Algebraic cycles on toric fibrations over abelian varieties. (English) Zbl 0956.14004

Künnemann, Klaus

Math. Z. 232, No. 3, 427-435 (1999).

Let \(X\) be a smooth projective variety defined over a finite field \(k\). Beilinson has conjectured that the cycle class map cl\(:\text{CH}^i(X) \otimes_Z\mathbb{Q}_\ell \to H^{2i}_{et}(X\otimes_k\overline k,\mathbb{Q}_\ell (i))\) is injective for any prime \(\ell\) different from \(\text{char} k\). Assuming this conjecture, Grothendieck’s standard conjecture and the Tate conjectures can be reformulated as follows.
Let \({\mathcal L}_X\) be an ample invertible sheaf on \(X\) and \(L\) the associated Lefschetz operator. Then
(i) the Lefschetz operator induces an isomorphism \(L^{d-2p}:\text{CH}^p (X)_\mathbb{Q} @>\cong>> \text{CH}^{d-p} (X)_\mathbb{Q}\), \(0\leq 2p\leq d=\dim X\);
(ii) define by \((\alpha,\beta)\mapsto(-1)^p\deg_X(\alpha.L^{d-2p}\beta)\) CH\(^p(X)_\mathbb{Q}\times\text{CH}^p(x)_\mathbb{Q}\to\mathbb{Q}\) asymmetric bilinear form. It is positive definite on the subspace of primitive elements;
(iii) the action of \(\Gamma_k =\text{Gal}(\overline k/k)\) on \(H^i_{et}(X\otimes_k\overline k, \mathbb{Q}_\ell (j))\) is semi-simple;
(iv) the cycle class map induces an isomorphism \(\text{CH}^i (X)\otimes_Z \mathbb{Q}_\ell @>\cong>> H^{2i}_{et} (X\otimes_k \overline k,\mathbb{Q}_\ell(i))^{\Gamma_k}\).
In this paper the author proves these conjectures for the contraction product \(P=G\times^TZ\), where \(G\) is an extension of an abelian variety \(A\) by an algebraic torus \(T\) which is split over \(k\) and \(Z\) is a smooth projective toric variety \(T\subset Z\), under the assumption \(\dim A\leq 2\). The strategy of his proof is to deform \(P\) into the product \(A\times Z\) for which the proof becomes easy. The restriction on the dimension of \(A\) comes from the fact that it relies on the validity of the case \(i=1\) of the part (iv) due to Tate, and on the Hodge index theorem for surfaces due to Fulton.

Reviewer: Fumio Hazama (Hatoyama)

MSC:

14C25	Algebraic cycles
14M25	Toric varieties, Newton polyhedra, Okounkov bodies
14K15	Arithmetic ground fields for abelian varieties
14C20	Divisors, linear systems, invertible sheaves
14G15	Finite ground fields in algebraic geometry

Keywords:

Grothendieck conjecture; Chow group; finite ground field; cycle class map; Tate conjectures; ample invertible sheaf; Lefschetz operator; extension of an abelian variety; toric variety; Hodge index theorem