# $\partial \bar{\partial}$ on a complex manifold

hallo,

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.

marco

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I don't think there is much literature on this topic, unless you are willing to assume that $\alpha$ be nonnegative definite and seek for $f$ plurisubharmonic. – YangMills Feb 10 '12 at 23:08
yes ok then lets assume that ... i will edit the first post ! what about that ... can you offer me some literature ? – william Feb 11 '12 at 6:12