We describe a notion of ampleness for line bundles on orbifolds with cyclic quotient singularities that is related to embeddings in weighted projective space, and prove a global asymptotic expansion for a weighted Bergman kernel associated to such a line bundle.

It is proved that a proper holomorphic mapping / between bounded complete Reinhardt domains extends holomorphically past the boundary and that if, in addition, /~'(0) = {0}, then / is a polynomial mapping. The proof is accomplished via a transformation rule for the Bergman kernel function under proper holomorphic mappings.

In this paper, we give both lower and upper bound estimates of the Bergman kernel for a degeneration of Riemann surfaces with constant curvature −1. As a result, we give a geometric proof of the Riemann–Rock theorem for a singular Riemann surface. Mathematics Subject Classification (2010). Primary 53C55; Secondary 32S30.

This expository article is an expanded version of the talk given by the author at the 2013 ICCM conference in Taipei. We report our recent work on explicit closed formulas of the asymptotic expansions of the Bergman kernel, and the relationship to deformation quantization on Kähler manifolds. These formulas are expressed as summations over strongly connected graphs.

Abstract In this paper, we obtain some interesting reproducing kernel estimates and Carleson properties that play an important role. We characterize the bounded compact Toeplitz operators on weighted Bergman spaces with Békollé-Bonami weights in terms of Berezin transforms. Moreover, estimate essential norm them assuming they are bounded.

It is proved that the family of Ahlfors extremal mappings of a multiply connected region in the plane onto the unit disc can be expressed as a rational combination of two fixed Ahlfors mappings in much the same way that the family of Riemann mappings associated to a simply connected region can be expressed in terms of a single such map. The formulas reveal that this family of mappings extends t...