K\"ahler-Einstein metrics and obstruction flatness of circle bundles

TL;DR

This paper investigates the conditions under which circle bundles over K"ahler manifolds admit obstruction flatness and complete K"ahler-Einstein metrics, linking geometric properties of the base manifold to the bundle's complex structure.

Contribution

It establishes new criteria for obstruction flatness of circle bundles over K"ahler manifolds with constant Ricci eigenvalues and provides conditions for the existence of complete K"ahler-Einstein metrics.

Findings

Circle bundles over manifolds with constant Ricci eigenvalues are obstruction flat.

Complete K"ahler-Einstein metrics exist on certain disk bundles under specified eigenvalue conditions.

A characterization of obstruction flatness for circle bundles over K"ahler surfaces with constant scalar curvature.

Abstract

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