Differential Chow varieties exist

TL;DR

This paper constructs differential Chow varieties, extending classical Chow varieties to differential algebraic varieties, using algebraic geometry, differential theory, and definability results, thus answering a question posed by Gao, Li, and Wei.

Contribution

It introduces the existence of differential Chow varieties, providing a foundational tool for the study of differential algebraic cycles.

Findings

Constructed differential Chow varieties for differential algebraic varieties.

Connected classical algebraic geometry with differential algebraic geometry.

Provided elementary proofs of key definability results.

Abstract

Chow varieties are a parameter space for cycles of a given variety of a given codimension and degree. We construct their analog for differential algebraic varieties with differential algebraic subvarieties, answering a question of Gao, Li and Wei. The proof uses the construction of classical algebro-geometric Chow varieties, the theory of characteristic sets of differential 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 an appendix by William Johnson.