# On hypercohomology of perverse sheaves

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 ?

• 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 Jul 9, 2021 at 7:28
• 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 $IC_X$. Jul 9, 2021 at 12:43