L\({}_ 2\)-cohomology of arithmetic varieties. (English) Zbl 0722.14009

Saper, Leslie; Stern, Mark

Ann. Math. (2) 132, No. 1, 1-69 (1990).

Suppose that G is the group of real points of a semi-simple algebraic group defined over the rationals, that K is a maximal compact subgroup of G and that the associated symmetric space \(D=G/K\) is Hermitian. The quotient X of D by an arithmetic subgroup of G which acts freely on D has a natural complete metric induced from the Bergman metric on D. It was conjectured by S. Zucker in Invent. Math. 70, 169-218 (1982; Zbl 0508.20020) that there is a natural isomorphism between the \(L_ 2\)- cohomology of X and the (middle perversity) intersection cohomology of the Satake-Baily-Borel compactification of X.
In the paper under review the authors present a proof of this conjecture which was announced earlier by them in Proc. Natl. Acad. Sci. USA 84, 5516-5519 (1987; Zbl 0653.14010). A different and independent proof has been given by E. Looijenga in Compos. Math. 67, No.1, 3-20 (1988; Zbl 0658.14010).

Reviewer: F.Kirwan (Oxford)

MSC:

14F43	Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
14G40	Arithmetic varieties and schemes; Arakelov theory; heights
14M17	Homogeneous spaces and generalizations

Keywords:

quotient by an arithmetic subgroup; middle perversity intersection cohomology; \(L_ 2\)-cohomology

Citations:

Zbl 0508.20020; Zbl 0653.14010; Zbl 0658.14010