The INI has a new website!

This is a legacy webpage. Please visit the new site to ensure you are seeing up to date information.

An Isaac Newton Institute Workshop

An Introduction to Recent Applications of Model Theory

Isotriviality criteria for families of non-algebraic compact K\" ahler manifolds, and model-theoretic nonmultidimensionality of the class C.

6th April 2005

Author: campana (universit\'e nancy 1)


A question raised by A.Pillay is whether the class $\calC$ of compact complex manifolds $F$ bimeromorphic to some compact K\" ahler manifold $F'$ (depending on $F$) is nonmultidimensional in the model theoretic sense.

Specialised to the case of {\it simple} manifolds $F$ (those which are not covered by proper compact analytic subsets, and of complex dimension at least $2$), this means that if $f:X\to S$ is a surjective holomorphic map with $X$ in $\calC$, and general smooth fibre $X_s$ simple, then $f$ is {\it isotrivial}, which means that any two such fibres are isomorphic.

We show that this is indeed the case for (most of) the known simple manifolds: the non-projective hyperk\" ahler manifolds, and the general complex tori.

The talk is intended for non-specialists in complex geometry.