Bergman Kernels and Symplectic Reduction

Book excerpt: The authors 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, they study the asymptotic expansion of the $G$-invariant Bergman kernel of the $\mathrm{spin}^c$ Dirac operator associated with high tensor powers of a positive line bundle on a symplectic manifold admitting a Hamiltonian action of a compact connected Lie group $G$. The authors also develop a way to compute the coefficients of the expansion, and compute the first few of them; especially, they 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 have 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, the authors establish some Toeplitz operator type properties in semi-classical analysis in the framework of geometric quantization. The method used is inspired by Local Index Theory, especially by the analytic localization techniques developed by Bismut and Lebeau.