Geodesic-Einstein metrics and nonlinear stabilities

Feng, Huitao; Liu, Kefeng; Wan, Xueyuan

doi:10.1090/tran/7658

Geodesic-Einstein metrics and nonlinear stabilities
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Trans. Amer. Math. Soc. 371 (2019), 8029-8049 Request permission

Abstract:

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 holomorphic 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.

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