Model-theoretic consequences of a theorem of Campana and Fujiki

Anand Pillay¹

¹ Department of Mathematics University of Illinois at Urbana-Champaign Altgeld Hall 1409 W. Green St. Urbana, IL 61801, U.S.A. Institut für Mathematik Humboldt Universität D-10099 Berlin, Germany

Fundamenta Mathematicae, Tome 174 (2002) no. 2, pp. 187-192.

Résumé

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.

Détail

DOI : 10.4064/fm174-2-3

Keywords: model theoretic interpretation result campana fujiki algebraicity certain spaces cycles compact complex spaces model theoretic interpretation language canonical bases says tuples elementary extension cal * structure cal compact complex manifolds canonical base internal sort mathbb * zilber dichotomy cal * follows immediately type u rank locally modular nonorthogonal field mathbb * algebraicity subvariety group definable cal * stab trivial