projective variety

projective variety (plural projective varieties)

1. (algebraic geometry) A Zariski closed subvariety of a projective space; the zero-locus of a set of homogeneous polynomials that generates a prime ideal.
   • 2005, Max K. Agoston, Computer Graphics and Geometric Modelling: Mathematics, Springer, page 724,

     Varieties are sometimes called closed sets and some authors call an open subset of a projective variety a quasiprojective variety. The latter term is in an attempt to unify the concept of affine and projective variety.

   • 2006, Werner Ballmann, Lectures on Kähler Manifolds, European Mathematical Society, page 16,

     A closed subset ${\displaystyle V\subset \mathbb {C} P^{n}}$ is called a (complex) projective variety if, locally, ${\displaystyle V}$ is defined by a set of complex polynomial equations. Outside of its singular locus, that is, away from the subset where the defining equations do not have maximal rank, the projective variety is a complex submanifold of ${\displaystyle \mathbb {C} P^{n}}$ .

   • 2015, Katsutoshi Yamanoi, Kobayashi Hyperbolicity and Higher-dimensional Nevanlinna Theory, Takushiro Ochiai, Toshiki Mabuchi, Yoshiaki Maeda, Junjiro Noguchi, Alan Weinstein (editors), Springer (Birkhäuser), Geometry and Analysis on Manifolds: In Memory of Professor Shoshichi Kobayashi, page 209,

     The central topic of this note is a famous open problem to characterize which projective varieties are Kobayashi hyperbolic.