The concept was introduced in the work of Joseph Bernstein, Alexander Beilinson, and Pierre Deligne (1982) as a consequence of the Riemann-Hilbert correspondence, which establishes a connection between the derived categories regular holonomic D-modules and constructible 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.
The Beilinson–Bernstein–Deligne 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.
For many applications in representation theory, perverse sheaves can be treated as a 'black box', a category with certain formal properties.
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 has real dimension at most 2i, for all i.
If X is a smooth complex algebraic variety and everywhere of dimension d, then is a perverse sheaf for any local system
[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
The geometric Satake equivalence identifies equivariant perverse sheaves on the affine Grassmannian
Conifolds are important objects in string theory: Brian Greene explains the physics of conifolds in Chapter 13 of his book The Elegant Universe —including the fact that the space can tear near the cone, and its topology can change.
These properties, known as the Kähler package (T. Hubsch, 1992), should hold for singular and smooth target spaces.
T. Hubsch and A. Rahman determined the (co)-homology of this ground state variety in all dimensions, found it compatible with Mirror symmetry and String Theory but found an obstruction in the middle dimension (T. Hubsch and A. Rahman, 2005).
This perverse sheaf proved the Hübsch conjecture for isolated conic singularities, satisfied Poincaré duality, and aligned with some of the properties of the Kähler package.
Satisfaction of all of the Kähler package by this Perverse sheaf for higher codimension strata is still an open problem.