By Paolo Francia, Fabrizio Catanese, C. Ciliberto, A. Lanteri, C. Pedrini, Mauro Beltrametti

Eighteen papers, many drawing from shows on the September 2001 convention in Genova, conceal a variety of algebraic geometry. specific realization is paid to raised dimensional types, the minimum version application, and surfaces of the final sort. a listing of Francia's courses is integrated. participants contain mathematicians from Europe, the USA, Japan, and Brazil

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For X a simplicial scheme we thus have ω : (X )an → (X )Zar as above and a Leray spectral sequence · · · q,r · · · · L2 = Hq (X , R r (ω )∗ (Z)) ⇒ Hq+r (X , Z) where ·)∗(Z) ∼= R r (ω HXr ·. 3 (Local-to-global). There are spectral sequences q,r L2 = H q (X, HXr ) ⇒ H q+r (Xan , Q) 46 L. Barbieri-Viale and · q,r · L2 = Hq (X , HXr ) ⇒ Hq+r (X , Q) · in the category of ∞-mixed Hodge structures. 4]. 4. Edge maps Recall that the classical cycle class maps can be obtained via edge homomorphisms in the coniveau spectral sequence.

1. 2. Extensions 32 2. 1. 2. 3. Hodge conjecture for singular varieties 38 3. 1. 2. 3. 4. Local-to-global properties 45 4. 1. 2. Coniveau ﬁltration 49 Algebraic Geometry. A Volume in Memory of Paolo Francia M. C. Beltrametti, F. Catanese, C. Ciliberto, A. Lanteri, C. ) © Walter de Gruyter 2002 26 L. 3. 4. K-cohomology and motivic cohomology 5. 1. 2. Srinivas’ example 51 53 55 55 56 0. Introduction Let X be an algebraic C-scheme. The singular cohomology groups H ∗ (X, Z(·)) carry a mixed Hodge structure, see [11, III].

For such a nonsingular variety X we then have NS 1 (X) = F 1 ∩ H 2 (X, Z) = HZ1,1 = H 1 (X, HX1 ) = N 1 H 2 (X, Z). For i = −1 and X possibly singular, the conjecture corresponds to the fact (proved 1 in [4]) that the abelian variety corresponding to gr W 1 H is ker 0 (Pic 0 (X0 ) → Pic 0 (X1 )). 4 is quite easily veriﬁed by checking the claimed compatibility of the extension class map. Such a statement then corresponds to a Lefschetz (1, 1)-theorem for complete varieties with arbitrary singularities.