On the integral Hodge conjecture for real varieties, II

Benoist, Olivier; Wittenberg, Olivier

doi:10.5802/jep.120

On the integral Hodge conjecture for real varieties, II
[Sur la conjecture de Hodge entière pour les variétés réelles, II]

Benoist, Olivier ¹ ; Wittenberg, Olivier ²

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

Journal de l’École polytechnique — Mathématiques, Tome 7 (2020), pp. 373-429.

Résumé
Abstract

Nous établissons la conjecture de Hodge entière réelle pour les 1-cycles pour diverses classes de solides uniréglés (fibrés en coniques, solides de Fano sans points réels, certaines fibrations en del Pezzo) et pour les fibrés en coniques sur des bases de dimension supérieure satisfaisant elles-mêmes la conjecture de Hodge entière réelle pour les 1-cycles. De plus, nous montrons que les solides rationnellement connexes sur les corps réels clos non archimédiens ne vérifient pas en général la conjecture de Hodge entière réelle et que sur de tels corps, le théorème EPT de Bröcker reste vrai pour les surfaces simplement connexes de genre géométrique nul mais tombe en défaut pour certaines surfaces K3.

We establish the real integral Hodge conjecture for $1$ -cycles on various classes of uniruled threefolds (conic bundles, Fano threefolds with no real point, some del Pezzo fibrations) and on conic bundles over higher-dimensional bases which themselves satisfy the real integral Hodge conjecture for $1$ -cycles. In addition, we show that rationally connected threefolds over non-archimedean real closed fields do not satisfy the real integral Hodge conjecture in general and that over such fields, Bröcker’s EPT theorem remains true for simply connected surfaces of geometric genus zero but fails for some K3 surfaces.

Reçu le : 2018-01-11
Accepté le : 2019-03-12
Publié le : 2020-03-06

Zbl

DOI : 10.5802/jep.120

Classification : 14C25, 14C30, 14P99, 14J30
Keywords: Real algebraic geometry, integral Hodge conjecture, real closed fields
Mot clés : Géométrie algébrique réelle, conjecture de Hodge entière, corps réels clos

Affiliations des auteurs :

Benoist, Olivier ¹ ; Wittenberg, Olivier ²

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Benoist, Olivier; Wittenberg, Olivier. On the integral Hodge conjecture for real varieties, II. Journal de l’École polytechnique — Mathématiques, Tome 7 (2020), pp. 373-429. doi : 10.5802/jep.120. http://www.numdam.org/articles/10.5802/jep.120/

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