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\begin{document}

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\section*{integral Hodge conjecture}

\hypertarget{statement}{}\subsection*{{Statement}}\label{statement}

Let $X$ be a smooth projective [[complex manifold]]. The \emph{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 \emph{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]].

\hypertarget{counterexamples}{}\subsection*{{Counterexamples}}\label{counterexamples}

The integral Hodge conjecture can fail in two ways:

\begin{itemize}%
\item There are [[torsion]] [[cohomology classes]] which are not [[algebraic cycle|algebraic]].
\item There are [[cohomology classes]] of infinite [[order of a group|order]] which are not [[algebraic cycle|algebraic]], but for which some multiple is algebraic.

\end{itemize}
Counterexamples of the first type were given first by \hyperlink{AtiyahHirzebruch61}{Atiyah-Hirzebruch 61}. They were later re-interpreted in terms of [[complex cobordism]] by \hyperlink{Totaro97}{Totaro 97}. These examples were so-called [[Godeaux-Serre varieties]], constructed in (\hyperlink{Serre58}{Serre 58}, section 20).

Counterexamples of the second type were first given by \hyperlink{Kollar90}{Kollar 90}.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Hodge conjecture]]
\item [[Tate conjecture]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[J.-P. Serre]], \emph{Sur la topologie des varietes algebriques en caracteristique p}, Symposium internacional de topologia algebraica, Mexico (1958), 24-53; in Oeuvres vol. 1, 501-530.


\item [[M. F. Atiyah]], [[F. Hirzebruch]], \emph{Analytic cycles on complex manifolds}, 1961, Topology Vol. 1, pp. 25-45, \href{http://hirzebruch.mpim-bonn.mpg.de/151/1/29_Analytic%20cycles%20on%20complex%20manifolds.pdf}{pdf}.


\item [[Burt Totaro]], \emph{Torsion algebraic cycles and complex cobordism}, J. Amer. Math. Soc. 10 (1997), no. 2, 467--493, \href{http://www.math.ucla.edu/~totaro/papers/public_html/bord.pdf}{pdf}.


\item [[J. Kollar]], \emph{Trento examples}, in Classification of irregular varieties, edited by E. Ballico, F. Catanese, C. Ciliberto, Lecture Notes in Math. 1515, Springer (1990).


\item [[C. Soulé]], [[C. Voisin]], \emph{Torsion cohomology classes and algebraic cycles on complex projective manifolds}, Adv. Math. 198 (2005), no. 1, 107--127, \href{http://arxiv.org/abs/math/0403254v2}{arXiv:math/0403254}.



\end{itemize}
These are summarized in section 2 of the following notes

\begin{itemize}%
\item [[Claire Voisin]], \emph{Some aspects of the Hodge conjecture}, notes from Takagi lectures, Kyoto 2006, \href{http://www.math.polytechnique.fr/~voisin/Articlesweb/takagifinal.pdf}{pdf}.

\end{itemize}
and section 4 of

\begin{itemize}%
\item [[Arnaud Beauville]], \emph{The Hodge conjecture}, \href{http://math.unice.fr/~beauvill/pubs/Hodge.pdf}{pdf}.

\end{itemize}
A [[motivic homotopy theory|motivic]] reinterpretation is discussed in

\begin{itemize}%
\item [[Alena Pirutka]], [[Nobuaki Yagita]], \emph{Note on the counterexamples for the integral Tate conjecture over finite fields}, \href{http://arxiv.org/abs/1401.1620}{arXiv:1401.1620}.

\end{itemize}
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

\begin{itemize}%
\item [[Jean-Louis Colliot-Thélène]], [[Tamás Szamuely]], \emph{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/{\tt \symbol{126}}colliot/CTSz21dec2009.pdf).}

\end{itemize}
and

\begin{itemize}%
\item [[Gereon Quick]], \emph{Torsion algebraic cycles and etale cobordism}, \href{http://arxiv.org/abs/0911.0584v3}{arXiv:0911.0584}.

\end{itemize}
New examples are in

\begin{itemize}%
\item [[Benjamin Antieau]], \emph{On the integral Tate conjecture for finite fields and representation theory}, \href{http://arxiv.org/abs/1504.04879}{arXiv:1504.04879}.

\end{itemize}


\end{document}