---
tag: JYW9312EE6
kind: definition
name: Perverse sheaf
msc: [32S60, 14F08, 55N33]
defines: 5WHXRZDZFK
uses: [HV6YBS0YDH, TB6TM6R4MD, 5ST7NVJFSM, 3F84AMVJNV]
sources:
  - url: https://en.wikipedia.org/wiki/Perverse_sheaf
---

## Definition

Let $X$ be a complex [[algebraic variety|HV6YBS0YDH]], with its classical topology.
A [[sheaf|TB6TM6R4MD]] of finite-dimensional $\C$-vector spaces on $X$ is *constructible* if it is locally constant on each piece of some finite partition of $X$.
The pieces must be locally closed subvarieties.
Let $D^b(X)$ be the bounded [[derived category|5ST7NVJFSM]] of sheaves of $\C$-vector spaces on $X$.
Let $D^b_c(X)$ be its full subcategory of objects $K$ whose cohomology sheaves $\mathcal{H}^i(K)$ are constructible.
Let $\mathbb{D} \colon D^b_c(X) \to D^b_c(X)$ be Verdier duality.
An object $K$ of $D^b_c(X)$ is a *perverse sheaf* if, for every integer $i$:

1. (support) $\dim_\C \operatorname{supp} \mathcal{H}^{-i}(K) \le i$;
2. (cosupport) $\dim_\C \operatorname{supp} \mathcal{H}^{-i}(\mathbb{D} K) \le i$.

Here $\operatorname{supp}$ is the closure of the set of points where the stalk is nonzero, and the empty set has dimension $-\infty$.

## Remarks

The same definition works for complex analytic spaces, with partitions into analytic strata, for example [[Whitney stratifications|3F84AMVJNV]], and for other coefficient fields.

Example: let $X$ be smooth and connected of dimension $d$, and $K = \C_X[d]$, the constant sheaf placed in degree $-d$.
Its only cohomology sheaf is $\mathcal{H}^{-d}(K) = \C_X$, with support of dimension $d$, so clause 1 holds.
Verdier duality gives $\mathbb{D}(\C_X) = \C_X[2d]$ on a smooth $X$, so $\mathbb{D} K = \C_X[d]$ and clause 2 holds too.

Non-example: for $d \ge 1$, the constant sheaf $\C_X$ in degree $0$ is not perverse.
Its cohomology sheaf $\mathcal{H}^0(\C_X) = \C_X$ has support of dimension $d > 0$, so clause 1 fails for $i = 0$.

The skyscraper sheaf $\C_p$ at a point, in degree $0$, is perverse, and so is the intersection cohomology complex of any closed irreducible subvariety, suitably shifted.
Perverse sheaves form an abelian category in which every object has finite length.