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Bochner Laplacian and Bergman kernel expansion of semi-positive line bundles on a Riemann surface

George Marinescu, Nikhil Savale

Abstract

We generalize the results of Montgomery for the Bochner Laplacian on high tensor powers of a line bundle. When specialized to Riemann surfaces, this leads to the Bergman kernel expansion and geometric quantization results for semi-positive line bundles whose curvature vanishes at finite order. The proof exploits the relation of the Bochner Laplacian on tensor powers with the sub-Riemannian (sR) Laplacian.

Bochner Laplacian and Bergman kernel expansion of semi-positive line bundles on a Riemann surface

Abstract

We generalize the results of Montgomery for the Bochner Laplacian on high tensor powers of a line bundle. When specialized to Riemann surfaces, this leads to the Bergman kernel expansion and geometric quantization results for semi-positive line bundles whose curvature vanishes at finite order. The proof exploits the relation of the Bochner Laplacian on tensor powers with the sub-Riemannian (sR) Laplacian.

Paper Structure

This paper contains 15 sections, 15 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. Statement of the main results
  3. Background and commentary
  4. Organization of the article
  5. sub-Riemannian geometry
  6. sR Laplacian
  7. sR heat kernel
  8. Bochner Laplacian on tensor powers
  9. Structure of $Y_{r}$
  10. Smallest eigenvalue
  11. Weyl law
  12. Expansion for the smallest eigenvalue
  13. Kodaira Laplacian on tensor powers
  14. Bergman kernel expansion
  15. Model operators

Key Result

Theorem 1

Let $(L,h^{L})\to\left(Y,g^{TY}\right)$, $(F,h^{F})\to\left(Y,g^{TY}\right)$ be Hermitian line and vector bundles on a compact Riemannian manifold with unitary connections $\nabla^{L}$, $\nabla^{F}$ . Assuming that the curvature $R^{L}$ vanishes to finite order at all points, with maximal order $r$e for some positive constant $C$. Moreover, the first eigenfunction concentrates on $Y_{r}$:

Theorems & Definitions (33)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • proof
  • Theorem 5
  • Proposition 6
  • proof
  • Proposition 7
  • proof
  • ...and 23 more