The first L2-Betti number of certain homogeneous spaces, including classifying spaces for variations of Hodge structures

Abstract

In this paper, we prove the vanishing of the first L-Betti number for certain complex homogeneous spaces D = G/V , where G is a semi-simple Lie group and V is a compact subgroup. L-cohomology is the appropriate extension of Hodge theory for harmonic forms to the case of a noncompact (complete) manifold X inasmuch as here every L-cohomology class can be represented by an L-harmonic form, see [1]. In particular, non-existence of non-trivial L-harmonic p-forms means that the p-th L-Betti number vanishes. Besides offering the possibility to extend Hodge theory to the noncompact case, it can be used to obtain topological information about compact quotients of X by the Lindex theorem of Atiyah [1]. Based on this theorem Dodziuk-Singer [4][12] suggested to use L-cohomology to approach the Hopf conjecture concerning the Euler characteristic of Riemannian manifolds of non-positive curvature, which has been verified in several special but important cases [2],[3],