Zu Chongzhi Mathematics Research Seminar

Date and Time (China standard time): Friday, October 13, 11:30 am-12:30 pm

Zoom: 974 5280 8060; Passcode: dkumath

Title: Kähler-Einstein metrics and obstruction flatness of circle bundles

Speaker: Ming Xiao

Bio: Dr. Ming Xiao is an associate professor from University of California, San Diego. He obtained his PhD from Rutgers University in 2015. After his graduation, Dr. Xiao worked at University of Illinois at Urbana-Champaign as a Doob instructor. He joined UCSD as an assistant professor in 2017 and has been there since then. Dr. Xiao’s research lies in the field of several complex variables and complex geometry/CR geometry which interact with PDE and geometric analysis. His work has been published in prestigious journals including American Journal of Mathematics, Advances in Mathematics, Journal für die Reine und Angewandte Mathematik, and Journal de Mathématiques Pures et Appliquées. In 2021, he was awarded NSF CAREER Award grant. 

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 −logu such that u is C∞ -smooth up to Σ. In general, u has only a finite degree of smoothness up to Σ. In this talk, we are interested in obstruction flatness of hypersurfaces Σ 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 Kähler 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. We also give a necessary and sufficient condition for obstruction flatness of S(L) in terms of the Kähler geometry of (M, g) in some special cases. The talk is based on a recent joint paper with Ebenfelt and Xu.