Abstract
The main purpose of this paper is to provide a structure theorem for codimension one singular transversely projective foliationson projective manifolds. To reach our goal, we firstly extend Corlette-Simpson's classification of rank two representationsof fundamental groups of quasiprojective manifolds by dropping the hypothesis of quasi-unipotency at infinity.Secondly we establish an analogue classification for rank two flat meromorphic connections.In particular, we prove that a rank two flat meromorphic connection with irregular singularities having non trivial Stokesprojectively factors through a connection over a curve.
Highlights
Let X be a smooth projective manifold over C
If F is defined by ω = aω where a is any rational function not identically zero, the corresponding connection matrix is
Given a local basis of ∇-horizontal sections B = ∈ SL2(O(U )) on some open set U, i.e., satisfying dB + A · B = 0, the ratio φ := b21/b22 provides such a local first integral for F ; changing to another basis B · B0 will have the effect of composing φ with a Moebius transformation
Summary
Let X be a smooth projective manifold over C. They are precisely those foliations whose Galois groupoid in the sense of Malgrange is small (see [8, 28]) They often occur as exceptions or counter examples [4, 22, 42] and played an important role in our study of foliations with numerically trivial canonical bundle [25]. For these foliations, one can define a monodromy representation by considering analytic continuation of distinguished germs of first integrals, making the transverse pseudo-group into a group (see [23]); it coincides with the (projectivization of the) monodromy of the flat connection ∇. One of the main ingredients that goes into the proof of our structure theorem is an extension of Corlette-Simpson’s classification of rank two representations of quasi-projective fundamental groups [13] which we proceed to explain
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