The mathematical term perverse sheaves refers to the objects of certain abelian categories associated to topological spaces, which may be a real or complex manifold, or more general topologically stratified spaces, possibly singular. The concept was introduced in the work of Joseph Bernstein, Alexander Beilinson, and Pierre Deligne and Ofer Gabber (1982) as a consequence of the Riemann-Hilbert correspondence, which establishes a connection between the derived category of regular holonomic D-modules and the derived category of constructible sheaves. Perverse sheaves are the objects in the latter that correspond to individual D-modules (and not more general complexes thereof); a perverse sheaf is in general represented by a complex of sheaves. The concept of perverse sheaves is already implicit in a 75's paper of Kashiwara on the constructibility of solutions of holonomic D-modules. A key observation was that the intersection homology of Mark Goresky and Robert MacPherson could be described using sheaf complexes that are actually perverse sheaves. 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 comes through rough translation of the French "faisceaux pervers". The justification is that perverse sheaves are complexes of sheaves which 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 theory, and its origin was explained by Goresky (2010). The Beilinson–Bernstein–Deligne-Gabber definition of a perverse sheaf proceeds through the machinery of triangulated categories in homological algebra and has a very strong algebraic flavour, although the main examples arising from Goresky–MacPherson theory are topological in nature because the simple objects in the category of perverse sheaves are the intersection cohomology complexes. 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.
Definition and examples A perverse sheaf is an object 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 ) ≠ 0 {\displaystyle H^{-i}(j_{x}^{*}C)\neq 0} or H i ( j x ! C ) ≠ 0 {\displaystyle H^{i}(j_{x}^{!}C)\neq 0}
has real dimension at most 2i, for all i. Here jx is the inclusion map of the point x. If X is a smooth complex algebraic variety and everywhere of dimension d, then
F [ d ] {\displaystyle {\mathcal {F}}[d]}
is a perverse sheaf for any local system F {\displaystyle {\mathcal {F}}} . If X is a flat, locally complete intersection (for example, regular) scheme over a henselian discrete valuation ring, then the constant sheaf shifted by dim X + 1 {\displaystyle \dim X+1} is an étale perverse sheaf.
A simple example Let X be a disk around the origin in C {\displaystyle \mathbb {C} } stratified so that the origin is the unique singular stratum. Then the category of perverse sheaves on X is equivalent to the category of diagrams of vector spaces V ⇄ v u W {\displaystyle V{\overset {u}{\underset {v}{\rightleftarrows }}}W} where id − u ∘ v {\displaystyle \operatorname {id} -u\circ v} and id − v ∘ u {\displaystyle \operatorname {id} -v\circ u} are invertible. More generally, quivers can be used to describe perverse sheaves.
Properties 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. The bounded derived category of perverse l-adic sheaves on a scheme X is equivalent to the derived category of constructible sheaves and similarly for sheaves on the complex analytic space associated to a scheme X/C.
Applications Perverse sheaves are a fundamental tool for the geometry of singular spaces. Therefore, they are applied in a variety of mathematical areas. In the Riemann-Hilbert correspondence, perverse sheaves correspond to regular holonomic D-modules. This application establishes the notion of perverse sheaf as occurring 'in nature'. The decomposition theorem, a far-reaching extension of the hard Lefschetz theorem decomposition, requires the usage of perverse sheaves. Hodge modules are, roughly speaking, a Hodge-theoretic refinement of perverse sheaves. The geometric Satake equivalence identifies equivariant perverse sheaves on the affine Grassmannian G r G {\displaystyle Gr_{G}} with representations of the Langlands dual group of a reductive group G - see Mirković & Vilonen (2007). A proof of the Weil conjectures using perverse sheaves is given in Kiehl & Weissauer (2001).
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