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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Abstract:

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.

Keywords:

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