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On Gromov–Yomdin type theorems and a categorical interpretation of holomorphicity

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Abstract

In topological dynamics, the Gromov–Yomdin theorem states that the topological entropy of a holomorphic automorphism f of a smooth projective variety is equal to the logarithm of the spectral radius of the induced map \(f^*\). In order to establish a categorical analogue of the Gromov–Yomdin theorem, one first needs to find a categorical analogue of a holomorphic automorphism. In this paper, we propose a categorical analogue of a holomorphic automorphism and prove that the Gromov–Yomdin type theorem holds for them.

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Acknowledgements

The authors would like to thank Ed Segal for suggesting that, passed through homological mirror symmetry, the notion of holomorphicity might be formulated in terms of preserving a stability condition. F.B. was supported by the European Research Council (ERC) under the European Union Horizon 2020 research and innovation programme (Grant Agreement No. 725010). J.K. was supported by the Institute for Basic Science (IBS-R003-D1).

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Authors and Affiliations

  1. Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, UK

    Federico Barbacovi

  2. Center for Geometry and Physics, Institute for Basic Science (IBS), Pohang, 37673, Republic of Korea

    Jongmyeong Kim

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  1. Federico Barbacovi
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  2. Jongmyeong Kim
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Correspondence to Jongmyeong Kim.

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Barbacovi, F., Kim, J. On Gromov–Yomdin type theorems and a categorical interpretation of holomorphicity. Sel. Math. New Ser. 29, 65 (2023). https://doi.org/10.1007/s00029-023-00870-x

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