Connection and Curvature on Bundles of Bergman and Hardy Spaces

Miroslav Englis ; Genkai Zhang

Documenta mathematica, Tome 25 (2020), pp. 189-217

Résumé

We consider a complex domain D×V in the space Cm×Cn and a family of weighted Bergman spaces on V defined by a weight e−kφ(z,w) for a pluri-subharmonic function φ(z,w) with a quantization parameter k. The weighted Bergman spaces define an infinite dimensional Hermitian vector bundle over the domain D. We consider the natural covariant differentiation ∇Z​ on the sections, namely the unitary Chern connections preserving the Bergman norm. We prove a Dixmier trace formula for the curvature of the unitary connection and we find the asymptotic expansion for the curvatures R(k)(Z,Z) for large k and for the induced connection [∇Z(k)​,Tf(k)​] on Toeplitz operators Tf​. In the special case when the domain D is the Siegel domain and the weighted Bergman spaces are the Fock spaces we find the exact formula for [∇Z(k)​,Tf(k)​] as Toeplitz operators. This generalizes earlier work of J. E. Andersen in [Commun. Math. Phys. 255, No. 3, 727–745 (2005; Zbl 1079.53136)]. Finally, we also determine the formulas for the curvature and for the induced connection in the general case of D×V replaced by a general strictly pseudoconvex domain V⊂Cm×Cn fibered over a domain D⊂Cm. The case when the Bergman space is replaced by the Hardy space on the boundary of the domain is likewise discussed.

Détail

DOI : 10.4171/dm/744

Classification : 32A36, 32L05, 32Q20, 47B10, 47B35
Mots-clés : Fock space, Bergman space, Toeplitz operator, bundle of Bergman spaces, Fock bundle, Siegel domain, Chern connection and curvature