\(L^ 2\)-cohomology of locally symmetric varieties. (English) Zbl 0658.14010

Looijenga, Eduard

Compos. Math. 67, No. 1, 3-20 (1988).

Let G be a connected semisimple algebraic group defined over \({\mathbb{Q}}\) and \(G({\mathbb{R}})\) the associated real group. Assume that \(G({\mathbb{R}})\) is hermitean, i.e. \(D=G({\mathbb{R}})/K({\mathbb{R}})\) carries the structure of a complex manifold with an invariant hermitean metric, where \(K({\mathbb{R}})\) is a maximal compact subgroup of \(G({\mathbb{R}})\). If \(\Gamma\) is an arithmetic subgroup of G(\({\mathbb{Q}})\) let \(X^ 0=\Gamma \setminus D\) and X the Satake-Baily-Borel compactification of \(X^ 0\). The space X is a projective variety with singularities. Let \({\mathcal L}_ X\) be the sheaf of locally square integrable forms (with locally square integrable exterior derivative) defined on the regular set \(X_{reg}\) of \(X^ 0\). Then \({\mathcal L}_ X\) is a complex and defines the \(L^ 2\)-cohomology of X.
The main result of this paper is a proof of Zucker’s conjecture that the \(L^ 2\)-cohomology of X agrees with the intersection cohomology of X, a cohomology theory coming from a subcomplex of the ordinary chain complex, defined via a piecewise linear structure on X. The theorem holds for local coefficients defined by a finite dimensional representation of G. The space X has a stratification and the proof of the main theorem proceeds through various reductions and an induction on the codimension of the strata.

Reviewer: J.Hilgert

MSC:

14F99	(Co)homology theory in algebraic geometry
32M15	Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
55N20	Generalized (extraordinary) homology and cohomology theories in algebraic topology
14M17	Homogeneous spaces and generalizations
22E15	General properties and structure of real Lie groups

Keywords:

\(L^ 2\)-cohomology; intersection cohomology

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