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.

arXiv commentCompos. Math. to appear

References in corpus (15)

• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• On triangulated orbit categories
• Wall-crossing, Hitchin Systems, and the WKB Approximation
• Flat surfaces and stability structures
• Stability conditions on a non-compact Calabi-Yau threefold
• Exchange graphs and Ext quivers
• Stability conditions and quantum dilogarithm identities for Dynkin quivers
• Decorated marked surfaces: spherical twists versus braid twists
• Stability conditions and the $A_2$ quiver
• Auslander orders over nodal stacky curves and partially wrapped Fukaya categories
• Derived categories of graded gentle one-cycle algebras
• Picard-Lefschetz theory and dilating C^*-actions
• $q$-Stability conditions via $q$-quadratic differentials for Calabi-Yau-$\mathbb{X}$ categories
• Global dimension function on stability conditions and Gepner equations
• The (he)art of gluing

Cited by in corpus (9)

• Stability conditions and the $A_2$ quiver
• Dimensional reduction in cohomological Donaldson-Thomas theory
• $q$-Stability conditions via $q$-quadratic differentials for Calabi-Yau-$\mathbb{X}$ categories
• Erratum to "Deformed Calabi-Yau completions"
• Global dimension function on stability conditions and Gepner equations
• Contractibility of space of stability conditions on the projective plane via global dimension function
• Graded decorated marked surfaces: Calabi-Yau-$\mathbb{X}$ categories of gentle algebras
• Contractible flow of stability conditions via global dimension function
• Decorated Marked Surfaces: Calabi-Yau categories and related topics