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On the Reductive Borel-Serre Compactification, III: Mixed Hodge Structures

Steven Zucker

2004
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Asian Journal of Mathematics
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We establish a procedure for constructing compatible mixed Hodge structures for the cohomology of various topological compactifications of locally symmetric varieties, notably ones that are not algebraic varieties. This is carried out in full for the case of the reductive Borel-Serre compactification, and conditionally for the excentric compactifications. Introduction. In [BS], Borel and Serre gave a construction that compactified the arithmetic quotient X of a symmetric space (also certain
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... general locally homogeneous spaces) to a manifold-with-corners X. With its faces fibered by nilmanifolds, it was both an easy and a useful little construction to collapse each fiber to a point, producing what is now called the reductive Borel-Serre compactification X red of X. For some of the uses of this space, see [Z1: §4], [GHM], [GT], [Z3]. The construction applies, in particular, to the Hermitian cases (i.e., locally symmetric varieties). Even there, the boundary faces are often of odd real dimension, which rules out the idea that X red , like X, be an algebraic variety over C or even a complex space. Still, Mark Goresky once made a casual remark concerning X red in the Hermitian case. I forget his exact words, but it was something like, "It thinks it's an algebraic variety." As a Hodge-theorist, I had the inevitable reaction: there should be a mixed Hodge structure on its cohomology. That was the impetus for this work. Here, we must remind the reader that the existence of a mixed Hodge structure on a vector space is not a significant statement; the selecting of one is. In this article, after eight largely expository sections, we construct in §9 mixed Hodge structures for H • (X red ) when X is Hermitian. We also sketch a construction of mixed Hodge structures for the cohomology of related compactifications of X. This includes first, in §10, X exc (the excentric Borel-Serre compactification), which is a quotient of X that maps to X red . As there are mappings (see [Z2], where the last space is the projective algebraic Baily-Borel compactification [BB], we show that we get a morphism of mixed Hodge structure H • (X * ) → H • (X red ), and conditionally also H • (X red ) → H • (X exc ). The condition here is completing the construction of the mixed Hodge structure on H • (X exc ); that seems to be obstructed by our inability to show that two mixed Hodge structures, on a cohomology group that enters, coincide (see (8.17)). In §12, we mention a possible resolution: generalizing the notion of mixed Hodge modules to spaces more general than complex varieties. In a different direction are the smooth projective toroidal compactifications X tor of [AMRT], which lie in the realm of classical Hodge theory. However, these too have an excentric quotient X tor,exc , which is generally not a complex space. On its *

doi:10.4310/ajm.2004.v8.n4.a29
fatcat:7lshyde5xzfhlbshcks6ivi7tm