Spectral convergence in geometric quantization on $K3$ surfaces

TL;DR

This paper investigates how spectral properties of geometric quantization on K3 surfaces converge as the complex structure varies, linking spectral analysis to Bohr-Sommerfeld fibers in a hyper-Kähler setting.

Contribution

It demonstrates spectral convergence of the ar-Laplacians on K3 surfaces with special Lagrangian fibrations approaching large complex structure limits.

Findings

Spectral convergence of ar-Laplacians to Bohr-Sommerfeld fiber structures

Analysis of hyper-Ka4hler structures in the large complex structure limit

Connection between spectral data and geometric quantization on K3 surfaces

Abstract

We study the geometric quantization on $K3$ surfaces from the viewpoint of the spectral convergence. We take a special Lagrangian fibrations on the $K3$ surfaces and a family of hyper-K\"ahler structures tending to large complex structure limit, and show a spectral convergence of the $\bar{\partial}$-Laplacians on the prequantum line bundle to the spectral structure related to the set of Bohr-Sommerfeld fibers.