Conjugate complex homogeneous spaces with non-isomorphic fundamental groups

TL;DR

This paper investigates the topological fundamental groups of conjugate complex homogeneous spaces, revealing non-isomorphic fundamental groups in certain cases, and provides explicit examples illustrating these phenomena.

Contribution

It characterizes the fundamental groups of complex homogeneous spaces and constructs examples where conjugation alters their topological fundamental groups.

Findings

Fundamental groups are nonabelian when H is nonabelian.

Explicit example of conjugate varieties with non-isomorphic fundamental groups.

Topological properties can change under automorphisms of the complex numbers.

Abstract

Let X=G/H be the quotient of a connected reductive algebraic C-group G defined over the field of complex numbers C by a finite subgroup H. We describe the topological fundamental group of the homogeneous space X, which is nonabelian when H is nonabelian. Further, we construct an example of a homogeneous space X and an automorphism s of C such that the topological fundamental groups of X and of the conjugate variety sX are not isomorphic.