An improvement of a theorem of Van de Ven

Abstract

Abstract In this paper we improve a classical result of Van de Ven which characterizes linear subspaces of as the only smooth closed subvarieties of for which the normal sequence splits (see [A. Van de Ven, A property of algebraic varieties in complex projective spaces. In: Colloque Géom. Diff. Globale (Bruxelles, 1958), 151–152, Centre Belge Rech. Math., Louvain 1959. MR0116361 (22 #7149) Zbl 0092.14004]). Precisely we prove the following: Let X be a submanifold of of dimension ≥ 3, and Y ⊆ X a submanifold of X of dimension ≥ 2. Assume that Span(Y) = Span(X), where Span(X) is the smallest linear subspace of containing X. Then the exact sequence: splits if and only if X is a linear subspace of .