Bergman Kernels and symplectic reduction. (English) Zbl 1171.32001

Ma, Xiaonan; Zhang, Weiping

Astérisque 318. Paris: Société Mathématique de France (ISBN 978-2-85629-255-6/pbk). viii, 154 p. (2008).

Authors’ abstract: “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 admitting a Hamiltonian action of a compact connected Lie group \(G\). 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 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, we establish some Toeplitz operator type properties in semi-classical analysis in 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.”
This work generalizes some results previously obtained by the first author and collaborators to the framework of geometric quantization [X. Dai, K. Liu and X. Ma, J. Differ. Geom. 72, No. 1, 1–41 (2006; Zbl 1099.32003); X. Ma and G. Marinescu, Adv. Math. 217, No. 4, 1756–1815 (2008; Zbl 1141.58018); J. Geom. Anal. 18, No. 2, 565–611 (2008; Zbl 1152.81030)]. The authors announced some of the results previously [X. Ma and W. Zhang, C. R., Math., Acad. Sci. Paris 341, No. 5, 297–302 (2005; Zbl 1085.58028), Nankai Tracts in Mathematics 10, 343–349 (2006; Zbl 1127.53069)].

Reviewer: Harold P. Boas (College Station)

MSC:

32A25	Integral representations; canonical kernels (Szegő, Bergman, etc.)
32-02	Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces
58-02	Research exposition (monographs, survey articles) pertaining to global analysis
53-02	Research exposition (monographs, survey articles) pertaining to differential geometry
58J37	Perturbations of PDEs on manifolds; asymptotics
53D50	Geometric quantization
53D20	Momentum maps; symplectic reduction
32L10	Sheaves and cohomology of sections of holomorphic vector bundles, general results

Keywords:

Bergman kernel; symplectic reduction; Dirac operator; Berezin-Toeplitz quantization

Citations:

Zbl 1099.32003; Zbl 1141.58018; Zbl 1152.81030; Zbl 1085.58028; Zbl 1127.53069