projective variety

plural projective varieties

1. algebraic geometryA Zariski closed subvariety of a projective space; the zero-locus of a set of homogeneous polynomials that generates a prime ideal.

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

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

   2 more examples

   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.

   2005, Max K. Agoston, Computer Graphics and Geometric Modelling: Mathematics, Springer, page 724

   A closed subset V#92;subset#92;mathbb#123;C#125;Pⁿ is called a (complex) projective variety if, locally, 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 #92;mathbb#123;C#125;Pⁿ.

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