Compact Subvarieties in Flag Domains

Abstract

A real reductive Lie group G 0 acts on complex flag manifolds G/Q, where Q is a parabolic subgroup of the complexification G of G 0. The open orbits D = G 0(x) include the homogeneous complex manifolds of the form G 0/V 0 where V 0= G 0∩Q x the centralizer of a torus; those are the G 0-homogeneous pseudo-kahler manifolds. For an appropriate choice K 0 maximal compact subgroup of G 0, the orbit Y=K 0(x) is a maximal compact complex submanifold of D. The “cycle space” M D =gY ∣ g ∈ G and gY ⊂ D, space of maximal compact linear subvarieties of D, has a natural complex structure. M D plays important roles in the theory of moduli of compact kahler manifolds and in automorphic cohomology theory. Here we sketch a brief exposition of this interesting mathematical topic.