course: On the homotopy type of Kaehler compact manifolds and the Kodaira problem

speaker: Claire Voisin (CNRS, Inst. Math. Jussieu and IHES)

abstract: Kodaira proved that a compact Kaehler surface admits small deformations of its complex structure which are projective. The analogous problem in higher dimension was known as the Kodaira problem. More generally, there are numerous restrictions on the topology of Kaehler manifolds, and it was not known up to recently if being projective implies supplementary restrictions.

We show that this is indeed the case, thus disproving in particular the Kodaira problem:

Theorem In any dimension n at least 4, there exist Kaehler compact manifolds of dimension n which do not have the homotopy type of projective complex manifolds, and in particular do not deform to a projective one. Simply connected such examples exist starting from dimension 6.

The examples we construct are obtained by blowing-up complex tori (in the non-simply connected case). A number of people thus asked for a birational version of the Kodaira problem:

Question Is any Kaehler compact manifold birational to a complex manifold which deforms to a projective one?

The answer to this question is also no, at least if the deformations considered are via families of manifolds in the class \cal C.

We will try first of all to explain the basic topological restrictions on Kaehler compact manifolds, coming from Hodge theory, and explain the relation between Kaehler and projective geometry (Kodaira characterization of projective complex manifolds). We will then turn to the proof of the theorem.

timetable:
Tue 12 Oct, 11:15 - 12:15, Sala Conferenze Centro De Giorgi
Wed 13 Oct, 10:00 - 11:15, Sala Conferenze Centro De Giorgi
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