Chern classes of automorphic vector bundles

Abstract

1.1. Suppose X is a compact n-dimensional complex manifold. Each partition I = {i1, i2, . . . , ir} of n corresponds to a Chern number c(X) = (ci1(X)∪ci2(X)∪. . .∪cir(X)∩[X]) ∈ Z where c(X) ∈ H(X;Z) are the Chern classes of the tangent bundle, [X] ∈ H2n(X;Z) is the fundamental class, and : H0(X;Z) → Z is the augmentation. Many invariants of X (such as its complex cobordism class) may be expressed in terms of its Chern numbers ([Mi], [St]). During the last 25 years, characteristic classes of singular spaces have been defined in a variety of contexts: Whitney classes of Euler spaces [Su], [H-T], [Ak], Todd classes of singular varieties [BFM], Chern classes of singular algebraic varieties [Mac], L-classes of stratified spaces with even codimension strata [GM1], Wu classes of singular spaces [Go2], [GP] (to name a few). However, these characteristic classes are invariably homology classes and as such, they cannot be multiplied with each other. In some cases it has been found possible to “lift” these classes from homology to intersection homology, where (some) characteristic numbers may be formed ([BBF], [BW], [Go2],[GP], [T]). The case of locally symmetric spaces is particularly interesting. Suppose Γ is a torsionfree arithmetic group acting on a complex n-dimensional Hermitian symmetric domain D = G/K, where G is the group of real points of a semisimple algebraic group G defined over Q with Γ ⊂ G(Q), and where K ⊂ G is a maximal compact subgroup. Then X = Γ\D is a Hermitian locally symmetric space. When X is compact, Hirzebruch’s proportionality theorem [Hr1] says that there is a number v(Γ) ∈ Q so that for every partition I = {i1, i2, . . . , ir} of n, the Chern number satisfies c(X) = v(Γ)c(Ď), where Ď = Gu/K is the compact dual symmetric space (and Gu is a compact real form of G(C) containing K). If X = Γ\D is noncompact, it has a canonical Baily-Borel (Satake) compactification, X. This is a (usually highly singular) complex projective algebraic variety. To formulate a proportionality theorem in the noncompact case, one might hope that the tangent bundle TX extends as a complex vector bundle over X, but this is false. In [Mu1], D. Mumford showed that TX has a particular extension EΣ → XΣ over any toroidal resolution τ : XΣ → X of the Baily-Borel compactification and that for any partition I of n, the resulting Chern numbers c(EΣ) = (c (EΣ) ∪ c(EΣ) ∪ . . . ∪ c(EΣ) ∩ [XΣ]) (1.1.1)