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

ISBN-10: 3110171805

ISBN-13: 9783110171808

Eighteen papers, many drawing from displays on the September 2001 convention in Genova, disguise a variety of algebraic geometry. specific consciousness is paid to better dimensional kinds, the minimum version software, and surfaces of the overall kind. a listing of Francia's courses is incorporated. participants contain mathematicians from Europe, the U.S., Japan, and Brazil

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2. 3. Towards Hodge mixed motives 30 1. 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].

Hodge 1-motives Let k be a ﬁeld, for simplicity, algebraically closed of characteristic zero. Consider the Q-linear abelian category 1 − Motk of 1-motives over k with rational coefﬁcients (see [11] and [4]). Denote MQ the isogeny class of a 1-motive M = [L → G]. The category 1 − Motk contains (as fully faithful abelian sub-categories) the tensor category of ﬁnite dimensional Q-vector spaces as well as the semi-simple abelian category of isogeny classes of abelian varieties. The Hodge realization (see [11] and [4]) is a fully faithful functor THodge : 1 − Mot C → MHS, MQ → THodge (MQ ) deﬁning an equivalence of categories between 1 − Mot C and the abelian sub-category of mixed Q-Hodge structures of type {(0, 0), (0, −1), (−1, 0), (−1, −1)} such that gr W −1 is polarizable.

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]. , those having non-zero Hodge numbers in the set {(0, 0), (0, −1), (−1, 0), (−1, −1)}. Therefore, these cohomological invariants of algebraic varieties would be algebraically deﬁned as 1-motives over arbitrary base ﬁelds or schemes. Note that a general theory of mixed motives can be regarded as an algebraic framework in order to deal with all mixed Hodge structures H ∗ (X, Z(·)).

### Algebraic Geometry: A Volume in Memory of Paolo Francia by Paolo Francia, Fabrizio Catanese, C. Ciliberto, A. Lanteri, C. Pedrini, Mauro Beltrametti

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