Arithmetic applications of o-minimality Ical Atom Mailing List

We will start reading the paper O-minimal GAGA and a conjecture of Griffiths

Chow’s theorem says that a compact analytic submanifold of projective space is algebraic. A generalization of this result (proved with Starchenko) says that a complex analytic subvariety of affine space which in addition is definable in some o-minimal structure over the reals is algebraic.

In these 2-3 talks I will speak about the history of this problem, several related and less known results, and the reasons to expect the theorem to be true (not a full proof).

Some minimal background on definability in structures, as well as basics of o-minimality will be assumed.

1. Coherence - classical.
2. Proof of Oka - sketch of classical (via Weierstrass preparation and division, etc).
3. Proof of definable Oka - sketch of steps and discussion of problematic points.

If time permits, I will report on the bourbaki text I sent during the previous lecture.

We will discuss the proof of 2.13 and continue reading section 2

We will go over sections 2.4 and 2.5 of the paper

Shimura varieties, the end of section 3 and possibly additional topics

We will start summarising the result, via the paper Hodge Theory and O-mininality