Theorems of Barth-Lefschetz type and Morse theory on the space of paths in homogeneous spaces

Abstract

Homotopy connectedness theorems for complex submanifolds of homogeneous spaces (sometimes referred to as theorems of Barth-Lefshetz type) have been established by a number of authors. Morse Theory on the space of paths lead to an elegant proof of homotopy connectedness theorems for complex submanifolds of Hermitian symmetric spaces. In this work we extend this proof to a larger class of compact complex homogeneous spaces.

Criteria for complete intersections

“…The table below gives the range of values of ℓ for which the isomorphism between the Picard groups is satisfied (cf. Barth-Lefschetz-type theorems [16]):…”

“…However, in many situations and especially for subvarieties of rational homogeneous varieties and for zero loci of sections in vector bundles, the isomorphism holds with Z-coefficients (cf. Barth-Larsen's Lefschetz-type theorems [16]).…”

Splitting Criteria for Vector Bundles Induced by Restrictions to Divisors

“…We have to show that the direct summands are restrictions of line bundles on X that is, PicpXq Ñ PicpY q is surjective. This is the content of the Barth-Lefschetz criterion (see [19,20]): the restriction homomorphism is an isomorphism for 2 ď pℓ ´2q{2, where ℓ is the rank of G. l 3.1. A remark on Voisin's question.…”