International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 64, Pages 4041-4056

Differential operators and flat connections on a Riemann surface

Indranil Biswas

School of Mathematics, Tata Institute of Fundamental Research, Bombay 400005, India

Received 21 December 2002

Copyright © 2003 Indranil Biswas. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We consider filtered holomorphic vector bundles on a compact Riemann surface X equipped with a holomorphic connection satisfying a certain transversality condition with respect to the filtration. If Q is a stable vector bundle of rank r and degree (1genus(X))nr, then any holomorphic connection on the jet bundle Jn(Q) satisfies this transversality condition for the natural filtration of Jn(Q) defined by projections to lower-order jets. The vector bundle Jn(Q) admits holomorphic connection. The main result is the construction of a bijective correspondence between the space of all equivalence classes of holomorphic vector bundles on X with a filtration of length n together with a holomorphic connection satisfying the transversality condition and the space of all isomorphism classes of holomorphic differential operators of order n whose symbol is the identity map.