On the integral Hodge conjecture for real varieties, I
- PDF / 867,376 Bytes
- 77 Pages / 439.37 x 666.142 pts Page_size
- 40 Downloads / 183 Views
On the integral Hodge conjecture for real varieties, I Olivier Benoist1 · Olivier Wittenberg2
Received: 9 January 2018 / Accepted: 27 February 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We formulate the “real integral Hodge conjecture”, a version of the integral Hodge conjecture for real varieties, and raise the question of its validity for cycles of dimension 1 on uniruled and Calabi–Yau threefolds and on rationally connected varieties. We relate it to the problem of determining the image of the Borel–Haefliger cycle class map for 1-cycles, with the problem of deciding whether a real variety with no real point contains a curve of even geometric genus and with the problem of computing the torsion of the Chow group of 1-cycles of real threefolds. New results about these problems are obtained along the way. Introduction One of the central problems in the study of algebraic cycles of codimension k on a smooth proper complex algebraic variety X consists in determining the 2k (X (C), Z) ⊆ H 2k (X (C), Z) formed by their cycle classes. subgroup Halg Hodge theory provides a chain of inclusions
B Olivier Benoist
[email protected] Olivier Wittenberg [email protected]
1
Institut de Recherche Mathématique Avancée, UMR 7501, Université de Strasbourg et CNRS, 7 rue René Descartes, 67000 Strasbourg, France
2
Département de mathématiques et applications, École normale supérieure, 45 rue d’Ulm, 75230 Paris Cedex 05, France
123
O. Benoist, O. Wittenberg 2k Halg (X (C), Z) ⊆ Hdg2k (X (C), Z) ⊆ H 2k (X (C), Z),
(0.1)
where Hdg2k (X (C), Z) denotes the set of those classes whose images in H 2k (X (C), C) have type (k, k) with respect to the Hodge decomposition. By the Hodge conjecture, the first inclusion should become an equality after tensoring with Q. It is customary to refer to the property that the first inclusion is itself an equality as the integral Hodge conjecture. Despite its name, this property can fail. Its study for specific X and k has nevertheless played a significant role in recent years (see [36,108], [111, Chapter 6] and the references therein, and Sect. 2.1 for a more detailed discussion). Let now X denote a smooth proper real algebraic variety, by which we mean a smooth proper scheme over R. With any algebraic cycle of codimension k on X , Borel and Haefliger [13] have associated a cycle class k (X (R), Z/2Z) ⊆ in H k (X (R), Z/2Z). The study of the subgroup Halg H k (X (R), Z/2Z) formed by these classes is a classical topic in real algebraic geometry (see [9, §11.3], [15], [84, Chapitres 3–4], [101, Chapter III] and the references therein), related to the problem of C ∞ approximation of submanifolds of X (R) by algebraic subvarieties. Despite the formal similarity between these two settings, one critical difference stands out: being expressed with torsion coefficients, the definition of the k (X (R), Z/2Z) misses any information that might come from subgroup Halg the Hodge theory of the underlying complex variety. The latter, however, does have an influence
Data Loading...
