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  "body": "\n## Definition\n\nLet $X$ be a complex [[algebraic variety|HV6YBS0YDH]], with its classical topology.\nA [[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 $X$.\nThe pieces must be locally closed subvarieties.\nLet $D^b(X)$ be the bounded [[derived category|5ST7NVJFSM]] of sheaves of $\\C$-vector spaces on $X$.\nLet $D^b_c(X)$ be its full subcategory of objects $K$ whose cohomology sheaves $\\mathcal{H}^i(K)$ are constructible.\nLet $\\mathbb{D} \\colon D^b_c(X) \\to D^b_c(X)$ be Verdier duality.\nAn object $K$ of $D^b_c(X)$ is a *perverse sheaf* if, for every integer $i$:\n\n1. (support) $\\dim_\\C \\operatorname{supp} \\mathcal{H}^{-i}(K) \\le i$;\n2. (cosupport) $\\dim_\\C \\operatorname{supp} \\mathcal{H}^{-i}(\\mathbb{D} K) \\le i$.\n\nHere $\\operatorname{supp}$ is the closure of the set of points where the stalk is nonzero, and the empty set has dimension $-\\infty$.\n\n## Remarks\n\nThe same definition works for complex analytic spaces, with partitions into analytic strata, for example [[Whitney stratifications|3F84AMVJNV]], and for other coefficient fields.\n\nExample: let $X$ be smooth and connected of dimension $d$, and $K = \\C_X[d]$, the constant sheaf placed in degree $-d$.\nIts only cohomology sheaf is $\\mathcal{H}^{-d}(K) = \\C_X$, with support of dimension $d$, so clause 1 holds.\nVerdier 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.\n\nNon-example: for $d \\ge 1$, the constant sheaf $\\C_X$ in degree $0$ is not perverse.\nIts cohomology sheaf $\\mathcal{H}^0(\\C_X) = \\C_X$ has support of dimension $d \u003e 0$, so clause 1 fails for $i = 0$.\n\nThe 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.\nPerverse sheaves form an abelian category in which every object has finite length.\n",
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