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q-stability conditions on Calabi–Yau-𝕏 categories

Part of: Complex manifolds

Published online by Cambridge University Press:  07 June 2023

Akishi Ikeda  and
Yu Qiu
Akishi Ikeda
Affiliation:
Department of Mathematics, Josai University, Saitama 338 8570, Japan akishi@josai.ac.jp
Yu Qiu
Affiliation:
Yau Mathematical Sciences Center and Department of Mathematical Sciences, Tsinghua University, 100084 Beijing, PR China yu.qiu@bath.edu Beijing Institute of Mathematical Sciences and Applications, Yanqi Lake, 101408 Beijing, PR China
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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 Grothendieck group of $\mathcal {D}_\mathbb {X}$ over $\mathbb {Z}[q^{\pm 1}]$. When $s=N$ is an integer, we show that $q$-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 the $s$ direction, we introduce the inducing method. Sufficient and necessary conditions are given, for a stability condition on an $\mathbb {X}$-baric heart (that is, a 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 (Keller's) 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 perfect derived categories can be realized as cluster-$\mathbb {X}$ categories for acyclic quivers.

Keywords

q-deformationstability conditionsCalabi–Yau-𝕏 categoriescluster-𝕏 categories

MSC classification

Secondary: 32Q26: Notions of stability
Type
Research Article
Information
Compositio Mathematica , Volume 159 , Issue 7 , July 2023 , pp. 1347 - 1386
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Copyright
© 2023 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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