# Trace Formulae for Curvature of Jet Bundles over Planar Domains

- 154 Downloads
- 2 Citations

## 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.

## References

- 1.Cowen, M.J., Douglas, R.G.: Complex geometry and operator theory. Acta Math.
**141**(3–4), 187–261 (1978)CrossRefzbMATHMathSciNetGoogle Scholar - 2.Cowen, M.J., Douglas, R.G., Operators possessing an open set of eigenvalues, Functions, series, operators, vol. I, II (Budapest, : Colloq. Math. Soc. János Bolyai, vol. 35. North-Holland, Amsterdam 1983, 323–341 (1980)Google Scholar
- 3.Chern, S.S.: Hermitian vector bundles and the equidistribution of the zeroes of their holomorphic sections. Acta Math.
**114**, 71–112 (1965)CrossRefzbMATHMathSciNetGoogle Scholar - 4.Chandler, K., Wong, P.-M.: Finsler geometry of holomorphic jet bundles. In: A sampler of Riemann–Finsler geometry, vol. 50, pp. 107–196. Math. Sci. Res. Inst. Publ., Cambridge Univ. Press, Cambridge (2004)Google Scholar
- 5.Donaldson, S.K.: Anti self-dual Yang–Mills connections over complex algebraic surfaces and stable vector bundles. Proc. London Math. Soc. (3)
**50**(1), 1–26 (1985)CrossRefzbMATHMathSciNetGoogle Scholar - 6.Gantmacher, F.R.: The theory of matrices, vol. 1. AMS Chelsea Publishing, Providence (1998) (Translated from the Russian by K. A. Hirsch, Reprint of the 1959 translation)Google Scholar
- 7.Kumaresan, S.: A course in differential geometry and Lie groups. In: Texts and Readings in Mathematics, vol. 22. Hindustan Book Agency, New Delhi (2002)Google Scholar
- 8.Ramachandra Rao, A., Bhimasankaram, P.: Linear algebra. In: Texts and Readings in Mathematics, vol. 19, 2nd edn. Hindustan Book Agency, New Delhi (2000)Google Scholar
- 9.Wells, Jr. R.O.: Differential analysis on complex manifolds. In: Graduate Texts in Mathematics, 3rd edn., vol. 65, Springer, New York (2008) (With a new appendix by Oscar Garcia-Prada)Google Scholar