Betti numbers of Hilbert modular varieties

Summary

INTRODUCTION

In this paper we give formulae for the Betti numbers of Hilbert modular varieties, as -well as show that any such variety is simply-connected. These results complement the work of [19], where we calculate the Chern numbers of modular varieties of complex dimension three. The goal is the classification of modular varieties up to diffeomorphism, birational equivalence, or biholomorphic isomorphism; see section three for further discussion.

Throughout the paper K will denote a totally real algebraic number field of degree n (>1), O its ring of integers, and G (= PSL₂(O)) its Hilbert modular group. We will say that a group Γ is of modular type (for K) if either Γ = G or Γ is a torsion free subgroup of G of finite index. If Γ is a principal congruence subgroup (or Γ = G), we say that Γ is of principal type.

The group G acts on H (the complex upper half plane) by linear fractional transformations. Thus, by means of the n distinct embeddings of in the real numbers, we obtain an action of G (and hence of Γ) on Hⁿ. We define Y_Γ = Hⁿ/Γ, the orbit space of this action. Y_Γ is a non-compact normal complex space with a finite number of isolated (“quotient”) singularities, the images of the elliptic fixed points of the action of Γ on Hⁿ.

Let h denote the number of parabolic orbits of Γ. (if Γ = G, h is simply the class number of K.)