Bergman Kernels and Symplectic Reduction
The authors generalize several recent results concerning asymptotic expansions of Bergman kernels to the framework of geometric quantization and establish an asymptotic symplectic identification property. For a positive line bundle L and an Hermitian vector bundle E over a symplectic manifold X, one can construct the so called spinc Dirac operator Dp acting on the space of holomorphic forms with values in Lp ⊗ E. The Bergmann kernel is the smooth kernel of the orthogonal projection onto the kernel of the spinc Dirac operator. The authors consider Hamiltonian action of a compact connected Lie group G on a compact symplectic manifold X and relate the asymptotic expansion for p→ ∞ of the G-invariant Bergmann kernel of the spinc Dirac operator with the Bergman kernel on the corresponding Marsden-Weinstein reduction XG. A method for computation of the coefficients in the expansions is presented and the first few of them are explicitly calculated. In particular, the authors obtain the scalar curvature of the reduction space from the G-invariant Bergman kernel on the total space. To prove the main result of the book, the authors use the analytic localization techniques of Bismut and Lebeau – the key observation is that the G-invariant Bergman kernel is the kernel of the orthogonal projection to the zero space of a deformation of Dp2 by the Casimir operator. To localize the problem, the spectral gap property and the finite propagation speed of solutions of hyperbolic equations play essential roles. As an application, the authors establish some properties of Toeplitz operators on XG. Moreover, the non-equivariant asymptotic expansion played a crucial role in the recent work of Donaldson on the stability of projective manifolds.