Perverse sheaf

The mathematical term perverse sheaves refers to a certain abelian category associated to a topological space "X", which may be a real or complex manifold, or a more general stratified space, usually singular. This concept was introduced by Joseph Bernstein, Alexander Beilinson, Pierre Deligne, and Ofer Gabber (1982) as a formalisation of the Riemann-Hilbert correspondence, which related the topology of singular spaces (intersection homology of Mark Goresky and Robert MacPherson) and the algebraic theory of differential equations (microlocal calculus and holonomic D-modules of Joseph Bernstein, Masaki Kashiwara and Takahira Kawai). It was clear from the outset that perverse sheaves are fundamental mathematical objects at the crossroads of algebraic geometry, topology, analysis and differential equations. They also play an important role in number theory, algebra, and representation theory.

Preliminary remarks

The name "perverse sheaf" is a bit of a misnomer: they are neither sheaves in the ordinary sense, nor are they perverse. ["Les faisceaux perverse n'etant ni des faisceaux, ni pervers, la terminologie requiert une explication." BBD, p. 10] The justification is that perverse sheaves have several features in common with sheaves: they form an abelian category, you can take their cohomology, and to construct one, it suffices to construct it locally everywhere. The adjective "perverse" originates in the intersection homology theory.

The BBD definition of a perverse sheaf proceeds through the machinery of triangulated categories in homological algebra and has very strong algebraic flavour, although the main examples arising from Goresky-MacPherson theory are topological in nature. This motivated MacPherson to recast the whole theory in geometric terms on a basis of Morse theory. For many applications in representation theory, perverse sheaves can be treated as a 'black box', a category with certain formal properties.

In the Riemann-Hilbert correspondence, perverse sheaves correspond to holonomic D-modules. This application established the notion of perverse sheaf as occurring 'in nature'.

Definition

A perverse sheaf is an element "C" of the bounded derived category of sheaves with constructible cohomology on a space "X" such that the set of points "x" with:$H^\{-i\}(j\_x^*C)\; e\; 0$ or $H^\{i\}(j\_x^!C)\; e\; 0$has dimension at most 2"i", for all "i". Here "j"_"x" is the inclusion map of the point "x".

The category of perverse sheaves is an abelian subcategory of the (non-abelian) derived category of sheaves,equal to the core of a suitable t-structure, and is preserved by Verdier duality.

Notes

ee also

*Triangulated category

References

* cite journal
last = Beilinson
first = A. A.
authorlink = Alexander Beilinson
coauthors = J. Bernstein, P. Deligne
year = 1982
title = Faisceaux pervers
journal = Astérisque
volume = 100
publisher = Société Mathématique de France, Paris
language = French
* cite paper
author = Robert MacPherson
title = Intersection Homology and Perverse Sheaves
date = December 15, 1990
format = unpublished manuscript