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.
The integral Hodge conjecture can fail in two ways:
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.
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
and section 4 of
A motivic reinterpretation is discussed in
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
and
New examples are in
Last revised on May 4, 2018 at 22:19:28. See the history of this page for a list of all contributions to it.