Seminar

Assume that Gamma is a discrete infinite group acting on a complex manifold M. The goal is to realize the quotient space Gamma\M as a definable object in an o-minimal structure. The following is sufficient: the existence of a definable (in some o-minimal structure) subset F of M such that (1) Gamma.F=M and (2) the set of {g in Gamma : g.F ∩ F is non empty} is finite.

Under additional topological assumptions the quotient of F by Gamma, call it M_F, can be endowed with a definable manifold structure, which is naturally biholomorphic to Gamma\M . It turns out that different definable fundamental sets can give rise to definable manifolds which are not definably bi-holomorphic. This can be easily seen by considering the (non-definable) sets Gamma.F in elementary extensions.

One now considers the structure obtained by endowing M_F with all definable analytic subsets of its Cartesian powers. In the basic case of (C,+) and the group of integers (“the exponential case”), the various fundamental sets give rise to three types of strongly minimal structures: trivial, linear and non-locally modular. We use a theorem of Bishop and o-minimality to establish a GAGA principle in this setting, even when the covering map is not definable on F.

(joint work-in-progress with S. Starchenko)