Kähler-Einstein metrics and obstruction flatness II: unit sphere bundles

Abstract

This paper concerns obstruction flatness of hypersurfaces $\Sigma$ that arise as unit sphere bundles $S(E)$ of Griffiths negative Hermitian vector bundles $(E,h)$ over Kähler manifolds $(M,g).$ We prove that if the curvature of $(E,h)$ satisfies a splitting condition and $(M,g)$ has constant Ricci eigenvalues, then $S(E)$ is obstruction flat. If, in addition, all these eigenvalues are strictly less than one and $(M,g)$ is complete, then we show that the corresponding ball bundle admits a complete Kähler-Einstein metric.

arXiv admin note: text overlap with arXiv:2208.13367