On the topology of a resolution of isolated singularities

TL;DR

This paper explores the topology of resolutions of isolated singularities in complex projective varieties, providing new proofs and conditions for the Decomposition Theorem's validity, linking it to bivariant theory.

Contribution

It offers a simplified proof of the Decomposition Theorem under certain vanishing conditions and establishes equivalences involving Gysin morphisms and cohomology maps.

Findings

Vanishing of specific cohomology maps implies the Decomposition Theorem.

The Decomposition Theorem holds for normal surfaces and certain blow-ups.

A Gysin morphism condition is equivalent to cohomology map vanishing and injectivity of pull-back.

Abstract

Let $Y$ be a complex projective variety of dimension $n$ with isolated singularities, $\pi:X\to Y$ a resolution of singularities, $G:=\pi^{-1}{\rm{Sing}}(Y)$ the exceptional locus. From Decomposition Theorem one knows that the map $H^{k-1}(G)\to H^k(Y,Y\backslash {\rm{Sing}}(Y))$ vanishes for $k>n$. Assuming this vanishing, we give a short proof of Decomposition Theorem for $\pi$. A consequence is a short proof of the Decomposition Theorem for $\pi$ in all cases where one can prove the vanishing directly. This happens when either $Y$ is a normal surface, or when $\pi$ is the blowing-up of $Y$ along ${\rm{Sing}}(Y)$ with smooth and connected fibres, or when $\pi$ admits a natural Gysin morphism. We prove that this last condition is equivalent to say that the map $H^{k-1}(G)\to H^k(Y,Y\backslash {\rm{Sing}}(Y))$ vanishes for any $k$, and that the pull-back $\pi^*_k:H^k(Y)\to H^k(X)$ is injective. This provides a relationship between Decomposition Theorem and Bivariant Theory.