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Perverse sheaves and the reductive Borel-Serre compactification. (English) Zbl 1386.14078

Ji, Lizhen (ed.), Hodge Theory and \(L^2\)-analysis. Somerville, MA: International Press; Beijing: Higher Education Press (ISBN 978-1-57146-351-7/pbk). Advanced Lectures in Mathematics (ALM) 39, 555-581 (2017).
As the author points out, this paper is meant to be a report on work in progress towards a better understanding of the category of perverse sheaves on the Baily-Borel compactification \(X^\ast\) of a locally symmetric variety \(X\) by studying perverse sheaves on its reductive Borel-Serre compactification \(\widehat{X}\). The latter was introduced by S. Zucker about 35 years ago in [Invent. Math. 70, 169–218 (1982; Zbl 0508.20020)], and has been intensively investigated ever since. In particular, S. Zucker has shown that there is a continuous quotient map \(\pi:\widehat{X}\to X^\ast\) over the identity of \(X\) so that \(\widehat{X}\) may be viewed as a partial desingularization of \(X^\ast\). In this context, using a natural stratification of the reductive Borel-Serre compactification \(\widehat{X}\), the author proves a variant of the so-called composition theorem of Beilinson-Bernstein-Deligne-Gabber-Saito for the above-mentioned map \(\pi:\widehat{X}\to X^\ast\) of compactifications with respect to a middle perversity. The author’s theorem says that the direct image of a certain perverse sheaf on \(\widehat{X}\) decomposes into the direct sum of shifted simple perverse sheaves supported on subvarieties of \(X^\ast\). Apart from a sketch of the proof of this main result, it is explained how to calculate extension groups Ext\(^1\) for two perverse sheaves on the reductive Borel-Serre compactification, and a few example computations are provided at the end of the paper. In order to keep the discussion as self-contained as possible, a very detailed introduction is accompanied by introductory sections on \(t\)-structures, perverse sheaves, simplicity of perverse sheaves, and a comparison with intersection cohomology à la Goresky-MacPherson.
For the entire collection see [Zbl 1375.14007].

MSC:

14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
14M27 Compactifications; symmetric and spherical varieties
32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects)
57T15 Homology and cohomology of homogeneous spaces of Lie groups

Citations:

Zbl 0508.20020
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