A note about connectedness theorems a la Barth

TL;DR

This paper establishes Barth-type connectedness theorems for certain smooth subvarieties within specific ambient spaces, using cycle class theory and properties like bigness to extend classical results.

Contribution

It introduces new connectedness results for low-codimension subvarieties in homogeneous and spherical varieties, leveraging cycle class bigness concepts.

Findings

Connectedness theorems proven for subvarieties in homogeneous and spherical varieties.

Use of cycle class bigness to establish geometric connectedness.

Extension of classical Barth theorems to new ambient spaces.

Abstract

We prove Barth-type connectedness results for low-codimension smooth subvarieties with good numerical properties inside certain "easy" ambient spaces (such as homogeneous varieties, or spherical varieties). The argument employs some basics from the theory of cones of cycle classes, in particular the notion of bigness of a cycle class.