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On stability of tangent bundle of toric varieties

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Abstract

Let X be a nonsingular complex projective toric variety. We address the question of semi-stability as well as stability for the tangent bundle T X. In particular, a complete answer is given when X is a Fano toric variety of dimension four with Picard number at most two, complementing the earlier work of Nakagawa (Tohoku. Math. J. 45 (1993) 297–310; 46 (1994) 125–133). We also give an infinite set of examples of Fano toric varieties for which TX is unstable; the dimensions of this collection of varieties are unbounded. Our method is based on the equivariant approach initiated by Klyachko (Izv. Akad. Nauk. SSSR Ser. Mat. 53 (1989) 1001–1039, 1135) and developed further by Perling (Math. Nachr. 263/264 (2004) 181–197) and Kool (Moduli spaces of sheaves on toric varieties, Ph.D. thesis (2010) (University of Oxford); Adv. Math. 227 (2011) 1700–1755).

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Acknowledgements

The authors would like to thank the anonymous referee for many useful comments and suggestions that have helped to improve the exposition. The first-named author is supported by a J. C. Bose Fellowship. The second-named author is supported by a SERB MATRICS research grant. The third-named author is supported by Narodowe Centrum Nauki 2018/30/E/ST1/00530. The last-named author has been supported in part by an SRP grant from METU NCC and a SERB MATRICS grant.

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Author notes

  1. Ozhan Genc

    Present address: Department of Mathematics and Computer Science, Jagiellonian University, ul. prof. Stanisława, Łojasiewicza 6, 30-348, Kraków, Poland

Authors and Affiliations

  1. School of Mathematics, Tata Institute of Fundamental Research, Mumbai, 400 005, India

    Indranil Biswas

  2. Department of Mathematics, Indian Institute of Technology Madras, Chennai, 600 036, India

    Arijit Dey

  3. Middle East Technical University, Northern Cyprus Campus, Guzelyurt, Mersin 10, Turkey

    Ozhan Genc

  4. Mathematics Department, Indian Institute of Science Education and Research, Pune, 411 008, India

    Mainak Poddar

Authors
  1. Indranil Biswas
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  2. Arijit Dey
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  3. Ozhan Genc
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  4. Mainak Poddar
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Correspondence to Mainak Poddar.

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Communicating Editor: Parameswaran Sankaran

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Biswas, I., Dey, A., Genc, O. et al. On stability of tangent bundle of toric varieties. Proc Math Sci 131, 36 (2021). https://doi.org/10.1007/s12044-021-00623-w

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