The generalized Hodge and Bloch conjectures are equivalent for general complete intersections (Q2854292)

Consider in \( \mathbb{P}^n\) a smooth variety \(X\) which is the complete intersection of hypersurfaces of degrees \(d_1\leq\ldots\leq d_r\), the following two conjectures descend from the generalized Bloch and Hodge conjectures respectively:NEWLINENEWLINE{ B-Conjecture. } The cycle map \(\mathrm{cl}: \mathrm{CH}_i(X)_\mathbb{Q}\rightarrow H^{2n-2r-2i}(X,\mathbb{Q})\) is injective for \(i\leq c-1\), if \(n\geq\sum_{i}d_i+(c-1)d_r\).NEWLINENEWLINE{ H-Conjecture. } The primitive cohomology \(H^{n-r}(X,\mathbb{Q})_{\text{prim}}\) vanishes on the complement of a closed algebraic subset \(Y\subset X\) of codimension \(c\), if \( n\geq\sum_{i}d_i+(c-1)d_r\).NEWLINENEWLINEVarious authors proved refinements of the diagonal decomposition theorem due to \textit{S. Bloch} and \textit{V. Srinivas} [Am. J. Math. 105, 1235--1253 (1983; Zbl 0525.14003)], it turns out that such a decomposition is a consequence of the first conjecture and that it implies the H-conjecture.NEWLINENEWLINEThe main object of the present paper is to establish the theorem that the converse statement holds, namely that \(H\) implies \(B\) for a certain class of varieties which are a generalization of the complete intersections above. The proof is in part conditional, since it needs the assumption that the S-conjecture below holds for cycles of codimension \(n-r\).NEWLINENEWLINEThe strategy of attack is based on the fact that in the case under consideration one has a cohomological decomposition for the diagonal of the general member of the family, that then this decomposition lifts to a cohomological decomposition on an open set on the universal family. Now on the variety at hand and in the range of interest Voisin shows that rational equivalence injects in cohomological equivalence, in this manner one finds a decomposition for the Chow class of the universal diagonal. Restriction provides then a convenient decomposition of the diagonal class in rational equivalence for every member in the family, from this fact the conclusion comes readily.NEWLINENEWLINEThe auxiliary conjecture is: {S-Conjecture.} Let \(X\) be a smooth complex algebraic variety, and let \(Y\subset X\) be a closed algebraic subset. Let \(Z\subset X\) be a codimension \(k\) algebraic cycle, and assume that the cohomology class \([Z]\in H^{2k}(X,\mathbb{Q})\) vanishes in \(H^{2k}(X\setminus Y,\mathbb{Q})\). Then there exists a codimension \(k\) cycle \(Z'\) on \(X\) with \(\mathbb{Q}\)-coefficients, which is supported on \(Y\) and such that \([Z']=[Z]\) in \(H^{2k}(X,\mathbb{Q})\). Voisin remarks that this conjecture is satisfied by cycles of codimension \(2\) and she proves that the Lefschetz conjecture for all smooth projective varieties \(X\) is equivalent to the conjunction of the Künneth conjecture and of the S-conjecture for all such \(X\).NEWLINENEWLINEThe same method described above yields a wealth of further results, pertaining for instance to the Chow groups of complete intersections with group action or to the \(0\)-cycles on self-products of Calabi-Yau hypersurfaces in projective space.