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



$\partial \bar{\partial}$ on a riemann surface
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.


