How do Hodge classes for Calabi-Yau 4-folds compare with the classes for tori? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T20:52:31Z http://mathoverflow.net/feeds/question/108023 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/108023/how-do-hodge-classes-for-calabi-yau-4-folds-compare-with-the-classes-for-tori How do Hodge classes for Calabi-Yau 4-folds compare with the classes for tori? 喻yuwei 2012-09-25T07:12:00Z 2012-09-25T09:01:40Z <p>Let $X$ be a Calabi-Yau 4-fold, i.e., a connected 4-dimensional compact Kahler manifold with $K_{X} \cong \mathscr{O}_{X}$ and $h^{i} (X,O_{X} )= 0$ for $0 \lt i \lt 4$. Given a general 4-dimensional Weil torus $T$, one has a Hodge class contained in $H^{2,2}( X )$. I would like to ask whether the following equality holds:</p> <p>$$Hg^{2} ( X)_{prim}= Hg^{2}\left ( T \right )_{prim}$$</p> <p>where $Hg^{2}\left ( X \right )_{prim}$ is a primitive Hodge class, and $Hg^2 ( \star )$ is defined as $H^4 ( \star ,\mathbb{Q} )\cap H^{2,2} ( \star )$ for $\star = X,T$. If the equality is faulty, I would like to know why. If Hodge class is not unique, how do we get the classification of Hodge classes？</p>