On the dimension of bundle-valued Bergman spaces on compact Riemann surfaces

Anne-Katrin Gallagher; Purvi Gupta; Liz Vivas

On the dimension of bundle-valued Bergman spaces on compact Riemann surfaces

Complex Variables 2022-06-16 v1

Authors: Anne-Katrin Gallagher , Purvi Gupta , Liz Vivas

Abstract

Given a holomorphic vector bundle $E$ over a compact Riemann surface $M$ , and an open set $D$ in $M$ , we prove that the Bergman space of holomorphic sections of the restriction of $E$ to $D$ must either coincide with the space of global holomorphic sections of $E$ , or be infinite dimensional. Moreover, we characterize the latter entirely in terms of potential-theoretic properties of $D$ .

Keywords

holomorphic vector bundle compact Riemann surface Bergman space holomorphic sections potential theory infinite dimensional open set

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@article{arxiv.2206.07120,
  title  = {On the dimension of bundle-valued Bergman spaces on compact Riemann surfaces},
  author = {Anne-Katrin Gallagher and Purvi Gupta and Liz Vivas},
  journal= {arXiv preprint arXiv:2206.07120},
  year   = {2022}
}

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9 pages

On the dimension of bundle-valued Bergman spaces on compact Riemann surfaces

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