Intersection-cohomology complex

Definition

Let $X$ be an irreducible complex algebraic variety of dimension $d$, let $j:U\hookrightarrow X$ be a smooth dense open subset, and let $\mathcal L$ be a local system on $U$. The intersection-cohomology complex is the perverse middle extension

It is characterized as the extension having no nonzero perverse subobject or quotient supported on $X\setminus U$. For the constant local system, its hypercohomology is the intersection cohomology of $X$.