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A new holomorphic invariant and uniqueness of Kähler-Ricci solitons

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Commentarii Mathematici Helvetici

Abstract.

In this paper, a new holomorphic invariant is defined on a compact Kähler manifold with positive first Chern class and nontrivial holomorphic vector fields. This invariant generalizes the Futaki invariant. We prove that this invariant is an obstruction to the existence of Kähler-Ricci solitons. In particular, using this invariant together with the main result in [TZ1], we solve completely the uniqueness problem of Kähler-Ricci solitons. Two functionals associated to the new holomorphic invariant are also discussed. The main result here was announced in [TZ2].

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Authors and Affiliations

  1. Massachusetts Institute of Technology, Department of Mathematics, Cambridge, MA 02139, USA, e-mail: tian@math.mit.edu , , , , , , US

    G. Tian

  2. The Australian National University, Centre for Math. and its Appl., School of Mathematical Sciences, Canberra, ACT 0200, Australia , , , , , , AU

    X. Zhu

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  1. G. Tian
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  2. X. Zhu
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Received: December 8, 2000

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Tian, G., Zhu, X. A new holomorphic invariant and uniqueness of Kähler-Ricci solitons. Comment. Math. Helv. 77, 297–325 (2002). https://doi.org/10.1007/s00014-002-8341-3

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  • DOI: https://doi.org/10.1007/s00014-002-8341-3

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