On a resolution of singularities with two strata

Let $X$ be a complex, irreducible, quasi-projective variety, and $pi:widetilde Xto X$ a resolution of singularities of $X$. Assume that the singular locus ${text{Sing}}(X)$ of $X$ is smooth, that the induced map $pi^{-1}({text{Sing}}(X))to {text{Sing}}(X)$ is a smooth fibration admitting a cohomology extension of the fiber, and that $pi^{-1}({text{Sing}}(X))$ has a negative normal bundle in $widetilde X$. We present a very short and explicit proof of the Decomposition Theorem for $pi$, providing a way to compute the intersection cohomology of $X$ by means of the cohomology of $widetilde X$ and of $pi^{-1}({text{Sing}}(X))$. Our result applies to special Schubert varieties with two strata, even if $pi$ is non-small. And to certain hypersurfaces of $mathbb P^5$ with one-dimensional singular locus.