Spectral convergence in geometric quantization on $K3$ surfaces

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.