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**Weights in the local cohomology of a Baily-Borel compactification.**
*(English)*
Zbl 0753.14016

Complex geometry and Lie theory, Proc. Symp., Sundance/UT (USA) 1989, Proc. Symp. Pure Math. 53, 223-260 (1991).

[For the entire collection see Zbl 0741.00047.]

Let \(M=\Gamma\backslash D\) be the quotient of a symmetric space \(D\) with a hermitian structure (arising from a semisimple algebraic group \(G\) over \(\mathbb{Q})\) by an arithmetic subgroup \(\Gamma\) of \(G(\mathbb{Q})\). Then \(M\) is an algebraic variety having a canonical compactification \(X\) (the Baily- Borel-Satake compactification) which is a projective variety. Let \(E\) be local system on \(M\) defined by a rational representation of \(G\), and denote by \(H(E)\) the intersection cohomology complex of \(X\) associated to \(E\).

The aim of this paper is two-fold. The first one is to relate various weight filtrations on the fiber of a cohomology sheaf of \(H(E)\) in a point of the boundary. The first named author had proved a conjecture of Zucker saying that the \(L^ 2\)-complex represents \(H(E)\) [see E. Looijenga, Compos. Math. 67, No. 1, 3-20 (1988; Zbl 0658.14010)]. This proof was based on a purity theorem involving weights with respect to a “local Hecke operator”. This purity result used in turn the elaborated theory of M. Sakai.

The second aim of this paper is to give another proof of the Zucker conjecture [which in fact is the third known proof after Looijenga’s proof mentioned above and the proof by L. Saper and M. Stern contained in Ann. Math., II. Ser. 132, No. 1, 1-69 (1990; Zbl 0653.14010)]. This new proof has the advantage of using more elementary results and is based on a purity lemma of Serre (which in fact was the starting point of the theory of weights).

Let \(M=\Gamma\backslash D\) be the quotient of a symmetric space \(D\) with a hermitian structure (arising from a semisimple algebraic group \(G\) over \(\mathbb{Q})\) by an arithmetic subgroup \(\Gamma\) of \(G(\mathbb{Q})\). Then \(M\) is an algebraic variety having a canonical compactification \(X\) (the Baily- Borel-Satake compactification) which is a projective variety. Let \(E\) be local system on \(M\) defined by a rational representation of \(G\), and denote by \(H(E)\) the intersection cohomology complex of \(X\) associated to \(E\).

The aim of this paper is two-fold. The first one is to relate various weight filtrations on the fiber of a cohomology sheaf of \(H(E)\) in a point of the boundary. The first named author had proved a conjecture of Zucker saying that the \(L^ 2\)-complex represents \(H(E)\) [see E. Looijenga, Compos. Math. 67, No. 1, 3-20 (1988; Zbl 0658.14010)]. This proof was based on a purity theorem involving weights with respect to a “local Hecke operator”. This purity result used in turn the elaborated theory of M. Sakai.

The second aim of this paper is to give another proof of the Zucker conjecture [which in fact is the third known proof after Looijenga’s proof mentioned above and the proof by L. Saper and M. Stern contained in Ann. Math., II. Ser. 132, No. 1, 1-69 (1990; Zbl 0653.14010)]. This new proof has the advantage of using more elementary results and is based on a purity lemma of Serre (which in fact was the starting point of the theory of weights).

Reviewer: L.Bădescu (Bucureşti)

### MSC:

14F43 | Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) |

14G35 | Modular and Shimura varieties |