Notation for the set of holomorphic differential forms

Question

I used to write $\Omega^p(M)$ for the set of $p$-forms on $M$ and $\Omega^{p,q}(M)$ for the set of $(p,q)$-forms on a complex manifold $M$.

Now some authors use $\Omega^p(M)$ to denote the set $\text{Ker}(\overline \partial:\Omega^{p,0}\to\Omega^{p+1,0})$ of holomorphic $p$-forms.

What can I do to avoid confusion?

Answer 1

In complex manifold theory, I think the most common convention is to use $\Omega^p(M)$ for the space of holomorphic $p$-forms, and some other notation like $\mathscr A^p(M)$ and $\mathscr A^{p,q}(M)$ (or $\mathscr E^p(M)$ and $\mathscr E^{p,q}(M)$) for smooth forms.

But if you want to stick with $\Omega^p(M)$ and $\Omega^{p,q}(M)$ for smooth forms, one alternative I've seen is $\mathcal O^p(M)$ for holomorphic $p$-forms, generalizing the standard notation $\mathcal O(M)$ for holomorphic functions.

Perhaps some day differential geometers will agree on one standard universal set of notation conventions. But we're not there yet!