Bochner Laplacians and Bergman kernels for families

We generalize the results of Marinescu-Savale to families of Bochner Laplacians. This particularly leads to the fiberwise expansion for families of Bergman kernels of horizontally semi-positive index bundles. The proof uses Ma-Zhang's description for the curvature of the index bundle as a fiberwise Toeplitz operator. Based on joint work with X. Ma and G. Marinescu.