Uniform K-stability and asymptotics of energy functionals in Kähler geometry

Abstract

Consider a polarized complex manifold $(X,L)$ and a ray of positive metrics on $L$ defined by a positive metric on a test configuration for $(X,L)$. For many common functionals in Kähler geometry, we prove that the slope at infinity along the ray is given by evaluating the non-Archimedean version of the functional (as defined in our earlier paper [BHJ17]) at the non-Archimedean metric on $L$ defined by the test configuration. Using this asymptotic result, we show that coercivity of the Mabuchi functional implies uniform K-stability, as defined in [Der15, BHJ17]. As a partial converse, we show that uniform K-stability implies coercivity of the Mabuchi functional when restricted to Bergman metrics.

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