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Serre dimension and stability conditions

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Abstract

We study relations between the Serre dimension defined as the growth of the entropy of the Serre functor and the global dimension of Bridgeland stability conditions due to Ikeda–Qiu. A fundamental inequality between the Serre dimension and the infimum of the global dimensions is proved. Moreover, we characterize Gepner type stability conditions on fractional Calabi–Yau categories via the Serre dimension, and classify triangulated categories of Serre dimension lower than one with a Gepner type stability condition.

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Acknowledgements

The authors would like to thank Tom Bridgeland for valuable discussions. K. Kikuta is supported by JSPS KAKENHI Grant Number JP17J00227 and the JSPS program “Overseas Challenge Program for Young Researchers”. G. Ouchi is supported by Interdisciplinary Theoretical and Mathematical Science Program (iTHEMS) in RIKEN and JSPS KAKENHI Grant number 19K14520. A. Takahashi is supported by JSPS KAKENHI Grant number 16H06337.

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Authors and Affiliations

  1. Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka, 560-0043, Japan

    Kohei Kikuta & Atsushi Takahashi

  2. Interdisciplinary Theoretical and Mathematical Sciences Program, RIKEN, 2-1 Hirosawa, Wako, Saitama, 351-0198, Japan

    Genki Ouchi

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  1. Kohei Kikuta
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  2. Genki Ouchi
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  3. Atsushi Takahashi
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Correspondence to Kohei Kikuta.

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Kikuta, K., Ouchi, G. & Takahashi, A. Serre dimension and stability conditions. Math. Z. 299, 997–1013 (2021). https://doi.org/10.1007/s00209-021-02718-6

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  • DOI: https://doi.org/10.1007/s00209-021-02718-6

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