On the topology of a resolution of isolated singularities, II

Let Y be a complex projective variety of dimension n with isolated singularities, π: X → Y a resolution of singularities, G:= π−1 (Sing(Y)) the exceptional locus. From the Decomposition Theorem one knows that the map Hk−1(G) → Hk (Y, Y Sing(Y)) vanishes for k > n. It is also known that, conversely, assuming this vanishing one can prove the Decomposition Theorem for π 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 π, involving only ordinary cohomology.