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, conceal a variety of algebraic geometry. specific cognizance is paid to better dimensional forms, the minimum version software, and surfaces of the overall style. a listing of Francia's guides is incorporated. individuals comprise mathematicians from Europe, the us, Japan, and Brazil

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P−1 p−1,p+1 p+1 is a quotient of H p−1 (X, HX In this case gr N H 2p (X)(p) = C∞ For example: gr 1N H 4 (X)(2) = H 1 (X, HX3 (2)). (p)). 2. Is F 2 ∩ H 1 (X, HX3 ) = 0? Case i = p and j = 2p + 1. Let X be a smooth C-scheme. , there is an edge map s p+1 −1 p+1 : H p (X, HX (p + 1)) → H 2p+1 (X)(p + 1) On algebraic 1-motives related to Hodge cycles 51 with image N p H 2p+1 (X)(p + 1). In this case the Grothendieck–Hodge conjecture characterize N p H 2p+1 (X) as the largest sub-Hodge structure of type {(p, p + 1), (p + 1, p)}.

3. Local Hodge theory See [11] for notations, deﬁnitions and properties of mixed Hodge structures. 1. , for A a noetherian subring of R such that A ⊗ Q is a ﬁeld, an object H of MHS is deﬁned as a triple H = (HA , W, F ) where HA is a ﬁnitely generated A-module, W is a ﬁnite increasing ﬁltration on HA ⊗ Q and F is a ﬁnite decreasing ﬁltration on HA ⊗ C such that W, F and F is a system of opposed ﬁltrations. 1. An ∞-mixed Hodge structure H is a triple (HA , W, F ) where HA is any A-module, W is a ﬁnite increasing ﬁltration on HA ⊗Q and F is a ﬁnite decreasing ﬁltration on HA ⊗ C such that W, F and F is a system of opposed ﬁltrations.

17]). If k = C it is easy to see that such extension G(C) exists trascendentally. 2. 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.

### 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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