Newton polyhedra and the Bergman kernel
The author studies the structure of singularities of the Bergman kernel at the boundary for pseudoconvex domains of finite type from the viewpoint of the theory of singularities. Assuming that the boundary \(\partial\Omega\) of \(\Omega\in\mathbb C^{n+1}\) is \(C^\infty\)-smooth, the Bergman kernel \(B(z)\) of \(\Omega\) takes the form near a boundary point \(p\): \[ B(z) =\frac{\Phi(w,p)}{\rho^{2+2/d_F}(\log(1/\rho))^{m_F-1}}, \] where \((w,p)\) is some polar coordinates on a nontangential cone \(\Lambda\) with apex at \(p\) and \(\rho\) means the distance from the boundary. Here \(\Phi\) admits some asymptotic expansion with respect to the variables \(\rho^{1/m}\) and \(\log(1/\rho)\) as \(\rho\to 0\) on \(\Lambda\). The values of \(d_F > 0\), \(m_F\in\mathbb Z_+\) and \(m\in\mathbb N\) are determined by geometrical properties of the Newton polyhedron of defining functions of domains and the limit of \(\Phi\) as \(\rho\to 0\) on \(\Lambda\) is a positive constant depending only on the Newton principal part of the defining function. Analogous results are obtained in the case of the Szegö kernel. The analysis is based on some integral formula for the Bergman kernel due to Haslinger. The Fourier transform gives a clear connection between the Bergman space of \(\Omega\) and an associated weighted Bergman space \(H_\tau\) in \(\mathbb C^n\) with a parameter \(\tau\). As a result, the Bergman kernel of \(\Omega\) is expressed as a superposition of the reproducing kernel of \(H_\tau\) by using the Laplace transform with respect to \(\tau\). From the Haslinger formula, the problem is reduced to the analysis of a Laplace integral of the following form: \(I(\tau)=\int_{\mathbb R_+^n}e^{-\tau f(x)}\varphi(x)\,dx,\) \(\tau>0\), where the phase \(f\) is associated with the defining function of \(\Omega\). In order to investigate the boundary behavior of the Bergman kernel, the author analyzes the behavior of the integral \(I(\tau)\) at infinity. The essential idea of the analysis is due to the work of Varchenko on the analysis of an oscillatory integral: \(\tilde I(\tau)=\int_{\mathbb R^n}e^{i\tau f(x)}\varphi(x)\,dx,\) \(\tau>0\). Assuming \(f(x) =\prod x_j^{p_j}\), \(p_j\geq0\), an asymptotic expansion of \(I(\tau)\) as \(\tau\to\infty\) is computed. The problem on the resolution of singularities of the phase \(f\) at a critical point \(x_0\) is considered by applying the theory of toric varieties. Toric resolutions are constructed in terms of the Newton polyhedron of the phase \(f\) at \(x_0\). As a result, the principal term of the asymptotic expansion of \(I(\tau)\) can be seen directly by using the geometry of the Newton diagram of \(f\). Moreover the form of the asymptotic expansion of \(I(\tau)\) is determined by quantitative information on the toric resolutions. From this result, the essential structure of singularities of the Bergman kernel appears in the symbol of its Laplace integral representation.
scientific article; zbMATH DE number 2078059