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∞-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(L) of negative Hermitian line bundles (L,h) over Kähler manifolds (M,g). We prove that if (M,g) has constant Ricci eigenvalues, then S(L) 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(L) when (M,g) is a Kähler surface (dim⁡M=2) with constant scalar curvature.