I'm a humble physicist, looking for a reference that will explain the Dolbeault cohomology of holomorphic functions of homogeneity $-2$ on the Riemann sphere. In particular, my geometry is sufficiently rusty that I can't work out what the general form of the functions in this space are. Could someone point me to a (very pedestrian) reference, crucially with explicit representatives of the cohomology?
Just so you know where I am at present, I'm fairly sure that $$ c \frac{ [\bar \lambda d \bar \lambda] } { [\lambda \bar \lambda]^2 }$$
is one option, where $c$ is a constant. Is it true that I can use any homogeneous function of $\lambda$ as my $c$ and this will give me the full cohomology? Or does the most compact general formula involve further $\bar \lambda$ terms that cancel out in the $\bar \partial$ derivative, as happens for the measure?
I guess that the notion "holomorphic functions of homogeneity $−2$" means sections of the line bundle $\mathcal{O}(-2)$, which is isomorphic to the line bundle of holomorphic $1$-forms $\Omega^1$ (look at its degree$\ldots$). So the Dolbeault resolution of $\mathcal{O}(-2)$ is the Dolbeault resolution of $\Omega^1$, which consists in degree $0$ of ($C^{\infty}$ sections of) $(1,0)$-forms and in degree $1$ of $(1,1)$-forms, which are just the $2$-forms. The Dolbeault operator between them coincides with the Cartan differential and its image are the exact $2$-forms (since $\text{H}^{(0,1)}$ is vanishing, every $1$-form is the sum of a $(1,0)$-form and of an exact $1$-form).
So a $2$-form represents a nontrivial cohomology class in $\text{H}^1(\mathbb{P}^1, \Omega^1)$ if and only if its integral over the projective line is nonvanishing. In Čech cohomology you can represent it by the $1$-form $\text{d}z/z$ on $\mathbb{P}^1 \setminus \{0,\infty\}$, the intersection of the disks around $0$ and $\infty$ (it is not a boundary, because it has nonvanishing residues at $0$ and $\infty$).