On the existence of Kähler metrics of constant scalar curvature

Details On the existence of Kähler metrics of constant scalar curvature On the existence of Kähler metrics of constant scalar curvature

2009-02-05

arxiv.org/abs/0902.0861v1

Mathematics

Differential Geometry

Scientific paper

For certain compact complex Fano manifolds $M$ with reductive Lie algebras of holomorphic vector fields, we determine the analytic subvariety of the second cohomology group of $M$ consisting of K\"ahler classes whose Bando-Calabi-Futaki character vanishes. Then a K\"ahler class contains a K\"ahler metric of constant scalar curvature if and only if the K\"ahler class is contained in the analytic subvariety. On examination of the analytic subvariety, it is shown that $M$ admits infinitely many nonhomothetic K\"ahler classes containing K\"ahler metrics of constant scalar curvature but does not admit any K\"ahler-Einstein metric.

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