---
tag: 5WHXRZDZFK
kind: concept
name: Perverse sheaf
msc: [32S60, 14F08, 55N33]
sources:
  - url: https://en.wikipedia.org/wiki/Perverse_sheaf
same_as:
  wikidata: Q17086746
  wikipedia: Perverse sheaf
---

## Summary

A perverse sheaf on a complex algebraic variety or analytic space is an object of the [[derived category|5ST7NVJFSM]] of constructible sheaves, see the [[definition|JYW9312EE6]].
It satisfies dimension conditions on its cohomology sheaves and on those of its dual.
Despite the name, a perverse sheaf is a complex of sheaves, not a single sheaf.
Perverse sheaves form an abelian category, the heart of the perverse [[t-structure|2GD9V7SA0E]], introduced by Beilinson, Bernstein, Deligne and Gabber.
They include the intersection cohomology complexes of Goresky and MacPherson, and by the [[Riemann–Hilbert correspondence|KMAEWSVGRS]] they correspond to regular holonomic D-modules.

## Explanations

### L1 human

On a smooth space, the constant sheaf, which assigns to each open set the locally constant functions on it, behaves well with respect to Poincaré duality.
On a singular space it does not.
The fix is to use complexes of sheaves with cohomology in several degrees, with rules about how large the support of each degree may be.
These rules are chosen so that the class of objects is preserved by duality.
On a smooth variety of dimension $d$, the constant sheaf placed in degree $-d$ is perverse, and so is the sheaf of a single point placed in degree $0$.