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

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

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Let X be a proper smooth C-scheme. Note that we have N 1 H j (X) = ker(H j (X) → H 0 (X, H j )) = { Zariski locally trivial classes in H j (X)}. Thus H j (X, Q) ∩ F 1 H j (X) = gr 0N H j (X) ∩ F 1 ⊆ H 0 (X, H j ) ∩ F 1 . N 1 H j (X) We remark that H j /F 1 is the constant sheaf associated to H j (X, OX ). Thus F 1 ∩ H 0 (X, H j ) ∼ = ker(H 0 (X, H j ) → H j (X, OX )). If j = 1 then H 1 (X) = H 0 (X, H 1 ) from (13) and (8) is trivially an equality. If j = 2 then F 1 ∩ H 0 (X, H 2 ) = 0 from the exponential sequence.

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 )• . p Deﬁne Fm∗ (CHp )• = (Fm∗ CHp )• . The extension (1) given by the ﬁltration Fm∗ CHp (Xi ) on each component Xi of the simplicial scheme X , for a ﬁxed p ≥ 0, yields the following short exact sequence of complexes def · 0 → gr 1Fm (CHp )• → (CHp )• /Fm2 → gr 0Fm (CHp )• → 0 (3) p Note that gr 1Fm (CHp )i contains the group of k-points of the abelian variety AXi /k and, moreover gr 0Fm (CHp )iQ is the ﬁnite dimensional vector space of codimension p cycles on Xi modulo homological (or numerical) equivalence.

The cokernels) of ϕQ and ϕC with induced ﬁltrations. Any morphism is strictly compatible with the ﬁltrations. The functors gr W and gr F are exacts. Note that if X = {∞} is the singleton then MH S ∞ is equal to MHS∞ . Examples of Q-mixed sheaves are clearly given by constant sheaves associated to Q-mixed Hodge structures, yielding a canonical fully faithful functor MHS∞ → MH S X . Stalks of a Q-mixed sheaf H are in MHS∞ , the ﬁltrations being induced stalkwise. In fact, the condition on the ﬁltrations given with any Q-mixed sheaf is local, at any point of X.