## 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.

### MSC:

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