# Perverse sheaf

The mathematical term

**perverse sheaves**refers to a certainabelian category associated to a topological space "X", which may be a real or complex manifold, or a more generalstratified space , usually singular. This concept was introduced byJoseph Bernstein ,Alexander Beilinson ,Pierre Deligne , andOfer Gabber (1982) as a formalisation of theRiemann-Hilbert correspondence , which related the topology of singular spaces (intersection homology ofMark Goresky andRobert MacPherson ) and the algebraic theory of differential equations (microlocal calculus and holonomicD-module s ofJoseph Bernstein ,Masaki Kashiwara andTakahira Kawai ). It was clear from the outset that perverse sheaves are fundamental mathematical objects at the crossroads ofalgebraic geometry ,topology , analysis anddifferential equations . They also play an important role innumber theory , algebra, andrepresentation 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 inhomological 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 ofMorse 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 holonomicD-module s. This application established the notion of perverse sheaf as occurring 'in nature'.**Definition**A

**perverse sheaf**is an element "C" of the boundedderived 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 byVerdier 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

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