Differential Chow varieties exist

Abstract

Abstract. The Chow variety is a parameter space for effective algebraic cycles on P n (or A n ) of given dimension and degree. We construct its analog for differential algebraic cycles on A n , answering a question of [12]. The proof uses the construction of classical algebro-geometric Chow varieties, the theory of characteristic sets of differential varieties and algebraic varieties, the theory of prolongation spaces, and the theory of differential Chow forms. In the course of the proof several definability results from the theory of algebraically closed fields are required. Elementary proofs of these results are given in the appendix.

Partial differential Chow forms and a type of partial differential Chow varieties

“…The set of Chow coordinates of all d-cycles in P n of degree e is a projective variety in the Chow coordinate space [3], called the Chow variety, and denoted by Chow n (d, e). However, the affine Chow variety of all d-cycles in A n of degree e is not closed in the Chow coordinate space, but it is always a constructible set [6,Proposition 3.4 Let C 1 be the subset consisting of all points a ∈ C such that a is the Chow coordinate of an irreducible variety W which is prolongation admissible and additionally satisfies the following conditions:…”

“…Here are some basic notions and results from model theory that we will be used in the proof of the main theorem. For more details and explantations, see [6].…”

“…Fact 6.3. [6] Relative to the theory of algebraically closed fields (ACF), we have the the following statements.…”

“…We also need to generalise results on prolongation admissible varieties [6] to the partial differential case. Notations τ l , ∇ l , B l should be specified beforehand.…”

“…The work on differential Chow forms [9,20] could be regarded as the beginning of such a systematic development, where the theory of differential Chow forms is established for ordinary differential varieties in both affine and projective cases and the existence of differential Chow varieties is proved in very special cases. Then the existence of ordinary differential Chow varieties in general cases is finally proved with a model-theoretical proof [6]. However, the theory of partial differential Chow forms is not yet developed for partial differential varieties.…”