Harmonic Metrics and Connections with Irregular Singularities

We identify the holomorphic de Rham complex of the minimal extension of a meromorphic vector bundle with connexion on a compact Riemann surface X with the L 2 complex relative to a suitable metric on the bundle and a complete metric on the punctured Riemann surface. Applying results of C. Simpson, we show the existence of a harmonic metric on this vector bundle, giving the same L complex. As a consequence, we obtain a Hard Lefschetz-type theorem. 1. Statement of the results Let X be a compact Riemann surface, D ⊂ X be a finite set of points and denote by j the open inclusion X def = X −D →֒ X . Let M be a locally free OX [∗D]-module of finite rank d, equipped with a connection ∇ : M → M⊗OX Ω 1 X which may have regular or irregular singularities at each point of D. Therefore, M is also a holonomic module on the ring DX of holomorphic differential operators on X . We call such a DX -module a meromorphic connection for short. There exists a unique holonomic DX -submodule Mmin ⊂ M satisfying (1) OX [∗D]⊗OX Mmin = M, (2) Mmin has no quotient supported on a subset of D. One says that Mmin is the minimal extension of M along D. If M denotes the dual DX -module, we have an exact sequence 0 −→ Mmin −→ M −→ [ H [D](M ) ]∗ −→ 0 where H [D](M ) denotes the torsion of M supported on D. Denote by M the local system of horizontal sections of M|X∗ and denote by DR(M) the de Rham complex (Ω•X ⊗OX M,∇). If M is regular at each point of D, we have DR(Mmin) = j∗M . In general, however, DR(Mmin) cannot be computed in terms of M only. When ∇ has only regular singularities, it follows from [11] that, when the meromorphic bundle with connection (M,∇) is irreducible, i.e. in this case when the local system M is so, there exists on M|X∗ a harmonic metric having a moderate behaviour near each point of D. Then, Zucker’s arguments in [13] show that the L complex of this bundle (associated with this harmonic metric and with a metric on X locally equivalent, near each point of D, to the Poincaré metric on the punctured disc) is isomorphic to the de Rham complex DR(Mmin) = j∗M , and its cohomology can be computed with L harmonic sections. It 1991 Mathematics Subject Classification. 32S40, 32S60, 32L10, 35A20, 35A27.