# 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 topologically stratified space, usually singular. This concept was introduced in the thesis of Zoghman Mebkhout, gaining more popularity after the (independent) work of Joseph Bernstein, Alexander Beilinson, and Pierre Deligne (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 Takahiro 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. The properties characterizing perverse sheaves already appeared in the 75's paper of Kashiwara on the constructibility of solutions of holonomic D-modules.

## Preliminary remarks

The name perverse sheaf comes through rough translation of the French "faisceaux pervers"[1]. 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 "pervers" originates in the intersection homology theory,[2] and its origin was explained by Goresky (2010).

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 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

${\displaystyle H^{-i}(j_{x}^{*}C)\neq 0}$ or ${\displaystyle H^{i}(j_{x}^{!}C)\neq 0}$

has dimension at most 2i, for all i. Here jx is the inclusion map of the point x.

If X is smooth and everywhere of dimension d, then

${\displaystyle {\mathcal {F}}[d]}$

is a perverse sheaf for any local system ${\displaystyle {\mathcal {F}}}$.[3] If X is a flat, locally complete intersection (for example, regular) scheme over a henselian discrete valuation ring, then the constant sheaf shifted by ${\displaystyle \dim X+1}$ is an étale perverse sheaf.[4]

## 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.[5]

## 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 ${\displaystyle Gr_{G}}$ 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).

## Notes

1. Les faisceaux pervers n'etant ni des faisceaux, ni pervers, la terminologie requiert une explication. BBD, p. 10
2. Beilinson, Bernstein & Deligne (1982, Proposition 2.2.2, §4.0)
3. Illusie (2003, Corollaire 2.7)
4. Beilinson (1987, Theorem 1.3)

## References

• Andrea de Cataldo, Mark; Migliorini, Luca (May 2010). "What is a perverse sheaf?" (PDF). Notices of the AMS. 57 (5).
• Beilinson, A. A. (1987), "On the derived category of perverse sheaves", K-theory, arithmetic and geometry (Moscow, 19841986), Lecture Notes in Math., 1289, Springer, Berlin, pp. 27–41, doi:10.1007/BFb0078365, ISBN 978-3-540-18571-0, MR 0923133
• Brasselet, Jean-Paul (2009), Introduction to intersection homology and perverse sheaves, Instituto Nacional de Matemática Pura e Aplicada (IMPA), MR 2533465