Theorems of Barth-Lefschetz type for complex subspaces of homogeneous complex manifolds

Abstract

Barth, Larsen, and others showed that complex submanifolds of complex projective space, CP(N), of small codimension strongly resemble CP(N) both homotopically and cohomologically. These results are generalized to yield analogous results for complex subspaces of arbitrary homogeneous complex manifolds. One very special corollary that gives the flavor of the results is:Corollary. Let A and B be complex submanifolds with B a connected submanifold of X, a simple Abelian variety, i.e., X has no proper sub-Abelian varieties. Then: [Formula: see text], where basepoints are suppressed for simplicity, and [Formula: see text]. Further, given any coherent sheaf on B, which may be assumed to be a connected local complete intersection, then: [Formula: see text] and given a coherent sheaf on X, then: [Formula: see text].