On the topology of a resolution of isolated singularities

Vincenzo Di Gennaro ; Davide Franco

Journal of Singularities, Tome 16 (2017), pp. 195-211

Résumé

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

Détail

DOI : 10.5427/jsing.2017.16j

Classification : 14B05, 14E15, 14F05, 14F43, 14F45, 32S20, 32S60, 58K15