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.

arXiv commentversion 2 is shorter, to appear in Mathematische Annalen

References in corpus (9)

• Toeplitz Quantization of Kähler Manifolds and $gl(N)$ $N\to \infty$
• Distribution of zeros of random and quantum chaotic sections of positive line bundles
• Quantum Chaotic Dynamics and Random Polynomials
• Generalized Bergman kernels on symplectic manifolds
• On the asymptotic expansion of Bergman kernel
• Asymptotic distribution of complex zeros of random analytic functions
• Asymptotics of spectral function of lower energy forms and Bergman kernel of semi-positive and big line bundles
• The SPIN${}^c$ Dirac operator on high tensor powers of a line bundle
• A survey on zeros of random holomorphic sections

Cited by in corpus (6)

• A semiclassical Birkhoff normal form for symplectic magnetic wells
• A direct approach to the analytic Bergman projection
• Spectrum and abnormals in sub-Riemannian geometry: the 4D quasi-contact case
• Kähler-Einstein Bergman metrics on pseudoconvex domains of dimension two
• Spectral asymptotics for the semiclassical Bochner Laplacian of a line bundle with constant rank curvature
• Geometric quantization results for semi-positive line bundles on a Riemann surface