Download Algebraic Geometry: A Volume in Memory of Paolo Francia ( De by Mauro C. Beltrametti, Fabrizio Catanese, Ciro Ciliberto PDF

By Mauro C. Beltrametti, Fabrizio Catanese, Ciro Ciliberto

The papers during this quantity conceal a large spectrum of algebraic geometry, from reasons conception to numerical algebraic geometry and are generally curious about greater dimensional types and minimum version application and surfaces of basic sort. part of the articles grew out of a convention in reminiscence of Paolo Francia held in Genova in September 2001 with nearly 70 members.

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Extra resources for Algebraic Geometry: A Volume in Memory of Paolo Francia ( De Gruyter Proceedings in Mathematics )

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2. Let AP(H) ~Ext (Z(- p). H;p- ) denote the abelian part of the compact complex torus JP(H). , the subgroup of Hf'P mapped to AP(H) under the extension class map eP. Define 38 L. Barbieri-Viale the mixed Hodge structure corresponding to the 1-motive defined from (7) above. Call this 1-motive the Hodge 1-motive of H. We remark that the mixed Hodge structure H" clearly corresponds to the following extension 0 ~ H;p-l ~ Hh ~ HP(H) ~ 0, obtained by pulling back HP(H) = FP n (H" 1H')z along the induced projection He~H"/H'in(1).

The main task of this paper is to define Hodge 1-motives of singular varieties and to state a corresponding cohomological Grothendieck-Hodge conjecture by dealing with their Hodge realizations. 1. A short survey of the subject The classical Hodge conjecture along with a tantalizing overview can be found in [ 13]. Recall that Grothendieck corrected the general Hodge conjecture in [19]. The book of Lewis [27] is a very good compendium of methods and results. Recall that Jannsen [23] formulated an homological version of the Hodge conjecture for singular varieties.

J{~ p ) ::::? , J{~ • ) is in the category MHS 00 • Let w : Xan ~ Xzar be the continuous map of sites induced by the identity mapping. (Z)) ::::? Hq+r (Xan. Z) of abelian groups. (Q) ~ Je~, these sheaves can be regarded as Qt-mixed sheaves and their Zariski cohomology carry oo-mixed Hodge structures as above. For X. a simplicial scheme we thus have w. : (X. )an ~ (X. , Rr (w. (Z)) ::::? , Z) where Rr (w. (Z) ~ Je~ •. 3 (Local-to-global). There are spectral sequences Li'r = Hq(X, Jl~)::::? Hq+r(Xan.

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