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Geometric quantization results for semi-positive line bundles on a Riemann surface

George Marinescu, Nikhil Savale

Abstract

In earlier work the authors proved the Bergman kernel expansion for semipositive line bundles over a Riemann surface whose curvature vanishes to atmost finite order at each point. Here we explore the related results and consequences of the expansion in the semipositive case including: Tian's approximation theorem for induced Fubini-Study metrics, leading order asymptotics and composition for Toeplitz operators, asymptotics of zeroes for random sections and the asymptotics of holomorphic torsion.

Geometric quantization results for semi-positive line bundles on a Riemann surface

Abstract

In earlier work the authors proved the Bergman kernel expansion for semipositive line bundles over a Riemann surface whose curvature vanishes to atmost finite order at each point. Here we explore the related results and consequences of the expansion in the semipositive case including: Tian's approximation theorem for induced Fubini-Study metrics, leading order asymptotics and composition for Toeplitz operators, asymptotics of zeroes for random sections and the asymptotics of holomorphic torsion.
Paper Structure (12 sections, 22 theorems, 178 equations)

This paper contains 12 sections, 22 theorems, 178 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. sR and Bochner Laplacians
  4. Kodaira Laplacian and its spectral gap
  5. Bergman kernel expansion
  6. Uniform estimates on the Bergman kernel
  7. Induced Fubini-Study metrics
  8. Toeplitz operators
  9. Branched coverings
  10. Random sections
  11. Holomorphic torsion
  12. Model operators

Key Result

Theorem 1

Let $Y$ be a compact Riemann surface and $(L,h^{L})\to Y$ a semipositive line bundle whose curvature $R^{L}$ vanishes to finite order at any point. Let $(F,h^{F})\to Y$ be another Hermitian holomorphic vector bundle. Then the Bergman kernel $\Pi_{k}\coloneqq\Pi_{k}^{0}$ has the pointwise asymptotic Here $c_{j}$ are sections of $\textrm{End}\left(F\right)$, with the leading term $c_{0}\left(y\righ

Theorems & Definitions (43)

  • Theorem 1: Marinescu-Savale18
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Proposition 6
  • proof
  • Proposition 7
  • proof
  • Corollary 8
  • ...and 33 more