Potential density of rational points on the variety of lines of a cubic fourfold

Abstract

Recall that for a variety X defined over a field k, rational points are said to be potentially dense in X if for some finite extension k ′ of k, X(k ′ ) is Zariski dense in X. For example, this is the case if X is a rational or a unirational variety.If k is a number field, a conjecture of Lang and Vojta, proved in dimension 1 by Faltings, but still open even in dimension two, predicts that varieties of general type never satisfy potential density over k. On the contrary, it is generally expected that when the canonical bundle K X is negative (i.e. X is a Fano variety) or trivial, rational points are always potentially dense in X (see for example [9] for a recent and precise conjecture).In the Fano case, there are many examples confirming this. First of all, many Fano varieties are unirational (in dimension ≤ 2, all of them are even rational over a finite extension of the definition field). Furthermore, Fano threefolds such as a three-dimensional quartic or a "double Veronese cone" (a hypersurface of degree 6 in P(1, 1, 1, 2, 3)) are birational to elliptic fibrations over P 2 , which makes it possible to prove potential density (see [13]).If the canonical bundle is trivial, the known examples are much less convincing. It is well-known that rational points on abelian varieties are potentially dense. But the simply-connected case remains largely unsolved.A number of results concerning potential density of rational points on K3 surfaces defined either over a number field or a function field appeared in the last years. Bogomolov and Tschinkel [6] proved potential density of rational points for K3 surfaces admitting an elliptic pencil, or an infinite automorphisms group. Remark that these K3 surfaces are rather special; in particular, their geometric Picard number is never equal to 1, whereas it is equal to 1 for a general projective K3.The case of K3-surfaces defined over a (complex) function field has been solved for certain types of families by Hassett and Tschinkel [14]. They prove in particular the existence of K3-surfaces defined over C(t), whose geometric Picard group is equal to Z, and which satisfy potential density. Their method can likely be adapted to produce examples over Q(t).However, no example of K3 surface defined over a number field, and with geometric Picard group equal to Z is known to satisfy potential density.In this paper, we consider the case of higher dimensional varieties which are as close as possible to K3 surfaces, namely Fano varieties of lines F of cubic 4-folds X. These are 4-dimensional varieties with trivial canonical bundles, and which even possess a non degenerate holomorphic 2-form which will play an important role in our study. Their similarity with K3 surfaces is shown by the work of Beauville and 1

The Fourier Transform for Certain HyperKähler Fourfolds

“…It is also expected from the Bloch-Beilinson conjectures that CH 2 (F ) 0,hom = 0 ; see Theorem 3. 3. In fact, for F = S [2] , this would essentially follow from the validity of Bloch's conjecture for S. Although we cannot prove such a vanishing, a direct consequence of Theorem 3 is that CH 1 (F ) · CH 2 (F ) 0,hom = 0 and CH 2 (F ) 0 · CH 2 (F ) 0,hom = 0.…”

“…A proof in the case of S [2] can be found in Proposition 12.9 and a proof in the case of the variety of lines on a cubic fourfold can be found in Proposition 20. 3. It is also expected from the Bloch-Beilinson conjectures that CH 2 (F ) 0,hom = 0 ; see Theorem 3.…”

“…This is embodied in Proposition 1. 3. Along the way, we express the Künneth projectors in terms of B ; see Corollary 1.7. We start by defining the Beauville-Bogomolov class B seen as an element of H 4 (F × F, Q).…”

“…Proposition 1. 3. Assume F is a compact hyperkähler manifold of dimension 4 which satisfies H 4 (F, Q) = Sym 2 (H 2 (F, Q)) and H 3 (F, Q) = 0.…”

“…The cohomological Fourier transform 13 2. The Fourier transform on the Chow groups of hyperkähler fourfolds 17 3. The Fourier decomposition is motivic 22 4.…”