Bergman spaces on algebraic curves

TL;DR

The paper extends Wiegerinck's dichotomy for Bergman spaces from domains in ${\mathbb C}^n$ to Bergman spaces on singular algebraic curves by introducing $D$-volume forms and divisorial Bergman spaces. Using normalization and pushforward techniques, it relates $A^2(U,\rho,E,h)$ to spaces of holomorphic sections $H^0(M,E\otimes L_{D_U})$, showing finite-dimensionality is equivalent to the boundary being locally polar. It develops both projective and affine theories, expressing results via the multiplicity-divisor $D_m$ and its affine analogue $D_m^{\mathbb{A}}$, and provides explicit examples and a discussion of the method's limitations. The work furnishes a coherent framework for singular curves, proving a Wiegerinck-type dichotomy for projective and affine cases while illustrating failures to generalize to all Stein varieties through concrete counterexamples.

Abstract

A theorem of Wiegerinck asserts that the Bergman space of an open subset of the complex numbers is either infinite-dimensional or trivial. Recently, this has been generalized to holomorphic vector bundles over the projective line by the third author and later to vector bundles over any compact Riemann surface by Gallagher, Gupta and Vivas. In the present paper we extend the above results to the case of certain singular metrics associated to divisors on a Riemann surface. As corollaries we obtain versions of Wiegerinck's theorem for both projective and affine algebraic curves.

Theorems & Definitions (91)

• Theorem 1.1
• Conjecture 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Definition 2.1
• Remark 2.2
• Lemma 2.3
• proof
• Corollary 2.4