Invariant metrics in complex analysis and a conjecture of Kobayashi and Lang

Abstract: A compact complex manifold $X$ is declared Kobayashi hyperbolic if every holomorphic map from the complex plane into $X$ is constant. Kobayashi hyperbolic manifolds have maintained a central role in our understanding of the landscape of complex manifolds since their introduction in 1967. One striking feature of complex geometry is the capacity to encode this highly transcendental notion of hyperbolicity in the coarse geometric language of distance functions that are invariant under the automorphism group and decrease under holomorphic maps. A long-standing conjecture of Kobayashi (1970) and Lang (1986) predicts that a compact Kobayashi hyperbolic Kähler manifold admits a Kähler—Einstein metric of negative Ricci curvature. We will present the most general evidence for the Kobayashi—Lang conjecture: A compact Kähler manifold with a pluriclosed metric of negative holomorphic curvature admits a unique Kähler—Einstein metric of negative Ricci curvature. This result is a joint work with James Stanfield and comes from the first general improvement on the Schwarz lemma for holomorphic maps between Hermitian manifolds since 1978.

differential geometrygeometric topologymetric geometry

Audience: researchers in the topic

( paper | video )

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Virtual seminar on geometry with symmetries

Series comments: Description: Research seminar in Lie group actions in Differential geometry.

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