Normal bundles of cycles in flag domains

TL;DR

This paper investigates the complex geometry of flag domains, focusing on the normal bundles of cycles, revealing their triviality relates to the holomorphic convexity of the domain, and classifies the domain's geometric nature.

Contribution

It provides a detailed analysis of the normal bundles of cycles in flag domains and establishes a criterion linking their triviality to the domain's holomorphic convexity.

Findings

Normal bundle E is trivial iff D is holomorphically convex.

D is a product of C and a Hermitian symmetric space when E is trivial.

Otherwise, D is pseudoconcave.

Abstract

A real semisimple Lie group G_0 embedded in its complexification G has only finitely many orbits in any G-fag manifold Z = G/Q. The complex geometry of its open orbits D (flag domains) is studied from the point of view of compact complex submanifolds C (cycles) which arise as orbits of certain distinguished subgroups. Normal bundles E of the cycles are analyzed in some detail. It is shown that E is trivial if and only if D is holomorphically convex, in fact a product of C and a Hermitian symmetric space, and otherwise D is pseudoconcave.