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THE INTEGRAL HODGE CONJECTURE FOR 3-FOLDS OF KODAIRA DIMENSION ZERO

Part of: Cycles and subschemes (Co)homology theory Surfaces and higher-dimensional varieties

Published online by Cambridge University Press:  18 February 2020

Burt Totaro Open the ORCID record for Burt Totaro [Opens in a new window]
Burt Totaro*
Affiliation:
UCLA, Department of Mathematics, Los Angeles, CA, USA ([email protected])
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Abstract

We prove the integral Hodge conjecture for all 3-folds $X$ of Kodaira dimension zero with $H^{0}(X,K_{X})$ not zero. This generalizes earlier results of Voisin and Grabowski. The assumption is sharp, in view of counterexamples by Benoist and Ottem. We also prove similar results on the integral Tate conjecture. For example, the integral Tate conjecture holds for abelian 3-folds in any characteristic.

Keywords

integral Hodge conjectureintegral Tate conjecture3-foldCalabi–Yau manifold

MSC classification

Primary: 14C30: Transcendental methods, Hodge theory , Hodge conjecture
Secondary: 14F20: Étale and other Grothendieck topologies and (co)homologies 14J30: $3$-folds 14J32: Calabi-Yau manifolds
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 20 , Issue 5 , September 2021 , pp. 1697 - 1717
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Copyright
© Cambridge University Press 2020

