }{\mathcal{D}}_{\mathbb{X}}$ 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 ${\mathbb{C}}^{\ast }$-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.

Compos. Math. to appear