Geodesic-Einstein metrics and Nonlinear Stabilities

ArticleinTransactions of the American Mathematical Society · October 2017with 66 Reads 
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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 to show that this functional attains its absolute minimum at geodesic-Einstein metrics, and we also discuss the relationships 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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