Speaker: 

Ming Xiao

Institution: 

UCSD

Time: 

Tuesday, November 15, 2022 - 3:00pm to 3:50pm

Host: 

Location: 

RH 306

{\bf 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 talk, we are interested in obstruction flatness of hypersurfaces $\Sigma$ that arise as unit circle bundles $S(L)$ of negative Hermitian line bundles $(L, h)$ over a complex manifold $M$, whose dual line bundle induces a K\"ahler metric $g$ on $M$. The main result we will discuss can be summarized as follows: 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 the corresponding disk bundle admits a complete K\"ahler-Einstein metric. Finally, we give a necessary and sufficient condition for obstruction flatness of $S(L)$ in terms of the K\"ahler geometry of $(M,g)$ in some special cases.
The talk is based on a recent joint paper with Ebenfelt and Xu.