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, hide a variety of algebraic geometry. specific cognizance is paid to raised dimensional kinds, the minimum version software, and surfaces of the overall sort. a listing of Francia's courses is integrated. participants contain mathematicians from Europe, the us, Japan, and Brazil

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In fact, the usual Hodge conjecture can be reformulated by saying that the Hodge realization of the algebraically deﬁned Q-vector space of codimension p algebraic cycles modulo numerical (or homological) equivalence is the 1-motivic part of H 2p (X, Q(p)). Moreover, the 1-motivic part of H 2p+1 (X, Q(p+1)) would be the Hodge realization of the isogeny class of the universal regular quotient. 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.

Moreover, p,p ep F p ∩ HZe = ker(HZ → J p (H )). p Now, if gr W 2p−1 H is (polarizable) of level 1 then the torus J (H ) is an abelian variety e and H is the Hodge realization of the 1-motive over C deﬁned by the extension class map ep above. , if 2p−1 def Ha = (H p−1,p + H p,p−1 )Z is the polarizable sub-structure of gr W 2p−1 H of those elements which are purely of the above type, then H is deﬁned by the following pull-back extension 2p−1 0 → W2p−2 H → H → Ha → 0, along the canonical projection W2p−1 H → →gr W 2p−1 H.

11], [16], [15], [22] and [14]). Let X be such a proper smooth simplicial scheme over the base ﬁeld k. By · · · · j functoriality, the ﬁltration Fm CHp on each component Xi of X yields a complex ∗ δi−1 δi∗ (Fm CHp )• : · · · → Fm CHp (Xi−1 ) → Fm CHp (Xi ) → Fm CHp (Xi+1 ) → · · · j j j j where δi∗ is the alternating sum of the pullback along the face maps ∂ik : Xi+1 → Xi for 0 ≤ k ≤ i + 1. The complex of Chow groups (CHp )• , induced from the simplicial structure as above, is ﬁltered by sub-complexes: 0 ⊆ (Fm CHp )• ⊆ · · · ⊆ (Fm1 CHp )• ⊆ (Fm0 CHp )• = (CHp )• .

### 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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