Motivic homology of semiabelian varieties (Q462631)

This paper generalises various related results about the rational Chow groups of abelian varieties to semi-abelian varieties. The work is over a perfect field \(k\). For this review (as in the article), all motives have rational coefficients.NEWLINENEWLINEAn abelian variety is a smooth projective group variety. A semi-abelian variety is a smooth variety which is an extension of a projective variety by a torus. If \(A\) is an abelian variety then it is classical that its (pure) motive decomposes into symmetric powers of its \(H^1\): \(M(A) = \bigoplus_i \mathrm{Sym}^i(M_1(A))\). It follows that its rational Chow groups decompose as \(\mathrm{CH}^q(A, \mathbb{Q}) = \bigoplus_i \mathrm{CH}^q(A, \mathbb{Q})^{(i)}\), where \(\mathrm{CH}^q(A, \mathbb{Q})^{(i)}\) is the \(n^i\)-eigenspace of the linear operator on \(\mathrm{CH}^q(A, \mathbb{Q})\) corresponding to multiplication by \(n\) in \(A\) (for all \(n\)). It is known that these eigenspaces vanish unless \(q \leq i \leq q + \dim{A}\).NEWLINENEWLINEIn work of Ancona-Enright-Ward-Huber it is proved that if \(G\) is a semi-abelian variety then its \textit{triangulated} motive \(M(G) \in DM(k, \mathbb{Q})\) also decomposes into symmetric powers of an associated one-motive. This paper explores consequences of this motivic decomposition similar to the eigenspace decomposition of Chow groups explained above. It is natural in this context to interpret the Chow groups of projective varieties as certain motivic cohomology groups \(H^{p,q}(X)\) (also known as higher Chow groups \(\mathrm{CH}^q(X, 2q - p)\)) or, via poincare duality, as motivic homology groups \(H_{p,q}(X)\). For non-projective varieties motivic homology and cohomology are not isomorphic, and so there are two distinct generalisations this paper establishes.NEWLINENEWLINEAs before one obtains formally that \(H^{p,q}(G, \mathbb{Q}) = \bigoplus_i H^{p,q}(G, \mathbb{Q})^{(i)}\) and \(H_{p,q}(G, \mathbb{Q}) = \bigoplus_i H_{p,q}(G, \mathbb{Q})^{(i)}\). The most significant contribution of this paper is to establish vanishing results for these eigenspaces (depending on \(p, q, \dim{G}\) and the rank of \(G\)). In case of rank zero (and \(p=2q\)) the two results coincide and recover the old results. The vanishing theorem is obtained by using transfers to reduce to an algebraically closed base field, and then using induction on the rank.NEWLINENEWLINEAs a corollary of the vanshing result a generalisation of a result of Bloch about the vanishing of iterated Pontriyagin products of degree zero zero-cycles is obtained.