(joint with B.Bakker and Y.Brunebarbe) One very fruitful way of studying
complex algebraic varieties is by forgetting the underlying algebraic
structure, and just thinking of them as complex analytic spaces. To this
end, it is a natural and fruitful question to ask how much the complex
analytic structure remembers. One very prominent result is Chows theorem,
stating that any closed analytic subspace of projective space is in fact
algebraic. A notable consequence of this result is that a compact complex
analytic space admits at most 1 algebraic structure - a result which is
false in the non-compact case. This was generalized and extended by Serre
in his famous GAGA paper using the language of cohomology.
We will explain how to extend Chow's theorem and its generalization to GAGA
to the non-compact case by working with complex analytic structures that
are "tame" in the precise sense defined by o-minimality. This leads to some
very general "algebraization" theorems, and we give applications to Hodge
theory.