Non-uniruled variety with level one Hodge structure. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T05:45:14Z http://mathoverflow.net/feeds/question/78435 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/78435/non-uniruled-variety-with-level-one-hodge-structure Non-uniruled variety with level one Hodge structure. IMeasy 2011-10-18T09:56:03Z 2011-10-18T20:35:10Z <p>I wonder if there exists one example of non-uniruled algebraic variety with level one Hodge structure. </p> http://mathoverflow.net/questions/78435/non-uniruled-variety-with-level-one-hodge-structure/78444#78444 Answer by Donu Arapura for Non-uniruled variety with level one Hodge structure. Donu Arapura 2011-10-18T11:18:48Z 2011-10-18T20:35:10Z <p>I edited the answer to expand it and add more context:</p> <p>The question was whether there exist nonuniruled smooth projective varieties with Hodge numbers \$h^{pq}=0\$ for all \$|p-q|>1\$. Of course, any curve of positive genus has this property. In dimension \$2\$, an Enriques surface, or any surface with \$p_g=0\$ and nonnegative Kodaira dimension, will work. Using Kunneth's formula, we can see also that by taking products of Enriques surfaces or products of such surfaces with a positive genus curve, we have an example in every dimension.</p>