Elementary construction of perverse sheaves
type: Item
instance of: scholarly article
title: Elementary construction of perverse sheaves
zbMATH Open document ID: 0597.18005
DOI: 10.1007/BF01388812
author: Robert MacPherson
author: Kari Vilonen
published in: Inventiones Mathematicae
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143344
review text: This paper presents a construction of the category of perverse sheaves which are constructible relative to a given stratification which is ''elementary'' in the sense that derived categories do not appear explicitly. The construction is inductive: if S is a closed stratum in X the categories for X and X-S are related by the following construction. One has two categories A and B, two functors \(F,G: A\to B\) and a natural transformation \(T: F\to G;\) and forms the category of factorizations FA\(\to B\to GA\) of maps TA. The theoretical construction of this data involves operations with derived direct and inverse image functors of categories of sheaves. However, these can be made more concrete, assuming all strata even dimensional and using the middle perversity. Calculations centre round a ''perverse link'' which can be constructed using the second barycentric subdivision of a triangulation of the stratified link or, in the complex analytic case, more directly: the definition is by conditions on its intersection homology. If S is 2- connected this data suffices for the construction; in general consideration of the ''perverse link bundle'' is required. The authors apply their results to give a topological proof of a theorem of \textit{M. Kashiwara} and \textit{T. Kawai} [Publ. Res. Inst. Math. Sci. 17, 813-979 (1981; Zbl 0505.58033)] giving (in the complex case) a simple criterion for the extension to be trivial. They also give a number of examples; in the simplest, the complex line stratified by the origin and its complement, the sheaf is characterized by the monodromy. In the other examples also the authors show the category is equivalent to the category of representations of some fixed quiver, and conjecture that this is always so for certain natural stratifications of the space of Borel subgroups of a reductive complex algebraic group.
Mathematics Subject Classification ID: 18F20
Mathematics Subject Classification ID: 16P10
Mathematics Subject Classification ID: 32S60
Mathematics Subject Classification ID: 14F40
Mathematics Subject Classification ID: 58J10
Mathematics Subject Classification ID: 14C17
zbMATH DE Number: 3961920
zbMATH Keywords: representations of quiver
zbMATH Keywords: construction of the category of perverse sheaves
zbMATH Keywords: closed stratum
zbMATH Keywords: middle perversity
zbMATH Keywords: barycentric subdivision
zbMATH Keywords: intersection homology
zbMATH Keywords: perverse link bundle
zbMATH Keywords: monodromy
zbMATH Keywords: Borel subgroups
zbMATH Keywords: reductive complex algebraic group
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3961920