Connection and curvature on bundles of Bergman and Hardy spaces

0524631 - MÚ 2021 RIV DE eng J - Článek v odborném periodiku
Engliš, Miroslav - Zhang, G.
Connection and curvature on bundles of Bergman and Hardy spaces.
Documenta Mathematica. Roč. 25, June (2020), s. 189-217. ISSN 1431-0643. E-ISSN 1431-0643
Institucionální podpora: RVO:67985840
Klíčová slova: Bergman space * bundle of Bergman spaces * Fock space * Toeplitz operator
Obor OECD: Pure mathematics
Impakt faktor: 0.815, rok: 2020
Způsob publikování: Open access
http://dx.doi.org/10.25537/dm.2020v25.189-217

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 [∇(k)Z,T(k)f] 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 [∇(k)Z,T(k)f] as Toeplitz operators. This generalizes earlier work of J. E. Andersen in [Commun. Math. Phys. 255, No. 3, 727--745]. 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.
Trvalý link: http://hdl.handle.net/11104/0308971