Elementary construction of perverse sheaves
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.
construction of the category of perverse sheaves
scientific article; zbMATH DE number 3961920
scientific article