nLab integral Hodge conjecture

Statement

Let $X$ be a smooth projective complex manifold. The integral Hodge conjecture states that every integral Hodge cycle, i.e. $2k$-degree cohomology class of $X$ which lies in the $(k,k)$-piece of the Hodge decomposition, is algebraic, i.e. the class of an algebraic cycle.

This conjecture is known to be false, hence the refinement of the Hodge conjecture to rational cohomology classes. However it is true for $k=1$ by the Lefschetz theorem on (1,1)-classes.

Counterexamples

The integral Hodge conjecture can fail in two ways:

• There are torsion cohomology classes which are not algebraic.
• There are cohomology classes of infinite order which are not algebraic, but for which some multiple is algebraic.

Counterexamples of the first type were given first by Atiyah-Hirzebruch 61. They were later re-interpreted in terms of complex cobordism by Totaro 97. These examples were so-called Godeaux-Serre varieties?, constructed in (Serre 58, section 20).

Counterexamples of the second type were first given by Kollar 90.

Related concepts

• Hodge conjecture
• Tate conjecture

References

• J.-P. Serre, Sur la topologie des varietes algebriques en caracteristique p, Symposium internacional de topologia algebraica, Mexico (1958), 24-53; in Oeuvres vol. 1, 501-530.

• M. F. Atiyah, F. Hirzebruch, Analytic cycles on complex manifolds, 1961, Topology Vol. 1, pp. 25-45, pdf.

• Burt Totaro, Torsion algebraic cycles and complex cobordism, J. Amer.

  Math. Soc. 10 (1997), no. 2, 467–493, pdf.

• J. Kollar?, Trento examples, in Classification of irregular varieties,

  edited by E. Ballico, F. Catanese, C. Ciliberto, Lecture Notes in Math. 1515, Springer (1990).

• C. Soulé, C. Voisin, Torsion cohomology classes and algebraic cycles on complex projective manifolds, Adv. Math. 198 (2005), no. 1, 107–127, arXiv:math/0403254.

These are summarized in section 2 of the following notes

• Claire Voisin, Some aspects of the Hodge conjecture, notes from Takagi lectures, Kyoto 2006, pdf.

and section 4 of

• Arnaud Beauville, The Hodge conjecture, pdf.

A motivic reinterpretation is discussed in

• Alena Pirutka?, Nobuaki Yagita?, Note on the counterexamples for the integral Tate conjecture over finite fields, arXiv:1401.1620.

The ideas of Atiyah-Hirzebruch and Totaro are shown to extend to positive characteristic, using etale cohomology and etale homotopy theory, to give counterexamples to the integral Tate conjecture, in

• Jean-Louis Colliot-Thélène?, Tamás Szamuely, Autour de la conjecture de Tate `a coefficients Z_l pour les vari'et'es sur les corps finis_, [arXiv:0902.1666](http://arxiv.org/abs/0902.1666), [pdf](http://www.math.u-psud.fr/~colliot/CTSz21dec2009.pdf).

and

• Gereon Quick, Torsion algebraic cycles and etale cobordism, arXiv:0911.0584.

New examples are in

• Benjamin Antieau, On the integral Tate conjecture for finite fields and representation theory, arXiv:1504.04879.