Kähler-Einstein metrics and obstruction flatness of circle bundles

Abstract

Obstruction flatness of a strongly pseudoconvex hypersurface Σ in a complex manifold refers to the property that any (local) Kähler-Einstein metric on the pseudoconvex side of Σ, complete up to Σ, has a potential $-\log u$ such that u is $C^{\infty }$-smooth up to Σ. In general, u has only a finite degree of smoothness up to Σ. In this paper, we study obstruction flatness of hypersurfaces Σ that arise as unit circle bundles $S\left(L\right)$ of negative Hermitian line bundles $(L,h)$ over Kähler manifolds $(M,g)$. We prove that if $(M,g)$ has constant Ricci eigenvalues, then $S\left(L\right)$ 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 disk bundle admits a complete Kähler-Einstein metric. Finally, we give a necessary and sufficient condition for obstruction flatness of $S\left(L\right)$ when $(M,g)$ is a Kähler surface $(\dim M=2$) with constant scalar curvature.