On a resolution of singularities with two strata

TL;DR

This paper provides an explicit and concise proof of the Decomposition Theorem for certain resolutions of singularities, enabling computation of intersection cohomology in specific geometric contexts with two strata.

Contribution

It introduces a simplified proof of the Decomposition Theorem applicable to resolutions with two strata, even when the resolution is non-small.

Findings

Applicable to special Schubert varieties with two strata

Effective for certain hypersurfaces in P^5 with one-dimensional singular locus

Provides a method to compute intersection cohomology from cohomology of related spaces

Abstract

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