Let G be a complex Lie group and H a discrete subgroup which is large in a certain sense, e.g. of finite covolume. We are interested in studying holomorphic vector bundles over the quotient manifold X=G/H. In general (i.e. if G is non-commutative) these manifolds are not algebraic and not even Kähler.
Thus in order to understand these manifolds it is necessary to employ methods other than those usually applied in algebraic and Kähler geometry. These methods are in particular group-theoretical methods like methode from Lie theory, representation theory, and the theory of algebraic and arithmetic groups. In order to be able to employ these group-theoretical methods we restrict our attention to holomorphic vector bundles which carry a flat holomorphic connection. Any such bundle is induced by a representation of the fundamental group. Special emphasis is put on what we call essentially antiholomorphic representations. For irreducible lattices H in semisimple complex Lie groups G of rank at least two it is an easy consequence of the celebrated arithmeticity results of Margulis et al. that every flat holomorphic vector bundle over G/H is isomorphic as a holomorphic vector bundle to a flat bundle induced by an essentially antiholomorphic representation. For lattices in arbitrary complex Lie groups there are always non-trivial essentially antiholomorphic representations. Moreover flat vector bundles induced by antiholomorphic representations arise naturally as higher direct images sheaves of the structure sheaf for fibrations of complex parallelizable manifolds. Therefore these results are useful to calculate Dolbeault cohomology groups via Leray spectral sequences associated to certain fibrations. In a separate paper we will exploit this to deduce results on cohomology groups, deformations and the Albanese variety of complex parallellizable manifolds.
We give a precise criterion determining when a flat vector bundle induced by an essentially antiholomorphic representation is trivial and moreover classify these bundles completely up to isomorphism as holomorphic vector bundles in the case G=[G,G].
We then proceed to study sections and subbundles of flat vector bundles. Essentially we prove that in general sections exist only to the extent to which the bundle is trivial and that generically every vector subbundle of a flat vector bundle on a complex parallelizable manifold is again flat.
As a by-product we can show that every compact complex manifold of dimension n>0 admits a non-trivial holomorphic vector bundle of rank at most n (improving our earlier result).
The assumption that H is of finite covolume is often used in a rather indirect way. For instance we prove that, given a lattice H in a complex Lie group G and a fixed base point p in G/H, the group G is always generated by those connected commutative complex Lie subgroups A for which the orbit through p is compact. We will also use the fact that G/H carries no non-constant plurisubharmonic function.
Forum math. 13, 795-815 (2001)