Global invariants for strongly pseudoconvex varieties with isolated singularities: Bergman functions
Let \(M\) be a strongly pseudoconvex \(n\)-dimensional manifold which is a resolution of a strongly pseudoconvex variety \(V\) with only isolated singularities. Let \(F\) be the set of all holomorphic \(n\)-forms \(\phi\) on \(M\) such that \(| \int_M\phi\wedge\overline{\phi}\;| \;<\infty\). Let \(\{\omega_j\}\) be a complete orthonormal basis of \(F\) and \(K(z, \overline{\omega}):=\sum \omega_j(z)\wedge\overline{\omega_j(w)}\). \(K(z, \overline{z})\) is a \(2n\)-form on \(M\), the so called Bergman kernel of \(M\). The Bergman kernel \(K_0 (z, \overline{z})\) vanishing on the exceptional set \(E\) of \(M\) is defined replacing \(F\) by \(F_0\), the set of all \(L^2\) integrable holomorphic \(n\)-forms vanishing on \(E\). The Bergman function \(B_M\) is defined to be \(K_0/K\). Pushing \(B_M\) down one obtains a Bergman function \(B_V\) on \(V\) which is a biholomorphic invariant of \(V\) and vanishes precisely on the singularities of \(V\). This Bergman function is able to distinguish the CR structures on the boundaries of \(V\). As an example \(V=V(xy-z^2)\) is treated.
scientific article; zbMATH DE number 2152378