$\partial \bar{\partial}$ on a complex manifold - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T02:08:08Z http://mathoverflow.net/feeds/question/87968 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/87968/partial-bar-partial-on-a-complex-manifold $\partial \bar{\partial}$ on a complex manifold william 2012-02-09T06:42:38Z 2012-02-11T06:13:58Z <p>hallo,</p> <p>i have the following question: let $M$ be a complex $n-$dimensional manifold and $R \subset M$ be a totally real, compact, $n-$dimensional (real) manifold. let $\alpha$ be a smooth nonnegative $(n,n)-$form on $M$. does there exists a smooth plurisubharmonic function $f : U \rightarrow \mathbb{R}$, where $U$ is a open neigbourhood of $R$ in $M$ such that the equation $(\partial \bar{\partial} f)^{n} = \alpha$ is satiesfied. So i am interested in smooth solution in a neigbourhood of the real manifold of the inhomogenous monge-ampere equation. does there exists such a solution ? if yes, can you give me some reference and if no what can be done to ave such a solution ? hope for answers. thanks in advance.</p> <p>marco </p>