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Complex Analysis and Operator Theory

, Volume 8, Issue 8, pp 1723–1740 | Cite as

Trace Formulae for Curvature of Jet Bundles over Planar Domains

  • Dinesh Kumar KeshariEmail author
Article

Abstract

For a domain \(\varOmega \) in \(\mathbb {C}\) and an operator \(T\) in \({\mathcal {B}}_n(\varOmega )\), Cowen and Douglas construct a Hermitian holomorphic vector bundle \(E_T\) over \(\varOmega \) corresponding to \(T\). The Hermitian holomorphic vector bundle \(E_T\) is obtained as a pull-back of the tautological bundle \(S(n,{\mathcal {H}})\) defined over \({\mathcal {G}}r(n,{\mathcal {H}})\) by a nondegenerate holomorphic map \(z\mapsto {\mathrm{ker}}(T-z),\;z\in \varOmega \). To find the answer to the converse, Cowen and Douglas studied the jet bundle in their foundational paper. The computations in this paper for the curvature of the jet bundle are rather intricate. They have given a set of invariants to determine if two rank \(n\) Hermitian holomorphic vector bundle are equivalent. These invariants are complicated and not easy to compute. It is natural to expect that the equivalence of Hermitian holomorphic jet bundles should be easier to characterize. In fact, in the case of the Hermitian holomorphic jet bundle \({\mathcal {J}}_k({\mathcal {L}}_f)\), we have shown that the curvature of the line bundle \({\mathcal {L}}_f\) completely determines the class of \({\mathcal {J}}_k({\mathcal {L}}_f)\). In case of rank \(n\) Hermitian holomorphic vector bundle \(E_f\), We have calculated the curvature of jet bundle \({\mathcal {J}}_k(E_f)\) and also obtained a trace formula for jet bundle \({\mathcal {J}}_k(E_f)\).

Keywords

Cowen-Douglas class Curvature Hermitian holomorphic vector bundle Jet bundle 

Mathematics Subject Classification (2010)

47B32 

Notes

Acknowledgments

Results of this paper appeared as a part of the author’s doctoral dissertation titled “Infinitely Divisible Metrics, Curvature Inequalities and Curvature Formulae” at Indian Institute of Science under the supervision of Prof. Gadadhar Misra. The author would like to thank Prof. Gadaghar Misra and Dr. Cherian Varghese for their valuable suggestions and numerous stimulating discussions relating to this paper.

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Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of ScienceBangalore India
  2. 2.Department of MathematicsTexas A&M UniversityCollege StationUSA

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