Several complex variables and analytic spaces Definition

Perverse sheaf

Builds on Algebraic variety Sheaf Derived category Whitney stratification

Definition

Let $X$ be a complex algebraic variety, with its classical topology. A sheaf 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 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 $i$ :

(support) $\dim_\C \operatorname{supp} \mathcal{H}^{-i}(K) \le i$ ;
(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, 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.

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Algebraic variety concept
Derived category concept
Sheaf concept
Whitney stratification concept
Perverse sheaf definition

Tag	`JYW9312EE6`
URI	`m2://JYW9312EE6`
Kind	definition
Writing	W0 stub
MSC	32S60, 14F08, 55N33
Defines	Perverse sheaf
Uses	Algebraic variety, Sheaf, Derived category, Whitney stratification

Perverse sheaf

Definition

Remarks

Reading path

Further reading

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