Kähler-Einstein metrics and obstruction flatness of circle bundles

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 $S(L)$ 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.

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