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On a complex manifold, if we are dealing with the $d$ operator, there's a pretty easy way of showing some form is not $d$-exact, simply by integrating in a closed loop. If you can find a loop that is closed, but gives a nonzero integral for your form, then you know the form is not $d$-exact.

However, I can't seem to find an equivalent test for $\overline\partial$-exact forms. My intuitive understanding of the complex vector space is a bit shaky, but it I think if we integrate $\overline\partial f$ in a closed loop, we will get $-\int\partial f$ in that loop, which we don't really know anything about. We can't come up with a loop on which $\int\partial f$ is necessarily zero, or necessarily anything specific.

Is there some way to salvage this method, or is there another way we can prove $\overline\partial$-exactness? Any overview about how you'd approach this sort of problem is also welcome.

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Certainly if $\alpha$ is a $(0,1)$-form which is $\overline{\partial}$-exact, it must be $\overline{\partial}$-closed as well, since $\overline{\partial}^2 = 0$. Thus if $\alpha$ is not $\overline{\partial}$-closed, it cannot be $\overline{\partial}$-exact.

Assume then that $\alpha$ is $\overline{\partial}$-closed. In this case, my understanding is that your question is deep, and the answer depends greatly on the complex geometry of the manifold you are working on (whereas for the $d$-operator, knowing whether a form is exact is essentially topological).

If your manifold is $\mathbb{C}^n$ or a polydisk, the $\overline{\partial}$-poincare lemma says that $\alpha$ is necessarily $\overline{\partial}$-exact. I'm no expert in this field, but my understanding is that if your manifold is strongly pseudoconvex, then $\alpha$ is necessarily $\overline{\partial}$-exact (this is Cartan theorem B, I think).

A central theorem for finding solutions to $\overline{\partial}u = \alpha$ is Hormander's theorem, which gives conditions that allow you to find a solution which has some nice $L^2$ properties.

Bo Berndtsson has some notes called "An introduction to things $\overline{\partial}$" which I find to be pretty readable. Dror Varolin also has some notes called "Three Variations on a Theme in Complex Analytic Geometry" which could be useful. You may also want to look at this manuscript by Demailly, but it seems like it could be a bit harder.

Hope some of this helps.

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  • $\begingroup$ The two sets of notes you mention are the first and third sections respectively of the book Analytic and Algebraic Geometry: Common Problems, Different Methods. $\endgroup$ – Michael Albanese Jun 12 '13 at 6:17
  • $\begingroup$ @froggie Further Details: The Dolbeault lemma states that, at least locally, every $\overline{\partial}$-closed form is $\overline{\partial}$-exact. See Chapter 1 of Gunning and Rossi's text for details of this. Determining whether these local representations may be glued together to form a global representation is measured by sheaf (or Dolbeault) cohomology. In the above reference, it is shown that if your domain is a simply connected polydomain in $\mathbb{C}^n$, then the Dolbeault cohomology groups vanish. In particular, there is always a global representation for domains of this type. $\endgroup$ – AmorFati Sep 10 '18 at 22:42
  • $\begingroup$ @froggie Cartan's theorem $B$ states exactly that $H^k(X, \mathscr{S})=0$ for all $k \geq 1$ and coherent analytic sheaves $\mathscr{S}$. Gunning and Rossi is the best text for a first look at this rather advanced material. A second look at this material is the classic text by Grauert and Remmert: Theory of Stein Spaces, but is substantially more difficult to read (but would still highly recommend). Moreover, if you would like more details than anyone could ask for, there is a great book by the Kaup brothers called Holomorphic Functions of Many Variables. $\endgroup$ – AmorFati Sep 10 '18 at 22:44

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