Variétés unirationnelles non rationnelles: au-delà de l'exemple d'Artin et Mumford

Chow Rings, Decomposition of the Diagonal, and the Topology of Families

“…It follows then from the work of Colliot-Thélène and Ojanguren [23] that there are rationally connected sixfolds X for which Z 4 (X) = 0. Such examples are not known in smaller dimensions, and we proved that they do not exist in dimension 3.…”

“…Unramified cohomology of X with value in A is defined by the formula (see [23]) H i nr (X, A) = H 0 (X Zar , H i X (A)).…”

“…The conclusion of the above-mentioned Theorems 6.5 and 6.24 is that for a uniruled threefold with CH 0 = Z, all the interesting (and birationally invariant) phenomena concerning codimension 2 cycles, namely the kernel of the AbelJacobi map (Mumford [71]), the Griffiths group (Griffiths [48]) and the group Z 4 (X) versus degree 3 unramified cohomology with torsion coefficients (Soulé and Voisin [93], Colliot-Thélène and Voisin [24]) are trivial. In the rationally connected case, the only interesting cohomological or Chow-theoretic invariant could be the Artin-Mumford invariant (or degree 2 unramified cohomology with torsion coefficients; see [23]), which is also equal to the Brauer group since Stated in words, this question asks for the existence of a universal codimension 2 cycle on J(Y ) × Y . The analogous question for codimension 1 cycles is well known to have an affirmative answer; this is the universal divisor L ∈ Pic(X) × Pic 0 (X), which is itself induced by pull-back via the Albanese map alb X : X → Alb(X) of the Poincaré divisor P ∈ Pic(Alb(X) × Pic 0 (X)) (see [72, pp.…”

“…Going further, in the paper [23], the authors exhibit unirational sixfolds with vanishing group H 2 nr (X, Q/Z) but nonvanishing group H 3 nr (X, Q/Z). Their example is reinterpreted in the recent paper [24] using the groups Z 4 (X) introduced in (6.2).…”

“…Unramified cohomology with torsion coefficients (that is, A = Z/nZ or A = Q/Z) plays an important role in the study of the Lüroth problem, that is, the study of unirational varieties that are not rational (see, for example, the papers [4], [23], and [84]). To start with, we have the following result concerning the invariant used by Artin and Mumford, which is the torsion in the group H Proof.…”