Quotients of non-classical flag domains are not algebraic

TL;DR

This paper demonstrates that quotients of non-classical flag domains by certain discrete groups lack an algebraic structure, highlighting fundamental differences from classical cases.

Contribution

It proves that non-classical flag domains cannot be algebraically realized when quotiented by infinite, finitely generated discrete subgroups.

Findings

Any two points in a non-classical domain can be connected by a chain of compact subvarieties.

Quotients of non-classical flag domains by such groups are not algebraic.

The result distinguishes non-classical flag domains from classical Hermitian symmetric spaces.

Abstract

A flag domain D = G/V for G a simple real non-compact group G with compact Cartan subgroup is non-classical if it does not fiber holomorphically or anti-holomorphically over a Hermitian symmetric space. We prove that any two points in a non-classical domain D can be joined by a finite chain of compact subvarieties of D. Then we prove that for F an infinite, finitely generated discrete subgroup of G, the analytic space F\D does not have an algebraic structure.