8 INTRODUCTION

dimensional of GC in the compact case or infinite dimensional of GR in the non-

compact case (including both GR/T ’s and Γ\GR/T ’s), and that in some generality

the connection between these different manifestations may be realized geometrically.

Although we here have informally mentioned some of these general results, for the

reasons stated above we have in this work focused on our examples.

It is the authors’ pleasure to thank Sarah Warren for a marvelous job of con-

verting an at best barely legible handwritten manuscript into mathematical text.

Outline

The following is an outline of the contents of the various sections of this paper.

We begin in section I.A with a general discussion of the homogeneous complex

manifolds that will be considered in this work. Here, and later, we emphasize the

distinction between equivalence of homogenous complex manifolds and homogeneous

vector bundles over them, rather than just equivalence as complex manifolds and

holomorphic vector bundles.

In section I.B we discuss our first example, which is the non-classical complex

structure on U(2, 1)/T := D, realized as one of the three open orbits of U(2, 1)

acting on the homogeneous projective variety of flags (0) ⊂ F1 ⊂ F2 ⊂ F3 = C3

where dim Fi = i. Here C3 has the important additional structure of being the

complexification of F3 where F = Q(

√

−d) is a quadratic imaginary number field.

It is this additional structure that leads to the realization of D as a Mumford-Tate

domain, thereby bringing Hodge theory into the story. The other two open GR-

orbits D and D are classical and may also be realized as Mumford-Tate domains,

or what is more relevant to this work, the set of Hodge flags asssociated to Mumford-

Tate domains consisting of polarized Hodge structures of weight one with additional

structure.

All three of the above domains have three descriptions: geometric, group-

theoretic and Hodge-theoretic. The interplay between these different perspectives

is an important part of the exposition. Especially important is the book-keeping

between the tautological, root and weight, and Hodge theoretic descriptions of the

U(2, 1)-homogeneous line bundles over the domains, which is given in section II.B.

In our first example the three domains may be pictured as

✟

✟

✟

✟

✟

✟

✟

✟

✟

✟

✟

✟

✟

L

p

l

P

B

Figure 1