On the topology of a resolution of isolated singularities, II

Vincenzo Di Gennaro and Davide Franco

Journal of Singularities
volume 20 (2020), 95-102

Received: 9 January 2020. Received in revised form: 25 February 2020

DOI: 10.5427/jsing.2020.20e

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Abstract:

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 H^{k-1}(G) -> H^k(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.

2010 Mathematical Subject Classification:

Primary 14B05; Secondary 14C30, 14E15, 14F05, 14F43, 14F45, 32S20, 32S35, 32S60, 58A14, 58K15

Key words and phrases:

Projective variety, Isolated singularities, Resolution of singularities, Derived category, Intersection cohomology, Decomposition Theorem, Hodge theory

Author(s) information:

Vincenzo Di Gennaro	Davide Franco
Università di Roma "Tor Vergata"	Università di Napoli "Federico II"
Dipartimento di Matematica	Dipartimento di Matematica e Applicazioni
Via della Ricerca Scientifica	"R. Caccioppoli"
00133 Roma, Italy	Via Cintia
	80126 Napoli, Italy
email: digennar@axp.mat.uniroma2.it	email: davide.franco@unina.it