By Walker R. J.

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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.

Exotic (1, 1)-classes Consider X singular. 4 for p = 1. Moreover we show that there are edge maps generalizing the cycle class maps constructed in the previous section. For X a proper irreducible C-scheme, consider the mixed Hodge structure on H 2+i (X, Z) modulo torsion. The extension (5) is the following 0 → H 1+i ((H 1 )• ) → W2 H 2+i (X)/W0 → H i ((H 2 )• ) → 0. (15) Since the complex (H 1 )• is made of level 1 mixed Hodge structures then H 2+i (X)h = H 2+i (X)e in our notation. If X is nonsingular then H 2+i (X) is pure and there are only two cases where this extension is non-trivial.

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.

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Algebraic Curves by Walker R. J.


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