Abstract.We continue our investigation of the category of algebraic D-varieties and algebraic D-groups from  and . We discuss issues of quantifier-elimination, completeness, as well as the D-variety analogue of “Moishezon spaces”.
Introduction and preliminaries.If (K, ∂) is a differentially closed field of characteristic 0, an algebraic D-variety over K is an algebraic variety over K together with an extension of the derivation ∂ to a derivation of the structure sheaf of X. Welet D denote the category of algebraic D-varieties and G the full subcategory whose objects are algebraic D-groups. D and G were essentially introduced by Buium and G was exhaustively studied in . The category D is closely related but not identical to the class of sets of finite Morley rank definable in the structure (K,+, ・, ∂) (which is essentially the class of “finitedimensional differential algebraic varieties”). However an object ofD can also be considered as a first order structure in its own right by adjoining predicates for algebraic D-subvarieties of Cartesian powers. In a similar fashion D can be considered as a many-sorted first order structure. In  it was shown that the many-sorted “reduct” G has quantifier-elimination. We point out in this paper an easy example showing:
PROPOSITION 1.1. D does not have quantifier-elimination.
The notion of a “complete variety” in algebraic geometry is fundamental. Over C these are precisely the varieties which are compact as complex spaces. Kolchin  introduced completeness in the context of differential algebraic geometry. This was continued by Pong  who obtained some interesting results and examples. We will give a natural definition of completeness for algebraic D-varieties. The exact relationship to the notion studied by Kolchin and Pong is unclear: in particular I do not know whether any of the examples of new “complete” ∂-closed sets in  correspond to complete algebraic D varieties in our sense.
Email your librarian or administrator to recommend adding this book to your organisation's collection.