Spectral convergence in geometric quantization on \(K3\) surfaces (Q6090237)

This work examines the geometric quantization of \(K3\) surfaces by investigating the spectral convergence of the \(\partial\)-Laplacian acting on sections of the prequantum line bundle. Specifically, the study focuses on special Lagrangian fibrations on \(K3\) surfaces and a collection of hyper-Kähler structures that approach the large complex structure limit. It demonstrates a spectral convergence of the \(\partial\)-Laplacians on the prequantum line bundle towards the spectral structure associated with the set of Bohr-Sommerfeld fibers. The key finding of this analysis is the convergence of the quantum Hilbert spaces, a phenomenon previously observed in various contexts. Subsequently, leveraging the Kodaira Vanishing Theorem, the study corroborates a result originally established by \textit{A. Tyurin} [``Geometric quantization and mirror symmetry'', Preprint, \url{arXiv:math/9902027}]. The intricate nature of the proofs for these results underscores their complexity and warrants further exploration.