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Two conjectures on Ricci-flat Kähler metrics

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Abstract

We propose two conjectures about Ricci-flat Kähler metrics: Conjecture 1: A Ricci-flat projectively induced metric is flat. Conjecture 2: A Ricci-flat metric on an n-dimensional complex manifold such that the \(a_{n+1}\) coefficient of the TYZ expansion vanishes is flat. We verify Conjecture 1 (see Theorem 1.1) under the assumptions that the metric is radial and stable-projectively induced and Conjecture 2 (see Theorem 1.2) for complex surfaces whose metric is either radial or complete and ALE. We end the paper by showing, by means of the Simanca metric, that the assumption of Ricci-flatness in Conjecture 1 and in Theorem 1.2 cannot be weakened to scalar-flatness (see Theorem 1.3).

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Authors and Affiliations

  1. Dipartimento di Matematica e Informatica, Università di Cagliari, Via Ospedale 72, 09124, Cagliari, Italy

    Andrea Loi, Filippo Salis & Fabio Zuddas

Authors
  1. Andrea Loi
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  2. Filippo Salis
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  3. Fabio Zuddas
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Correspondence to Andrea Loi.

Additional information

A. Loi was supported by Prin 2015—Real and Complex Manifolds; Geometry, Topology and Harmonic Analysis—Italy, by INdAM. GNSAGA—Gruppo Nazionale per le Strutture Algebriche, Geometriche e le loro Applicazioni and also by GESTA—Funded by Fondazione di Sardegna and Regione Autonoma della Sardegna.

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Loi, A., Salis, F. & Zuddas, F. Two conjectures on Ricci-flat Kähler metrics. Math. Z. 290, 599–613 (2018). https://doi.org/10.1007/s00209-017-2033-6

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