| Tue, Apr 25, 2023
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Analysis Seminar
11:30 AM
MSCS 509 | | Kähler geometry and obstruction flat CR manifolds
Peter Ebenfelt, UCSD
Host: Jiri Lebl / Sean Curry
| | | Abstract: Let $X=X^{2n+1}$ be a ${C}^\infty$-smooth strictly pseudoconvex CR hypersurface in a complex manifold of dimension $n+1$. There exists a local defining function $\rho$ with a finite degree of smoothness, namely $C^{n+3-\epsilon}$, up to $X$ such that $\omega=i\partial\bar\partial\log \rho$ defines a Kähler-Einstein metric on the pseudoconvex side of $X$. In general, higher degree smoothness of $\rho$ up to $X$ is obstructed by a log-term whose coefficient is a local CR invariant of $X$. The CR manifold $X$ is said to be obstruction flat if this obstruction invariant vanishes. It is easy to see that if $X$ is spherical, then it is obstruction flat. For $n=1$, there is ample evidence that the converse holds, i.e., if $X=X^3$ is compact and obstruction flat, then it is spherical; this is sometimes referred to as the Strong Ramadanov Conjecture for reasons that will be explained in the talk. For $n\geq 2$, this is no longer true and the classification of compact, obstruction flat CR manifolds of higher dimension is wide open. In this talk, we shall discuss the special case where the CR manifold $X$ arises as the unit circle bundle of a negative Hermitian line bundle over a Kähler manifold. |
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