$\partial \bar{\partial}$ on a riemann surface hallo, 
i have the following question. let $M$ be a Riemann surface and $R \subset M$ be a compact totally real manifold (1 dimensional real). Furthermore assume there is a holomorphic 2-form $\alpha$ on $M$. Can one find a solution $\varphi$ of the equation $\partial \bar{\partial} \varphi = \alpha$ in a neighbourhood of $R$? Locally there is always a solution , but now in a neighbourhood of $R$ ? Hope for answers. Thanks in advance.
Marco
 A: Because you are only interested in a neighborhood of $R$, you might assume that $M$ is (connected and) noncompact. A function $f$ satisfies $\partial \bar{\partial} f=0$ iff it is harmonic. Now pick an open cover $(U_i)$ of $M$ and local solutions $\phi_i$ of your problem.
The differences $\phi_{ij}:=\phi_i - \phi_j$ are harmonic and form a cocycle, representing a cohomology class in $H^1 (M; \mathcal{H})$, where $\mathcal{H}$ is the sheaf of harmonic functions.
There is a short exact sequence of sheaves $0 \to \mathbb{R} \to \mathcal{O} \to \mathcal{H} \to 0$ (take real parts). Now $H^1 (M; \mathcal{O})=0$ (a deep result, holds for noncompact Riemann surfaces) and $H^2 (M;\mathbb{R})=0$ (not so hard, also since $M$ is noncompact. Thus $H^1 (M; \mathcal{H})=0$. 
Thus there are harmonic functions $f_i$ on $U_i$ with $f_i - f_j = \phi_{ij}$. The functions $\phi_i - f_i$ still are local solutions for your problem, and they coincide on intersections, therefore define a global solution.
A: In a comment above, marco asks whether this is true for larger $n$: That is to say, $M$ a complex $n$-fold, $R$ a totally real sub-real-$n$-fold and $\alpha$ a $(1,1)$-form on $R$. The answer is no for $n=2$.
Basic reason: Suppose that $\alpha = \partial \bar{\partial} f$. Then $\alpha = d ( \bar{\partial} f)$, so $\alpha$ is exact. In particular, when $n=2$, we should have $\int_R \alpha=0$. 
Counter-example: Let $M = (\mathbb{C}^{\ast})^2$; write $w$ and $z$ for the coordinates on $M$. Our $R$ will be $|w|=|z|=1$. Our $\alpha$ will be $\frac{dw \wedge d \bar{z}}{w \bar{z}}$. So
$$\int_R \alpha = \int_{|w|=1} \frac{dw}{w} \cdot \int_{|z|=1} \frac{d\bar{z}}{\bar{z}} = (2 \pi i) (- 2 \pi i) = - 4 \pi^2 \neq 0.$$
A: yes sure. So my question is the following: Let $M$ be a complex $n-$dimensional manifold and $R \subset M$ be a totally real, compact , n-dimensional (real) manifold. Assume one has a smooth $(n,n)-$form on $M$,say $\alpha$. Does there exists a smooth function $f : U \rightarrow \mathbb{R}$, where $U$ is an arbitralily small open neighbourhood of $R$ such that $(\partial \bar{\partial} f)^{n} = \alpha$ is satisfied?  if yes can you give me some referance . 
marco
