Model-theoretic consequences of a theorem of Campana and Fujiki
Volume 174 / 2002
Abstract
We give a model-theoretic interpretation of a result by Campana and Fujiki on the algebraicity of certain spaces of cycles on compact complex spaces. The model-theoretic interpretation is in the language of canonical bases, and says that if $b,c$ are tuples in an elementary extension ${\cal A}^{*}$ of the structure ${\cal A}$ of compact complex manifolds, and $b$ is the canonical base of ${\rm tp}(c/b)$, then ${\rm tp}(b/c)$ is internal to the sort $({\mathbb P}^{1})^{*}$. The Zilber dichotomy in ${\cal A}^{*}$ follows immediately (a type of $U$-rank $1$ is locally modular or nonorthogonal to the field ${\mathbb C}^{*}$), as well as the “algebraicity” of any subvariety $X$ of a group $G$ definable in ${\cal A}^{*}$ such that ${\rm Stab}(X)$ is trivial.
