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Transactions of the American Mathematical Society

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Central Kähler metrics

Author: Gideon Maschler
Journal: Trans. Amer. Math. Soc. 355 (2003), 2161-2182
MSC (2000): Primary 53C55, 53C25, 58E11
Published electronically: January 31, 2003
MathSciNet review: 1973986
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Abstract: The determinant of the Ricci endomorphism of a Kähler metric is called its central curvature, a notion well-defined even in the Riemannian context. This work investigates two types of Kähler metrics in which this curvature potential gives rise to a potential for a gradient holomorphic vector field. These metric types generalize the Kähler-Einstein notion as well as that of Bando and Mabuchi (1986). Whenever possible the central curvature is treated in analogy with the scalar curvature, and the metrics are compared with the extremal Kähler metrics of Calabi. An analog of the Futaki invariant is employed, both invariants belonging to a family described in the language of holomorphic equivariant cohomology. It is shown that one of the metric types realizes the minimum of an $L^2$ functional defined on the space of Kähler metrics in a given Kähler class. For metrics of constant central curvature, results are obtained regarding existence, uniqueness and a partial classification in complex dimension two. Consequently, on a manifold of Fano type, such metrics and Kähler-Einstein metrics can only exist concurrently. An existence result for the case of non-constant central curvature is stated, and proved in a sequel to this work.

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Additional Information

Gideon Maschler
Affiliation: Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 3G3

Received by editor(s): November 1, 1999
Published electronically: January 31, 2003
Additional Notes: Partially supported by the Edmund Landau Center for research in Mathematical Analysis and Related Areas, sponsored by the Minerva Foundation (Germany).
Article copyright: © Copyright 2003 American Mathematical Society

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