Mathematical Research Letters

Volume 11 (2004)

Number 6

Global invariants for strongly pseudoconvex varieties with isolated singularities: Bergman functions

Pages: 809 – 832



Stephen S.-T. Yau (University of Illinois)


Let $M$ be a strongly pseudoconvex manifold which is a resolution of strongly pseudoconvex variety $V$ with only isolated singularities. We define a Bergman function $B_M$ on $M$ which is a biholomorphic invariant of $M$. The Bergman function $B_M$ vanishes precisely on the exceptional set of $M$. Hence $B_M$ can be pushed down and we obtain a Bergman function $B_V$ which is a biholomorphic invariant of $V$ and vanishes precisely on the singularities of $V$. This Bergman function not only can distinguish analytic structures of isolated singularities, but it can also distinguish the CR structures of the boundaries of $V$. As an application, we define a continuous numerical invariant on strongly pseudoconvex CR manifolds in $V=\{(x,y,z)\in {\Bbb C}^3:xy=z^2\}$. We show that our invariant varies continuously in ${\Bbb R}$ when the CR structure of strongly pseudoconvex CR manifold changes in $V$. Our global numerical invariant is explicitly computable. Moreover we show that the Bergman function allows us to determine the automorphism groups of these CR manifolds.

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