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

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.


32L05 Holomorphic bundles and generalizations
30F10 Compact Riemann surfaces and uniformization
58A20 Jets in global analysis
Full Text: DOI EuDML