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.