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| 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 [[topologically stratified space]], usually singular. This concept was introduced by [[Zoghman Mebkhout]] in his thesis and then further developed 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 (mathematician)|Robert MacPherson]]) and the algebraic theory of differential equations ([[microlocal calculus]] and holonomic [[D-module]]s 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]].
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| == Preliminary remarks ==
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| The name ''perverse sheaf'' requires explanation: they are not sheaves in the mathematical (or any other) sense, nor are they perverse.<ref>''Les faisceaux pervers n'etant ni des faisceaux, ni pervers, la terminologie requiert une explication.'' BBD, p. 10</ref> The justification is that perverse sheaves have several features in common with sheaves: they form an abelian category, they have [[cohomology]], and to construct one, it suffices to construct it locally everywhere. The adjective "perverse" originates in the [[Intersection homology#Perversities|intersection homology]] theory, and its origin was explained by {{harvtxt|Goresky|2010}}.
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| The Beilinson-Bernstein-Deligne 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.
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| In the [[Riemann-Hilbert correspondence]], perverse sheaves correspond to holonomic [[D-module]]s. This application establishes the notion of perverse sheaf as occurring 'in nature'.
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| ==Definition==
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| A '''perverse sheaf''' is an object ''C'' of the bounded [[derived category]] of sheaves with [[Constructible sheaf|constructible]] cohomology on a space ''X'' such that the set of points ''x'' with
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| :<math>H^{-i}(j_x^*C)\ne 0 </math> or <math>H^{i}(j_x^!C)\ne 0</math>
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| has dimension at most 2''i'', for all ''i''. Here ''j''<sub>''x''</sub> is the inclusion map of the point ''x''.
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| The category of perverse sheaves is an abelian subcategory of the (non-abelian) derived category of sheaves,
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| equal to the core of a suitable [[t-structure]], and is preserved by [[Verdier duality]].
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| == Notes ==
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| <references />
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| ==See also==
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| *[[Triangulated category]]
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| ==References==
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| * {{cite journal
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| | url = http://www.ams.org/notices/201005/rtx100500632p.pdf
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| | last = Andrea de Cataldo
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| | first = Mark
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| | coauthors = Luca Migliorini
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| | title=What is a perverse sheaf?
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| | journal = Notices of the AMS
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| | date=May 2010
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| | volume = 57
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| | issue = 5
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| | ref = harv
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| }}
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| * {{cite journal
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| | last = Beilinson
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| | first = A. A.
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| | authorlink = Alexander Beilinson
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| | coauthors = [[Joseph Bernstein|J. Bernstein]], [[Pierre Deligne|P. Deligne]]
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| | year = 1982
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| | title = Faisceaux pervers
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| | journal = Astérisque
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| | volume = 100
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| | publisher = Société Mathématique de France, Paris
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| | language = French
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| | ref = harv
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| }}
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| * {{Cite web | last1=Goresky | first1=Mark | authorlink1=Mark Goresky | title=What is the etymology of the term "perverse sheaf"? | url=http://mathoverflow.net/questions/44149 | year=2010 | ref=harv | postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}}
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| * {{cite journal
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| | author = Robert MacPherson
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| | title = Intersection Homology and Perverse Sheaves
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| | date = December 15, 1990
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| | format = unpublished manuscript
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| | url = http://faculty.tcu.edu/gfriedman/notes/ih.pdf
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| | ref = harv
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| | authorlink = Robert MacPherson (mathematician)
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| }}
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| * {{cite arxiv
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| | eprint=math.RT/0307349.pdf
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| | title=An introduction to perverse sheaves
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| | author=Konstanze Rietsch
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| | year=2003
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| }}
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| [[Category:Homological algebra]]
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| [[Category:Morse theory]]
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