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Complex Analysis and Geometry Seminar

K¨ahler-Einstein metrics and obstruction flatness of circle bundles

Ming Xiao (UC-San Diego)

Location:  Hill Center Room 705
Date & time: Friday, 02 December 2022 at 10:30AM - 11:30AM

Abstract: Title: K¨ahler-Einstein metrics and obstruction flatness of circle bundles Abstract: Obstruction flatness of a strongly pseudoconvex hypersurface ? in a complex manifold refers to the property that any (local) K¨ahler-Einstein metric on the pseudoconvex side of ?, complete up to ?, has a potential ? log u such that u is C?-smooth up to ?. In general, u has only a finite degree of smoothness up to ?. In this talk, we are interested in obstruction flatness of hypersurfaces ? 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