On the Deligne–Beilinson cohomology sheaves

Abstract

We are showing that the Deligne-Beilinson cohomology sheaves H q+1 (Z(q) D ) are torsion free by assuming Kato's conjectures hold true for function fields. This result is 'effective' for q = 2; in this case, by dealing with 'arithmetic properties' of the presheaves of mixed Hodge structures defined by singular cohomology, we are able to give a cohomological characterization of the Albanese kernel for surfaces with p g = 0.

Degree 4 unramified cohomology with finite coefficients and torsion codimension 3 cycles

“…Concerning the structure of the sheaves H i (Z), we have the following result, which is a consequence of the Bloch-Kato conjecture recently proved by Rost and Voevodsky (we refer to [5], [8], [2] for more explanations concerning the way the very important result below is deduced from the Bloch-Kato conjecture).…”

“…of an hypersurface X 0 ⊂ P 1 × P 3 of bidegree (3,2), and furthermore the curve C 0 comes from a curve C 0 ⊂ X 0 . To see this, it suffices to recall that…”

Torsion 1-cycles and the coniveau spectral sequence

“…(1) This proof was inspired by the argument of Voisin [21] in the H 4 nr case. On the other hand, if we add the present argument to Voisin's proof, we obtain an exact sequence [11], [3]), the first term is equal to tor (H 5 (X, Z)/N 2 H 5 (X, Z)). This gives a refinement of Voisin's result which was proved under the assumption that H 5 (X, Z)/N 2 H 5 (X, Z) has no torsion.…”

“…Let K p and K M p be the Zariski sheaves on X associated to the Quillen K-theory and the Milnor K-theory respectively. As a consequence of the Bloch-Kato conjecture proved by Rost-Voevodsky [20] and the work of Kerz [15], we have an isomorphism K M p /n ≃ H p (Z/n) for every n > 1 (see [11] p.745 and [3] p.9). Theorem 5.1.…”

Unramified cohomology, integral coniveau filtration and Griffiths group

“…The edge map E p,p 2 → H 2p (X, Z) coincides with the cycle map. When A = Z, the sheaf H q (Z) is torsion-free ( [6], [3]), as a consequence of the Bloch-Kato conjecture proved by Voevodsky and Rost [16]. This implies that ( [6]), for each natural number n, we have a short exact sequence of sheaves 0 → H q (Z)…”