On the topology of a resolution of isolated singularities, II

Let $Y$ be a complex projective variety of dimension $n$ with isolated singularities, $\pi :X\to Y$ a resolution of singularities, $G:={\pi }^{-1}(\mathrm{S}\mathrm{i}\mathrm{n}\mathrm{g}(Y))$ the exceptional locus. From the Decomposition Theorem one knows that the map $H^{k-1}(G)\to H^k(Y,Y\backslash {\rm{Sing}}(Y))$vanishesfor$k>n$. It is also known that, conversely, assuming this vanishing one can prove the Decomposition Theorem for $\pi$ in few pages. The purpose of the present paper is to exhibit a direct proof of the vanishing. As a consequence, it follows a complete and short proof of the Decomposition Theorem for $\pi$, involving only ordinary cohomology.