Central Kähler metrics with non-constant central curvature

Hwang, Andrew; Maschler, Gideon

doi:10.1090/S0002-9947-03-03158-1

Central Kähler metrics with non-constant central curvature
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by Andrew D. Hwang and Gideon Maschler

Trans. Amer. Math. Soc. 355 (2003), 2183-2203

DOI: https://doi.org/10.1090/S0002-9947-03-03158-1

Published electronically: January 31, 2003

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Abstract:

The central curvature of a Riemannian metric is the determinant of its Ricci endomorphism, while the scalar curvature is its trace. A Kähler metric is called central if the gradient of its central curvature is a holomorphic vector field. Such metrics may be viewed as analogs of the extremal Kähler metrics defined by Calabi. In this work, central metrics of non-constant central curvature are constructed on various ruled surfaces, most notably the first Hirzebruch surface. This is achieved via the momentum construction of Hwang and Singer, a variant of an ansatz employed by Calabi (1979) and by Koiso and Sakane (1986). Non-existence, real-analyticity and positivity properties of central metrics arising in this ansatz are also established.

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