Intersection complex ($I{C}_{\bar{p}}$)

The degree-zero normalized Deligne intersection complex extends the constant sheaf from the regular stratum by successive derived direct images and cohomological truncations at degree $\bar{p}(k)$ when codimension-$k$ strata are added. Its hypercohomology defines intersection cohomology. On a cone with vertex of codimension $k$, its local cohomology is the link intersection cohomology in degrees at most $\bar{p}(k)$ and zero above.

For an isolated singular point of an $N$-dimensional oriented pseudomanifold, the unshifted middle intersection complex restricts to $\mathbb{Q}$ on the regular stratum and has no negative cohomology. Its support condition is $H^j(i_x^{\ast }I{C}_{\bar{p}})=0$ for $j>\bar{p}(N)$; its cosupport condition is $H^j(i_x^{!}I{C}_{\bar{p}})=0$ for $j\le \bar{p}(N)+1$. These conditions, together with constructibility and normalization, characterize it. On a Witt space they are exchanged by $IC\cong D(IC)[-N]$.