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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

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  • $\begingroup$ You don't mean to say that $\alpha$ is holomorphic, just smooth. $\endgroup$
    – Ben McKay
    Commented Feb 7, 2012 at 22:09
  • $\begingroup$ no i mean holomorphic ... i mean if its holomorphic then its also smooth. but does a solution $\varphi$ exist (on an open neighbourhood of $R$) ? $\endgroup$
    – william
    Commented Feb 7, 2012 at 22:13
  • $\begingroup$ I am pretty sure what you want is that $\alpha$ is a closed $(1,1)$ form. In particular, this includes that $\bar{\partial} \alpha=0$, since $d \alpha = \partial \alpha + \bar{\partial} \alpha$ and the two summands, being a $(2,1)$ form and a $(1,2)$ form, cannot sum to zero unless they are both $0$. $\endgroup$
    – David E Speyer
    Commented Feb 8, 2012 at 0:59

3 Answers 3

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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.

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  • $\begingroup$ is this true also in higher dimensions, when $M$ is n-dim. complex and $R$ is n-dim, compact, totally real??? $\endgroup$
    – william
    Commented Feb 7, 2012 at 23:09
  • $\begingroup$ Tiny correction: You want $0 \to \mathbb{R} \to \mathcal{O} \to \mathcal{H} \to 0$. Otherwise, very nice answer! $\endgroup$
    – David E Speyer
    Commented Feb 8, 2012 at 0:37
  • $\begingroup$ what i mean is: if $\alpha$ is a smooth $(n,n)-$form on $M$ (where $M$ is a n-dimensional complex manifold and $R$ is a compact, n-dim, totally real submanifold) does there exists in a neighbourhood of $R$ a solution (smooth) $f$ to the equation $(\partial \bar{\partial} f)^{n} = \alpha$ ???? $\endgroup$
    – william
    Commented Feb 8, 2012 at 6:25
  • $\begingroup$ this is somehow related to the inhomogeneous monge-ampere equation. $\endgroup$
    – william
    Commented Feb 8, 2012 at 6:26
  • $\begingroup$ Are you sure that's what you want to ask? It's highly nonlinear, for $n>1$. $\endgroup$
    – David E Speyer
    Commented Feb 8, 2012 at 13:12
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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 $$\int_R \alpha = \int_{|w|=1} \frac{dw}{w} \cdot \int_{|z|=1} \frac{d\bar{z}}{\bar{z}} = (2 \pi i) (- 2 \pi i) = - 4 \pi^2 \neq 0.$$

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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

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  • $\begingroup$ A helpful suggestion about using this site: The pages are organized hierarchically with three levels: Question, Answer and Comment. A change in the whole question being asked belongs up top in the Question. You can edit your question by clicking on the edit link which you should see to the lower left of your question, near the dg.differential-geometry flag. $\endgroup$
    – David E Speyer
    Commented Feb 8, 2012 at 15:01

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