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Quot-Scheme Limit of Fubini–Study Metrics and Its Applications to Balanced Metrics

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Birational Geometry, Kähler–Einstein Metrics and Degenerations (BGKEMD 2019, BGKEMD 2019, BGKEMD 2019)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 409))

Abstract

We present some results that complement our prequels [27, 28] on holomorphic vector bundles. We apply the method of the Quot-scheme limit of Fubini–Study metrics developed therein to provide a generalisation to the singular case of the result originally obtained by X. W. Wang for the smooth case, which states that the existence of balanced metrics is equivalent to the Gieseker stability of the vector bundle. We also prove that the Bergman 1-parameter subgroups form subgeodesics in the space of Hermitian metrics. This paper also contains a review of techniques developed in [27, 28] and how they correspond to their counterparts developed in the study of the Yau–Tian–Donaldson conjecture.

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Acknowledgements

Both authors thank Yuji Odaka for helpful discussions, and the anonymous referees for helpful comments. The first author is partially supported by JSPS KAKENHI (Grant-in-Aid for Early-Career Scientists), Grant Number JP19K14524. The second author is supported by an NSERC Discovery Grant.

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Authors and Affiliations

  1. Department of Mathematics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo, 152-8551, Japan

    Yoshinori Hashimoto

  2. Département de Mathématiques, Université du Québec à Montréal (UQÀM), C.P. 8888, Succ. Centre-Ville, Montréal, QC, H3C 3P8, Canada

    Julien Keller

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  1. Yoshinori Hashimoto
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Correspondence to Julien Keller .

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Editors and Affiliations

  1. School of Mathematics, University of Edinburgh, Edinburgh, UK

    Ivan Cheltsov

  2. Mathematics Department, Stony Brook University, Stony Brook, NY, USA

    Xiuxiong Chen

  3. Department of Mathematics, University of Miami, Coral Gables, FL, USA

    Ludmil Katzarkov

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

    Jihun Park

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Hashimoto, Y., Keller, J. (2023). Quot-Scheme Limit of Fubini–Study Metrics and Its Applications to Balanced Metrics. In: Cheltsov, I., Chen, X., Katzarkov, L., Park, J. (eds) Birational Geometry, Kähler–Einstein Metrics and Degenerations. BGKEMD BGKEMD BGKEMD 2019 2019 2019. Springer Proceedings in Mathematics & Statistics, vol 409. Springer, Cham. https://doi.org/10.1007/978-3-031-17859-7_14

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