Small codimension subvarieties in homogeneous spaces

TL;DR

This paper establishes Bertini-type theorems and connectedness properties for subvarieties in homogeneous spaces, particularly isotropic Grassmannians, using advanced geometric techniques.

Contribution

It introduces new Bertini theorems and connectedness results for inverse images of Schubert varieties and diagonal subvarieties in homogeneous spaces.

Findings

Connected inverse images of Schubert varieties are generally connected.

Subvarieties of isotropic Grassmannians are simply connected.

Transplanting theorems for Picard and Neron-Severi groups are proved for specific homogeneous spaces.

Abstract

We prove Bertini type theorems for the inverse image, under a proper morphism, of any Schubert variety in an homogeneous space. Using generalisations of Deligne's trick, we deduce connectedness results for the inverse image of the diagonal in $X^2$ where $X$ is any isotropic grassmannian. We also deduce simple connectedness properties for subvarieties of $X$. Finally we prove transplanting theorems {\`a} la Barth-Larsen for the Picard group of any isotropic grassmannian of lines and for the Neron-Severi group of some adjoint and coadjoint homogeneous spaces.