How "frequent" are smooth projective varieties with trivial canonical bundle? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T19:42:44Z http://mathoverflow.net/feeds/question/81820 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/81820/how-frequent-are-smooth-projective-varieties-with-trivial-canonical-bundle How "frequent" are smooth projective varieties with trivial canonical bundle? norondion 2011-11-24T16:24:32Z 2011-11-24T19:29:34Z <p>How "frequent" are smooth projective varieties $X$ with trivial canonical bundle $\omega_X = \bigwedge^d \Omega^1_{X/k}$?</p> <p>E.g. for curves $C/k$, the canonical bundle is trivial iff the genus $g(C) = 1$ (elliptic curves). What is the situation like in the higher dimensional case?</p> http://mathoverflow.net/questions/81820/how-frequent-are-smooth-projective-varieties-with-trivial-canonical-bundle/81824#81824 Answer by Balazs for How "frequent" are smooth projective varieties with trivial canonical bundle? Balazs 2011-11-24T16:55:02Z 2011-11-24T19:29:34Z <p>If $X$ is a complex projective manifold with trivial canonical bundle, then by a theorem of Bogomolov, there is a finite unramified cover $\tilde X$ of $X$ which decomposes into a product $A\times X_1\times\ldots\times X_n\times Y$. Here $A$ is an abelian variety; $X_i$ are irreducible holomorphic symplectic manifolds (simply-connected, with a unique non-vanishing holomorphic 2-form) and $Y$ is a "strict Calabi-Yau" (simply-connected, with no holomorphic 2-form but a non-vanishing holomorphic top-form). Quite how frequent they are depends on your definition of frequent; for example, it's not known whether there are finitely many or infinitely many deformation types of strict Calabi-Yaus in three dimensions. </p>