On the topology of a resolution of isolated singularities

Vincenzo Di Gennaro and Davide Franco

Journal of Singularities
volume 16 (2017), 195-211

Received: 7 April 2017. Received in revised form: 14 October 2017

DOI: 10.5427/jsing.2017.16j

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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.


Projective variety, Isolated singularities, Resolution of singularities, Derived category, Intersection cohomology, Decomposition Theorem, Bivariant Theory, Gysin morphism, Cohomology manifold

2010 Mathematical Subject Classification:

Primary 14B05; Secondary 14E15, 14F05, 14F43, 14F45, 32S20, 32S60, 58K15

Author(s) information:

Vincenzo Di Gennaro Davide Franco
Dipartimento di Matematica Dipartimento di Matematica e Applicazioni
Università di Roma, "Tor Vergata" "R. Caccioppoli"
Via della Ricerca Scientifica Università di Napoli, "Federico II"
00133 Roma, Italy P.le Tecchio 80
80125 Napoli, Italy
email: digennar@axp.mat.uniroma2.it email: davide.franco@unina.it