Real primitive of a complex form on a CR manifold - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T09:29:16Z http://mathoverflow.net/feeds/question/22843 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/22843/real-primitive-of-a-complex-form-on-a-cr-manifold Real primitive of a complex form on a CR manifold Andrea Altomani 2010-04-28T12:37:38Z 2010-04-28T16:35:15Z <p>I am looking for a characterization of (0,1)-forms on a CR manifold M that admit a real primitive, i.e. those can be written as:</p> <p>$\omega=\overline\partial_M f$</p> <p>for a real function f.</p> <p>If M is a complex manifold, by expanding ddf=0 one obtains the following characterization:</p> <p>$\overline\partial\omega=0$, $\partial\overline\omega=0$, $\partial\omega+\overline\partial\overline\omega=0$.</p> <p>In the CR case however, there is not a good substitute for $\partial$, and also the symmetry between (0,1) forms and (0,1)-forms fails.</p> <p><strong>Edit:</strong> One easy condition is $\overline\partial_M\omega=0$. In general it is a difficult problem even to say if $\omega=\overline\partial_M g$ for some <em>complex</em> function g.</p> <p>My question should be rephrased as follows: <em>Assuming that there exist a</em> complex <em>solution g to</em> $\omega=\overline\partial_M g$, <em>when is it possible to choose g real?</em></p>