et 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.

DI GENNARO, V., Franco, D. (2017). On the topology of a resolution of isolated singularities. JOURNAL OF SINGULARITIES, 16, 195-211 [10.5427/jsing.2017.16j].

### On the topology of a resolution of isolated singularities

#### Abstract

et 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.
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Settore MAT/03 - Geometria
English
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DI GENNARO, V., Franco, D. (2017). On the topology of a resolution of isolated singularities. JOURNAL OF SINGULARITIES, 16, 195-211 [10.5427/jsing.2017.16j].
DI GENNARO, V; Franco, D
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/2108/198039
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