Semi-regular varieties and variational Hodge conjecture

Ananyo Dan; Inder Kaur

doi:10.1016/j.crma.2016.01.005

Algebraic geometry

Semi-regular varieties and variational Hodge conjecture

Presented by: Claire Voisin

Ananyo Dan ¹; Inder Kaur ²

¹ Humboldt Universität Zu Berlin, Institut für Mathematik, Unter den Linden 6, Berlin 10099, Germany
² Freie Universität Berlin, FB Mathematik und Informatik, Arnimallee 3, 14195 Berlin, Germany

Comptes Rendus. Mathématique, Volume 354 (2016) no. 3, pp. 297-300.

Abstract
Résumé

Following [1,2], we know that semi-regular sub-varieties satisfy the variational Hodge conjecture, i.e., given a family of smooth projective varieties $π : X \to B$ , a special fiber $X_{o}$ and a semi-regular subvariety $Z \subset X_{o}$ , the cohomology class corresponding to Z remains a Hodge class (as $X_{o}$ deforms along B) if and only if Z remains an algebraic cycle. In this article, we investigate examples of such sub-varieties. In particular, we prove that any smooth projective variety Z of dimension n is a semi-regular sub-variety of a smooth projective hypersurface in $P^{2 n + 1}$ of large enough degree.

D'après [1,2] nous savons que les sous-variétés semi-régulières satisfont la conjecture de Hodge variationnelle, c'est-à-dire qu'étant données une famille de variétés projectives lisses $π : X \to B$ , une fibre spéciale $X_{o}$ et une sous-variété semi-régulière $Z \subset X_{o}$ , la classe de cohomologie correspondant à Z reste une classe de Hodge si et seulement si Z reste un cycle algébrique (lorsque $X_{o}$ se déforme le long de B). Nous étudions ici des exemples de telles sous-variétés. En particulier, nous montrons que toute variété projective lisse Z de dimension n est une sous-variété semi-régulière d'une hypersurface projective lisse de $P^{2 n + 1}$ de degré suffisamment grand.

Received: 2015-12-18
Accepted: 2016-01-20
Published online: 2016-02-09

DOI: 10.1016/j.crma.2016.01.005

Author's affiliations:

Ananyo Dan ¹; Inder Kaur ²

¹ Humboldt Universität Zu Berlin, Institut für Mathematik, Unter den Linden 6, Berlin 10099, Germany
² Freie Universität Berlin, FB Mathematik und Informatik, Arnimallee 3, 14195 Berlin, Germany

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     author = {Ananyo Dan and Inder Kaur},
     title = {Semi-regular varieties and variational {Hodge} conjecture},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {297--300},
     publisher = {Elsevier},
     volume = {354},
     number = {3},
     year = {2016},
     doi = {10.1016/j.crma.2016.01.005},
     language = {en},
}

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Ananyo Dan; Inder Kaur. Semi-regular varieties and variational Hodge conjecture. Comptes Rendus. Mathématique, Volume 354 (2016) no. 3, pp. 297-300. doi : 10.1016/j.crma.2016.01.005. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.01.005/

References
Cited by

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