International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 64, Pages 4041-4056
doi:10.1155/S0161171203212187
Differential operators and flat connections on a Riemann surface
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.
Abstract
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 (1−genus(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.