Tohoku Mathematical Journal
2009
June
SECOND SERIES VOL. 61, NO. 2

Tohoku Math. J.
61 (2009), 241-252

Title ON THE EXISTENCE OF KÄHLER METRICS OF CONSTANT SCALAR CURVATURE

Author Kenji Tsuboi

(Received February 7, 2008, revised December 17, 2008)
Abstract. 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 Kähler classes whose Bando-Calabi-Futaki character vanishes. Then a Kähler class contains a Kähler metric of constant scalar curvature if and only if the Kähler class is contained in the analytic subvariety. On examination of the analytic subvariety, it is shown that $M$ admits infinitely many nonhomothetic Kähler classes containing Kähler metrics of constant scalar curvature but does not admit any Kähler-Einstein metric.

2000 Mathematics Subject Classification. Primary 53C25; Secondary 53C55.
Key words and phrases. Kähler manifold, constant scalar curvature, Bando-Calabi-Futaki character.

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