# 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)

### Abstract

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.