Superconnection and family Bergman kernels

We establish an asymptotic expansion for families of Bergman kernels. The key idea is to use the superconnection as in the local family index theorem.Comment: C. R. Math. Acad. Sci. Pari

Bergman kernels and symplectic reduction

We generalize several recent results concerning the asymptotic expansions of Bergman kernels to the framework of geometric quantization and establish an asymptotic symplectic identification property. More precisely, we study the asymptotic expansion of the $G$-invariant Bergman kernel of the spin^c Dirac operator associated with high tensor powers of a positive line bundle on a symplectic manifold. We also develop a way to compute the coefficients of the expansion, and compute the first few of them, especially, we obtain the scalar curvature of the reduction space from the $G$-invariant Bergman kernel on the total space. These results generalize the corresponding results in the non-equivariant setting, which has played a crucial role in the recent work of Donaldson on stability of projective manifolds, to the geometric quantization setting. As another kind of application, we generalize some Toeplitz operator type properties in semi-classical analysis to the framework of geometric quantization. The method we use is inspired by Local Index Theory, especially by the analytic localization techniques developed by Bismut and Lebeau.Comment: 132 page

Berezin-Toeplitz quantization and its kernel expansion

We survey recent results about the asymptotic expansion of Toeplitz operators and their kernels, as well as Berezin-Toeplitz quantization. We deal in particular with calculation of the first coefficients of these expansions.Comment: 34 page

Berezin-Toeplitz quantization on Kaehler manifolds

We study the Berezin-Toeplitz quantization on Kaehler manifolds. We explain first how to compute various associated asymptotic expansions, then we compute explicitly the first terms of the expansion of the kernel of the Berezin-Toeplitz operators, and of the composition of two Berezin-Toeplitz operators. As application we estimate the norm of Donaldson's Q-operator.Comment: 45 pages, footnote at page 3 and Remark 0.5 added; v.3 is a final update to agree with the published pape