complex geometry - Dolbeault Cohomology of $mathbb{P}^1$

I wrote a blog post about almost exactly this question. I'll give a summary here:

Since $H^{1,1}(X)$ is one dimensional, I could answer your question by giving anythng with the correct integral. However, I'll try to give you the kind of cocycle which actually comes out of the proof of Dolbeaut-Cech equality.

Your cocycle isn't $dz/z$ but, rather, $dz/z$ with a specific choice of open cover of $X$. Lets say your choice is $U_1 cup U_2$, where $U_1 = { z : z neq infty }$ and $U_2 = { z : z neq 0 }$. Refine your cover to $V_1 cup V_2$, where $V_1 = { z : |z| > r }$ and $V_2 = { z : |z| < r^{-1} }$ for some $r < 1$.

Let $theta_1$ and $theta_2$ be $1$ forms on $V_1$ and $V_2$ such that $theta_1|_{V_1} - theta_2|_{V_2} = dz/z$. Then $overline{partial} theta_1$ and $overline{partial} theta_2$ have equal restrictions to $V_1 cap V_2$. The $(1,1)$-form you are looking for is their common value, which I'll call $omega$.

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Let's first do a fake solution. A real solution would look like a $C^{infty}$ smearing out of this one.

We'll work in the degenerate case $r=1$, so we are only gluing along a circle, not an annulus. We'll take $theta_1 = (1/2) overline{z} dz$ and $theta_2 = -(1/2) dz / (overline{z} z^2)$. Notice that both $theta_1$ and $theta_2$ restrict to $dz/z$ on the unit circle, but are constructed to extend smoothly to the appropriate discs.

So $overline{partial} theta_1 = (1/2) d overline{z} d z$ and $overline{partial} theta_2 = (1/2) dz d overline{z} / (overline{z}^2 z^2)$. Our $omega$ is formed by gluing these two differential forms together.

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A genuine smooth solution would be like this, but would interpolate smoothly between these two. If you push forward in a brute force manner, you'll get something with bump functions in it.

If you are more clever, you may discover the solution
$$theta_1 = frac{dz}{z} left( 1- frac{1}{1+z overline{z}} right)$$
and
$$theta_2 = - frac{dz}{z} left( frac{1}{1+z overline{z}} right).$$

You should check that $theta_1|_{V_1} - theta_2|_{V_2} = dz/z$ and that $theta_i$ is smooth and well-defined on $U_i$.

Then
$$overline{partial} theta_1 = overline{partial} theta_2 = frac{dz doverline{z}} {(1+z overline{z})^2}.$$

This is, as Scott guessed, the Fubini-Study form.