On a multiplicative version of Bloch’s conjecture

Abstract

A theorem of Esnault, Srinivas and Viehweg asserts that if the Chow group of 0-cycles of a smooth complete complex variety decomposes, then the top-degree coherent cohomology group decomposes similarly. In this note, we prove that (a weak version of) the converse holds for varieties of dimension at most 5 that have finite-dimensional motive and satisfy the Lefschetz standard conjecture. The proof is based on Vial's construction of a refined Chow-Künneth decomposition for these varieties.

A remark on the Chow ring of some hyperkähler fourfolds

“…Moreover, S has finite-dimensional motive [3,Theorem 4.13]. The main result of [17] then implies that i S : A 1 (S) ⊗ A 1 (S) → A 2 (S)…”

“…is surjective [8]. The conjectural converse statement is studied in [17]. To complete the picture, we propose the following conjecture: Conjecture 1.1.…”

Zero-cycles on self-products of surfaces: some new examples verifying Voisin's conjecture

“…Proof. Surjectivity of C, combined with finite-dimensionality of the motive of S, ensures that intersection product induces a surjection [39]. The assumption that C is an isomorphism implies that p g (S) = q(S)…”

On complete intersections in varieties with finite-dimensional motive

“…Using Proposition 4.11, the main result of[27] can be extended to arbitrary dimension, provided one replaces Vial's filtration N • in the statement of[27, Theorem 3] by the filtration N • .…”