On a resolution of singularities with two strata

Let $X$ be a complex, irreducible, quasi-projective variety, and $\pi :\tilde{X}\to X$ 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 $\tilde{X}$. 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 $\tilde{X}$ 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.