Integral Fourier transforms and the integral Hodge conjecture for one-cycles on abelian varieties

Research output: Contribution to journalArticleResearchpeer review

Authors

  • Thorsten Beckmann
  • Olivier De Gaay Fortman

Research Organisations

External Research Organisations

  • Max Planck Institute for Mathematics
View graph of relations

Details

Original languageEnglish
Pages (from-to)1188-1213
Number of pages26
JournalCompositio mathematica
Volume159
Issue number6
Early online date15 May 2023
Publication statusPublished - 2023

Abstract

We prove the integral Hodge conjecture for one-cycles on a principally polarized complex abelian variety whose minimal class is algebraic. In particular, the Jacobian of a smooth projective curve over the complex numbers satisfies the integral Hodge conjecture for one-cycles. The main ingredient is a lift of the Fourier transform to integral Chow groups. Similarly, we prove the integral Tate conjecture for one-cycles on the Jacobian of a smooth projective curve over the separable closure of a finitely generated field. Furthermore, abelian varieties satisfying such a conjecture are dense in their moduli space.

Keywords

    Chow rings, cohomology, integral Hodge conjecture, subvarieties of abelian varieties

ASJC Scopus subject areas

Cite this

Integral Fourier transforms and the integral Hodge conjecture for one-cycles on abelian varieties. / Beckmann, Thorsten; De Gaay Fortman, Olivier.
In: Compositio mathematica, Vol. 159, No. 6, 2023, p. 1188-1213.

Research output: Contribution to journalArticleResearchpeer review

Beckmann T, De Gaay Fortman O. Integral Fourier transforms and the integral Hodge conjecture for one-cycles on abelian varieties. Compositio mathematica. 2023;159(6):1188-1213. Epub 2023 May 15. doi: 10.48550/arXiv.2202.05230, 10.1112/S0010437X23007133
Beckmann, Thorsten ; De Gaay Fortman, Olivier. / Integral Fourier transforms and the integral Hodge conjecture for one-cycles on abelian varieties. In: Compositio mathematica. 2023 ; Vol. 159, No. 6. pp. 1188-1213.
Download
@article{77ea74363a3b41efaf0340c6623b4f7c,
title = "Integral Fourier transforms and the integral Hodge conjecture for one-cycles on abelian varieties",
abstract = "We prove the integral Hodge conjecture for one-cycles on a principally polarized complex abelian variety whose minimal class is algebraic. In particular, the Jacobian of a smooth projective curve over the complex numbers satisfies the integral Hodge conjecture for one-cycles. The main ingredient is a lift of the Fourier transform to integral Chow groups. Similarly, we prove the integral Tate conjecture for one-cycles on the Jacobian of a smooth projective curve over the separable closure of a finitely generated field. Furthermore, abelian varieties satisfying such a conjecture are dense in their moduli space.",
keywords = "Chow rings, cohomology, integral Hodge conjecture, subvarieties of abelian varieties",
author = "Thorsten Beckmann and {De Gaay Fortman}, Olivier",
note = "Funding Information: The first author was supported by the IMPRS program of the Max–Planck Society. The second author was supported by the European Union's Horizon 2020 research and innovation programme under the Marie Sk{\l}odowska-Curie grant agreement No. 754362. ",
year = "2023",
doi = "10.48550/arXiv.2202.05230",
language = "English",
volume = "159",
pages = "1188--1213",
journal = "Compositio mathematica",
issn = "0010-437X",
publisher = "Cambridge University Press",
number = "6",

}

Download

TY - JOUR

T1 - Integral Fourier transforms and the integral Hodge conjecture for one-cycles on abelian varieties

AU - Beckmann, Thorsten

AU - De Gaay Fortman, Olivier

N1 - Funding Information: The first author was supported by the IMPRS program of the Max–Planck Society. The second author was supported by the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 754362.

PY - 2023

Y1 - 2023

N2 - We prove the integral Hodge conjecture for one-cycles on a principally polarized complex abelian variety whose minimal class is algebraic. In particular, the Jacobian of a smooth projective curve over the complex numbers satisfies the integral Hodge conjecture for one-cycles. The main ingredient is a lift of the Fourier transform to integral Chow groups. Similarly, we prove the integral Tate conjecture for one-cycles on the Jacobian of a smooth projective curve over the separable closure of a finitely generated field. Furthermore, abelian varieties satisfying such a conjecture are dense in their moduli space.

AB - We prove the integral Hodge conjecture for one-cycles on a principally polarized complex abelian variety whose minimal class is algebraic. In particular, the Jacobian of a smooth projective curve over the complex numbers satisfies the integral Hodge conjecture for one-cycles. The main ingredient is a lift of the Fourier transform to integral Chow groups. Similarly, we prove the integral Tate conjecture for one-cycles on the Jacobian of a smooth projective curve over the separable closure of a finitely generated field. Furthermore, abelian varieties satisfying such a conjecture are dense in their moduli space.

KW - Chow rings

KW - cohomology

KW - integral Hodge conjecture

KW - subvarieties of abelian varieties

UR - http://www.scopus.com/inward/record.url?scp=85160861532&partnerID=8YFLogxK

U2 - 10.48550/arXiv.2202.05230

DO - 10.48550/arXiv.2202.05230

M3 - Article

AN - SCOPUS:85160861532

VL - 159

SP - 1188

EP - 1213

JO - Compositio mathematica

JF - Compositio mathematica

SN - 0010-437X

IS - 6

ER -