Vol. 242, No. 2, 2009

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Exponents of a meromorphic connection on a compact Riemann surface

Eduardo Corel

Vol. 242 (2009), No. 2, 259–279

We give a general definition of the exponents of a meromorphic connection on a holomorphic vector bundle of rank n over a compact Riemann surface X. We prove that they can be computed as invariants of a vector bundle L canonically attached to , which we construct and call the Levelt bundle of , and whose degree (equal to the sum of the exponents) we estimate by upper and lower bounds (Fuchs’ relations). We use this definition to construct, for every linear differential equation on a compact Riemann surface (with regular or irregular singularities), the companion bundle of the equation, a vector bundle endowed with a meromorphic connection that is equivalent to the given equation and has precisely the same singularities and the same set of exponents.

meromorphic connections, exponents, Fuchs’ relations, Levelt bundle
Mathematical Subject Classification 2000
Primary: 14F05, 34A30, 34M45
Secondary: 34M50, 12H05, 30F30
Received: 22 August 2007
Revised: 20 July 2009
Accepted: 21 July 2009
Published: 1 October 2009
Eduardo Corel
Georg-August Universität Göttingen
Goldschmidtstr. 1
37077 Göttingen