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Geodesic-einstein metrics and nonlinear stabilities

Journal article
Authors H. Feng
K. Liu
Xueyuan Wan
Published in Transactions of the American Mathematical Society
Volume 371
Issue 11
Pages 8029-8049
ISSN 0002-9947
Publication year 2019
Published at Department of Mathematical Sciences
Pages 8029-8049
Language en
Subject categories Mathematical Analysis


In this paper, we introduce notions of nonlinear stabilities for a relative ample line bundle over a holomorphic fibration and define the notion of a geodesic-Einstein metric on this line bundle, which generalize the classical stabilities and Hermitian-Einstein metrics of holomorphic vector bundles. We introduce a Donaldson type functional and show that this functional attains its absolute minimum at geodesic-Einstein metrics, and we also discuss the relations between the existence of geodesic-Einstein metrics and the nonlinear stabilities of the line bundle. As an application, we will prove that a holomor- phic vector bundle admits a Finsler-Einstein metric if and only if it admits a Hermitian-Einstein metric, which answers a problem posed by S. Kobayashi. © 2018 American Mathematical Society.

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