# Connection and Curvature on Bundles of Bergman and Hardy Spaces

### Miroslav Englis

Mathematics Institute, Silesian University in Opava, 74601 Opava, and Mathematics Institute, Czech Academy of Sciences, 11567 Prague 1, Czech Republic### Genkai Zhang

Mathematical Sciences, Chalmers University of Technology, and Göteborg University, SE-412 96 Göteborg, Sweden

## Abstract

We consider a complex domain $D×V$ in the space $C_{m}×C_{n}$ 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 $[∇_{Z},T_{f}]$ on Toeplitz operators $T_{f}$. 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 $[∇_{Z},T_{f}]$ 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 $V⊂C_{m}×C_{n}$ fibered over a domain $D⊂C_{m}$. The case when the Bergman space is replaced by the Hardy space on the boundary of the domain is likewise discussed.

## Cite this article

Miroslav Englis, Genkai Zhang, Connection and Curvature on Bundles of Bergman and Hardy Spaces. Doc. Math. 25 (2020), pp. 189–217

DOI 10.4171/DM/744