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References

Barth, W., Counting singularities of quadratic forms on vector bundles, in Vector Bundles and Differential Equations (Nice, 1979) (ed. Hirschowitz, A.), pp. 1–19 (Birkhäuser, Boston, 1980).Google Scholar
Beauville, A., Quelques remarques sur la transformation de Fourier dans l’anneau de Chow d’une variété abélienne, in Algebraic Geometry (Tokyo/Kyoto, 1982), Lecture Notes in Mathematics, Volume 1016, pp. 238–260 (Springer, Berlin, 1983).Google Scholar
Beauville, A., Variétés Kähleriennes dont la première classe de Chern est nulle, J. Diff. Geom. 18 (1984), 755–782.Google Scholar
Benoist, O. and Ottem, J., Failure of the integral Hodge conjecture for threefolds of Kodaira dimension zero, Comment. Math. Helv., to appear. arXiv:1802.01845.Google Scholar
Benoist, O. and Wittenberg, O., On the integral Hodge conjecture for real varieties, II, J. de l’École Polytechnique, to appear. arXiv:1801.00873.Google Scholar
Brion, M., Some basic results on actions of non-affine algebraic groups, in Symmetry and Spaces (ed. Campbell, H., Helminck, A., Kraft, H. and Wehlau, D.), pp. 1–20 (Birkhäuser, Boston, 2010).Google Scholar
Deligne, P., Poids dans la cohomologie des variétés algébriques, Actes ICM Vancouver I (1974), 79–85.Google Scholar
Druel, S., A decomposition theorem for singular spaces with trivial canonical class of dimension at most five, Invent. Math. 211 (2018), 245–296.Google Scholar
du Bois, P., Complexe de de Rham filtré d’une variété singulière, Bull. SMF 109 (1981), 41–81.Google Scholar
Faltings, G., Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73 (1983), 349–366.Google Scholar
Fulton, W., On the topology of algebraic varieties, in Algebraic Geometry (Bowdoin, 1985), Proceedings of Symposia in Pure Mathematics, Volume 46, Part 1, pp. 15–46 (American Mathematical Society, Providence, 1987).Google Scholar
Fulton, W., Intersection Theory (Springer, Berlin, 1998).Google Scholar
Grabowski, C., On the integral Hodge conjecture for 3-folds, PhD Thesis. Duke University (2004).Google Scholar
Greb, D., Guenancia, H. and Kebekus, S., Klt varieties with trivial canonical class – holonomy, differential forms, and fundamental groups, Geom. Topol. 23 (2019), 2051–2124.Google Scholar
Griffiths, P. and Harris, J., Principles of Algebraic Geometry (Wiley, New York, 1978).Google Scholar
Guillen, F., Navarro Aznar, V., Pascual Gainza, P. and Puerta, F., Hyperrésolutions cubiques et descente cohomologique, Lecture Notes in Mathematics, Volume 1335, (Springer, Berlin, 1988).Google Scholar
Hartshorne, R., Algebraic Geometry (Springer, New York–Heidelberg, 1977).Google Scholar
Höring, A. and Peternell, T., Algebraic integrability of foliations with numerically trivial canonical bundle, Invent. Math. 216 (2019), 395–419.Google Scholar
Höring, A. and Voisin, C., Anticanonical divisors and curve classes on Fano manifolds, Pure Appl. Math. Q. 7 (2011), 1371–1393.Google Scholar
Huber, A. and Jörder, C., Differential forms in the h-topology, Algebraic Geom. 1 (2014), 449–478.Google Scholar
Kawamata, Y., Abundance theorem for minimal threefolds, Invent. Math. 108 (1992), 229–246.Google Scholar
Kollár, J., Trento examples, in Classification of Irregular Varieties (Trento, 1990), Lecture Notes in Mathematics, Volume 1515, pp. 134–135 (Springer, Berlin, 1992).Google Scholar
Kollár, J. and Kovács, S., Log canonical singularities are Du Bois, J. Amer. Math. Soc. 23 (2010), 791–813.Google Scholar
Kollár, J. and Mori, S., Birational Geometry of Algebraic Varieties (Cambridge University Press, Cambridge, 1998).Google Scholar
Milne, J. S., Algebraic Groups (Cambridge University Press, Cambridge, 2017).Google Scholar
Moonen, B. and Polishchuk, A., Divided powers in Chow rings and integral Fourier transforms, Adv. Math. 224 (2010), 2216–2236.Google Scholar
Mukai, S., Finite groups of automorphisms of K3 surfaces and the Mathieu group, Invent. Math. 94 (1988), 183–221.Google Scholar
Mumford, D., Abelian Varieties (Tata Institute of Fundamental Research, Mumbai, 2012).Google Scholar
Nikulin, V., Finite groups of automorphisms of Kählerian K3 surfaces, Trudy Mosk. Mat. Obsch. 38 (1979), 75–137. Trans. Moscow Math. Soc. 38 (1980), 71–135.Google Scholar
Reid, M., Young person’s guide to canonical singularities, in Algebraic Geometry (Bowdoin, 1985), Volume 1, pp. 345–414 (American Mathematical Society, Providence, 1987).Google Scholar
Schoen, C., An integral analog of the Tate conjecture for one-dimensional cycles on varieties over finite fields, Math. Ann. 311 (1998), 493–500.Google Scholar
Steenbrink, J., Mixed Hodge structures associated with isolated singularities, in Singularities (Arcata, 1981), Proceedings of Symposia in Pure Mathematics, Volume 40, Part 2, pp. 513–536 (American Mathematical Society, Providence, 1981).Google Scholar
Steenbrink, J., Du Bois invariants of isolated complete intersection singularities, Ann. Inst. Fourier 47 (1997), 1367–1377.Google Scholar
Tankeev, S. G., Surfaces of type K3 over number fields, and l-adic representations, Izv. Akad. Nauk SSSR 52 (1988), 1252–1271, 1328. translation in Math. USSR Izv. 33 (1989), 575–595.Google Scholar
Tate, J., Endomorphisms of abelian varieties over finite fields, Invent. Math. 2 (1966), 134–144.Google Scholar
Tate, J., Conjectures on algebraic cycles in l-adic cohomology, in Motives, Proceedings of Symposia in Pure Mathematics, Volume 55, Part 1, pp. 71–83 (American Mathematical Society, Providence, 1994).Google Scholar
Totaro, B., On the integral Hodge and Tate conjectures over a number field, Forum Math. Sigma 1 (2013), e4 (13 pages).Google Scholar
Voisin, C., Hodge Theory and Complex Algebraic Geometry I (Cambridge University Press, Cambridge, 2002).Google Scholar
Voisin, C., On integral Hodge classes on uniruled or Calabi–Yau threefolds, in Moduli Spaces and Arithmetic Geometry, Advanced Studies in Pure Mathematics, Volume 45, pp. 43–73 (Mathematical Society of Japan, Tokyo, 2006).Google Scholar
Voisin, C., Some aspects of the Hodge conjecture, Japanese J. Math. 2 (2007), 261–296.Google Scholar