ENGLIŠ, Miroslav and Genkai ZHANG. Connection and curvature on bundles on Bergman and Hardy spaces. Documenta Mathematica. Berlin (Germany): Deutsche Mathematiker-Vereinigung e.V., 2020, vol. 25, February, p. 189-217. ISSN 1431-0643. Available from: https://dx.doi.org/10.25537/dm.2020v25.189-217.
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Basic information
Original name Connection and curvature on bundles on Bergman and Hardy spaces
Authors ENGLIŠ, Miroslav (203 Czech Republic, guarantor, belonging to the institution) and Genkai ZHANG (752 Sweden).
Edition Documenta Mathematica, Berlin (Germany), Deutsche Mathematiker-Vereinigung e.V. 2020, 1431-0643.
Other information
Original language English
Type of outcome Article in a journal
Field of Study 10101 Pure mathematics
Country of publisher Germany
Confidentiality degree is not subject to a state or trade secret
WWW Documenta Mathematica
RIV identification code RIV/47813059:19610/20:A0000067
Organization unit Mathematical Institute in Opava
Doi http://dx.doi.org/10.25537/dm.2020v25.189-217
UT WoS 000592702600007
Keywords in English Bergman space; bundle of Bergman spaces; Fock space; Fock bundle; Siegel domain; Chern connection and curvature; Toeplitz operator
Tags MÚ
Tags International impact, Reviewed
Links GA16-25995S, research and development project.
Changed by Changed by: Mgr. Aleš Ryšavý, učo 28000. Changed: 22/4/2021 13:01.
Abstract
We consider a complex domain D x V in the space C-m x C-n and a family of weighted Bergman spaces on V defined by a weight e(-k phi(z , w)) for a pluri-subharmonic function phi(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 del(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 [del((k))(Z), T-f((k))] 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 [del((k))(Z), T-f((k))] as Toeplitz operators. This generalizes earlier work of J.E. Andersen in Comm. Math. Phys. 255 (2005), 727-745. Finally, we also determine the formulas for the curvature and for the induced connection in the general case of D x V replaced by a general strictly pseudoconvex domain V subset of C-m x C-n fibered over a domain D subset of C-m. The case when the Bergman space is replaced by the Hardy space on the boundary of the domain is likewise discussed.
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