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\(L_ 2\)-cohomology of arithmetic varieties. (English) Zbl 0653.14010

In 1982, S. Zucker [Invent. Math. 70, 169-218 (1982; Zbl 0508.20020)] conjectured that the \(L_ 2\)-cohomology groups of an arithmetic quotient of a bounded symmetric domain with respect to its natural complete Kähler metric are naturally isomorphic to the middle intersection cohomology groups of its Baily-Borel compactification. In this paper the authors give an outline of their proof of this conjecture (which has been proved recently by Looijenga for local coefficient systems of geometric origin).
Reviewer: F.Kirwan

MSC:

14F99 (Co)homology theory in algebraic geometry
55N35 Other homology theories in algebraic topology
32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
14G99 Arithmetic problems in algebraic geometry; Diophantine geometry

Citations:

Zbl 0508.20020
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