Logarithmic singularity of the Szegö kernel and a global invariant of strictly pseudoconvex domains. (English) Zbl 1107.32013

Let \(\Omega\) be a relatively compact, smoothly bounded, strictly pseudoconvex domain in a complex manifold \(M,\) and \(d\sigma=\theta\wedge (d\theta)^{n-1}\) stands for surface element of \(\partial\Omega.\) The Szegö kernel \(S_\theta(z,\overline{w})\) is a reproducing kernel of Hardy space \(A(\partial\Omega, d\sigma)\) consisting of the boundary values of holomorphic fnctions on \(\Omega\) satisfying \(\int_{\partial\Omega}| f| ^2\,d\sigma<\infty.\) Given a smooth defining function \(\rho\) of the domain under some natural conditions one has \(S_\theta(z,\overline{z})=\varphi_\theta(z)\rho(z)^{-n}+\psi_\theta(z){\mathrm {log }} \rho(z).\)
As one of the main results the author proves the following theorem (Theorem 1), that, in particular, shows the invariance of the integral of the logarithmic singularity coefficient, and that this invariant remains unchanged under perturbation of the domain.
Theorem 1: (i) The integral \[ L(\partial\Omega,\theta)=\int_{\partial\Omega}\psi_\theta \theta\wedge (d\theta)^{n-1} \] is independent of the choice of a pseudo Hermitian structure \(\theta\) of \(\partial\Omega.\) Thus \(L(\partial\Omega)=L(\partial\Omega, \theta).\) (ii) Let \(\{\Omega_t\}_{t\in R}\) be a \(C^\infty\) family of strictly pseudoconvex domains in \(M.\) Then \(L(\partial\Omega_t)\) is independent of \(t.\)
As a next main result the author shows that the same invariant appears as the coefficient of the logarithmic term of the volume expansion of the domain with respect to the Bergman volume element.
Theorem 2: For any volume element \(dv\) on \(M\) and any defining function \(\rho\) of \(\Omega,\) the volume Vol\((\Omega_\varepsilon)\) admits an expansion \[ \text{Vol}(\Omega_\varepsilon)=\sum_{j=0}^{n-1}C_j\varepsilon^{j-n}+L(\partial\Omega) \log\varepsilon + O(1), \] where \(C_j\) are constants, \(L(\partial\Omega)\) is the invariant given in Theorem 1 and \(O(1)\) is a bounded term.


32T15 Strongly pseudoconvex domains
32T27 Geometric and analytic invariants on weakly pseudoconvex boundaries
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