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Blowing up extremal Kähler manifolds II

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Abstract

This is a continuation of the work of Arezzo–Pacard–Singer and the author on blowups of extremal Kähler manifolds. We prove the conjecture stated in Székelyhidi (Duke Math J 161(8):1411–1453, 2012), and we relate this result to the K-stability of blown up manifolds. As an application we prove that if a Kähler manifold \(M\) of dimension \(>\)2 admits a constant scalar curvature (cscK) metric, then the blowup of \(M\) at a point admits a cscK metric if and only if it is K-stable, as long as the exceptional divisor is sufficiently small.

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Acknowledgments

I would like to thank Frank Pacard and Michael Singer for several useful discussions.

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Authors and Affiliations

  1. Department of Mathematics, University of Notre Dame, Notre Dame, IN, 46615, USA

    Gábor Székelyhidi

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  1. Gábor Székelyhidi
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Correspondence to Gábor Székelyhidi.

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Székelyhidi, G. Blowing up extremal Kähler manifolds II. Invent. math. 200, 925–977 (2015). https://doi.org/10.1007/s00222-014-0543-y

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