The generalized Hodge and Bloch conjectures are equivalent for general complete intersections. II (Q2788599)

The decomposition of the diagonal is a most effective machine for the study of algebraic cycles, one of the few which are so far available. In recent years it has been at the core of Claire Vosin's research, with great success, cf. [\textit{C. Voisin}, Chow rings, decomposition of the diagonal, and the topology of families. Annals of Mathematics Studies 187. Princeton, NJ: Princeton University Press (2014; Zbl 1288.14001)]. The present paper is one more instance of the power of this method. The theme is the search for a proof of the equivalence of the generalized Hodge conjecture ( geometric coniveau = Hodge coniveau ) to the generalized Bloch conjecture (the Chow groups \(\mathrm{CH}_i\) are trivial for \(i\) smaller than the Hodge coniveau of the transcendental cohomology). The article is part \(2\), the first part being a preceding work by the same name, [Ann. Sci. Éc. Norm. Supér. (4) 46, No. 3, 449--475 (2013; Zbl 1282.14015)]. The abstract explains that the main new theorem is an unconditional (but slightly weakened) version of the principal result of [loc. cit.], which was, starting from dimension \(4\), conditional to the Lefschetz standard conjecture. One deals with a variety \(X\) with trivial Chow groups, (i.e. the cycle class map to cohomology is injective on \(\mathrm{CH}(X)_\mathbb{Q}\)), here it is proved that if the cohomology of a general hypersurface \(Y\) in \(X\) is ``parameterized by cycles of dimension \(c\)'', then the Chow groups \(\mathrm{CH}_{i}(Y)_\mathbb{Q}\) are trivial for \(i\leq c-1\). Voisin's definition is the following: let \(Y\) be smooth projective of dimension \(m\), the primitive part of the degree \(m\) cohomology of \(Y\) is parameterized by algebraic cycles of dimension \(c\) if: a) There exist a smooth projective variety \(T\) of dimension \(m-2c\) and a correspondence \(P\in \mathrm{CH}^{m-c}(T\times Y)_\mathbb{Q}\), such that \(P^*:H^{m}(Y,\mathbb{Q})\rightarrow H^{m-2c}(T,\mathbb{Q}_{\mathrm{prim}}) \) is injective and furthermore b) \(P^*\) is compatible up to a coefficient with the intersection forms: for some rational number \(N\not=0\), \(<P^*\alpha,P^*\beta>_T=N<\alpha,\beta>_Y\) for any \(\alpha,\,\beta\in H^m(Y,\mathbb{Q})_{\mathrm{prim}}\). It is essentially correct to understand parametrized by algebraic cycles in the naive way, indeed Voisin explains that for the general member point b) follows in practice from a).NEWLINENEWLINEIn the second section the author shows that her result can be used very concretely, by demonstrating the remarkable fact that the smooth hyperplane sections \(Y\) of the Grassmannian \(G(3,10)\) satisfy \(\mathrm{CH}_i(Y)_{\mathbb{Q},\hom}=0\) for \(i<9\). The final part of the paper is equally interesting. It contains several comments on the need to relax the very ampleness assumption. Variants of the main theorem above are presented, they apply with weaker hypotheses, one important case being the situation of hypersurfaces which are invariant under the action of a finite group. The paper concludes with the description of the challenge which is represented by an example descending from the geometry of sextic hypersurfaces in \(\mathbb P^5\) with an involution.