Model-theoretic consequences of a theorem of Campana and Fujiki

Volume 174 / 2002

Anand Pillay Fundamenta Mathematicae 174 (2002), 187-192 MSC: 03C45, 32J27. DOI: 10.4064/fm174-2-3


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.


  • Anand PillayDepartment 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

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