I was watching this Youtube video on a lecture given by J.P. Brasselet on perverse sheaves.

At around 54:37 he mentioned the following result:

Let $X=\cup X_\alpha$ be a Whitney stratified space of dimension $n$. Let $p$ be a perversity function, and let $q=n-2-p_\alpha$ be the complementary perversity. Define $p^{-1}(j)=\min \{\alpha |j\leq p_\alpha\}$. A constructible sheaf complex $S^\bullet$ is perverse if $$ \dim \{x\in X| H^j(i_x^* S^\bullet)\neq 0\}\leq n- q^{-1} (n-j)\\ \dim \{x\in X| H^j(i_x^! S^\bullet)\neq 0\}\leq n- p^{-1} (n-j) $$ holds. Then for any perverse sheaf $S^\bullet$, one has $$ \mathbb{H}^j(X, S^\bullet)=IH^{p, cl} _{n-i}(X)\\ \mathbb{H}^j_c(X, S^\bullet)=IH^p_{n-i}(X)\\ $$

This basically says that, by taking hypercohomology, all perverse sheaves give intersection cohomology. Is this correct ? If true, are there any reference for this result ?

  • 2
    $\begingroup$ I'd imagine this is a standard fact for which you can find many resources, but for a detailed proof you can check out proposition 4.9, chapter V section 4, from "Intersection cohomology" by A. Borel $\endgroup$
    – user127776
    Jul 9, 2021 at 7:28
  • 3
    $\begingroup$ The correct statement is not that for all perverse sheaves $S$, one has $H(X,S)=IH(X)$, it is that there exists a perverse sheaf with this property: the intersection complex $IC_X$. $\endgroup$ Jul 9, 2021 at 12:43


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