Chow variety

In mathematics, particularly in the field of algebraic geometry, a Chow variety is an algebraic variety whose points correspond to effective algebraic cycles of fixed dimension and degree on a given projective space. More precisely, the Chow variety^[1] ${\displaystyle \operatorname {Gr} (k,d,n)}$ is the fine moduli variety parametrizing all effective algebraic cycles of dimension ${\displaystyle k-1}$ and degree ${\displaystyle d}$ in ${\displaystyle \mathbb {P} ^{n-1}}$.

The Chow variety ${\displaystyle \operatorname {Gr} (k,d,n)}$ may be constructed via a Chow embedding into a sufficiently large projective space. This is a direct generalization of the construction of a Grassmannian variety via the Plücker embedding, as Grassmannians are the ${\displaystyle d=1}$ case of Chow varieties.

Chow varieties are distinct from Chow groups, which are the abelian group of all algebraic cycles on a variety (not necessarily projective space) up to rational equivalence. Both are named for Wei-Liang Chow (周煒良), a pioneer in the study of algebraic cycles.