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Existence of twisted constant scalar curvature Kähler metrics with a large twist

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Abstract

Suppose that there exist two Kähler metrics \(\omega \) and \(\alpha \) such that the metric contraction of \(\alpha \) with respect to \(\omega \) is constant, i.e. \(\varLambda _{\omega } \alpha = \text {const}\). We prove that for all large enough \(R>0\) there exists a twisted constant scalar curvature Kähler metric \(\omega '\) in the cohomology class \([ \omega ]\), satisfying \(S(\omega ' ) - R \varLambda _{\omega ' } \alpha = \text {const}\). We discuss its implication to K-stability and the continuity method recently proposed by Chen.

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Notes

  1. How large R must be depends on m, but this will not concern us since m will be fixed later.

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Acknowledgements

The author is very grateful to Xiuxiong Chen for kindly acknowledging this work in [6] although at that time this work was only in a draft state, and also for encouragements. The author thanks Ruadhaí Dervan for many helpful discussions on twisted cscK metrics and helpful comments. Thanks are also due to the author’s former supervisor Jason Lotay and the anonymous referee for helpful comments.

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  1. Yoshinori Hashimoto

    Present address: Dipartimento di Matematica e Informatica “U. Dini”, Università degli Studi di Firenze, Viale Morgagni 67/A, 50134, Firenze, Italy

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

    Yoshinori Hashimoto

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  1. Yoshinori Hashimoto
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Correspondence to Yoshinori Hashimoto.

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Most of this work was carried out when the author was a Ph.D. student at the Department of Mathematics of the University College London, which he thanks for the financial support. Part of this work was carried out in the framework of the Labex Archimède (ANR-11-LABX-0033) and of the A*MIDEX project (ANR-11-IDEX-0001-02), funded by the “Investissements d’Avenir” French Government programme managed by the French National Research Agency (ANR).

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Hashimoto, Y. Existence of twisted constant scalar curvature Kähler metrics with a large twist. Math. Z. 292, 791–803 (2019). https://doi.org/10.1007/s00209-018-2133-y

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