Differential operators and flat connections on a Riemann surface. (English) Zbl 1050.32012

Biswas, Indranil

Int. J. Math. Math. Sci. 2003, No. 64, 4041-4056 (2003).

Summary: 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 \(J^n(Q)\) satisfies this transversality condition for the natural filtration of \(J^n(Q)\) defined by projections to lower-order jets. The vector bundle \(J^n(Q)\) 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.

MSC:

32L05	Holomorphic bundles and generalizations
30F10	Compact Riemann surfaces and uniformization
58A20	Jets in global analysis

Keywords:

holomorphic vector bundles; holomorphic connections; transversality