A Lefschetz (1,1) theorem for normal projective complex varieties

“…Proof. To see the first assertion, we may apply Ext * (−, G m ) to (4.6) and then apply [10,Lemma 2.8] to the corresponding diagram. Here is a concrete description of this argument: via the map of Lemma 4.1.5, u ′ 2 g goes to the following pushout…”

Section: We May Viewmentioning

confidence: 99%

On the Theory of 1-Motives

Barbieri-Viale¹

2007

Algebraic Cycles and Motives

104

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We consider the category of Deligne 1-motives over a perfect field k of exponential characteristic p and its derived category for a suitable exact structure after inverting p. As a first result, we provide a fully faithful embedding into anétale version of Voevodsky's triangulated category of geometric motives. Our second main result is that this full embedding "almost" has a left adjoint, that we call LAlb. Applied to the motive of a variety we thus get a bounded complex of 1-motives, that we compute fully for smooth varieties and partly for singular varieties. As an application we give motivic proofs of Roǐtman type theorems (in characteristic 0).

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“…This has been proved recently by Biswas-Srinivas [8,Theorem 1.1]. They also give an example of a non-normal variety for which this Lefschetz theorem fails.…”

Section: Introductionmentioning

confidence: 63%

“…Lemma 2. Let X • be a smooth proper simplicial scheme over C. The Leray spectral sequence (8) induces an exact sequence…”

Section: Lefschetz 1-motivementioning

confidence: 99%