On a Resolution of Singularities with Two Strata

Abstract

Let X be a complex, irreducible, quasi-projective variety, and $\pi :{\widetilde{X}}\rightarrow X$ a resolution of singularities of X. Assume that the singular locus ${\text {Sing}}(X)$ of X is smooth, that the induced map $\pi ^{-1}({\text {Sing}}(X))\rightarrow {\text {Sing}}(X)$ 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.