Recently, I was reading the book of J.M. Lee Introduction to smooth manifolds, more precisely, chapters 12, 14, 16 where (covariant) tensor fields on smooth manifolds, differential forms and finally integrals of differential forms were defined.
While reading I thought about line integrals from classical analysis, in other words, integrals $\int_{\gamma} f(z)\,\mathrm{d}z$, where $\gamma$ is a smooth curve in complex plane.
Here, $f(z)\,\mathrm{d}z$ can be viewed as "holomorphic 1-form". So, I was wondering whether one can generalize notion of integral for "holomorphic $n$-forms", e.g., forms on complex $n$-manifold having a local formula $f(z)\,\mathrm{d}z_1 \wedge\mathrm{d}z_2\wedge\dots\wedge\mathrm{d}z_n$ for $f$ complex-valued holomorphic map?
This is not obvious for me, at least approach in the book I mentioned can not be generalized directly, for instance, because it uses smooth partitions of unity, while we can't expect analytic ones exist (however, this is not the only case that in my opinion can not be generalized directly).
Best Answer
On the one hand what you're suggesting is rather easy to do. Take a complex manifold $X$ of complex dimension $n$ (so real dimension $2n$). The set of "things that look like $f dz_1 \wedge \cdots \wedge dz_n$ locally" is a very important object, called the canonical line bundle of $X$ and denoted $K_X$. As Didier noted, the things in $K_X$ ("the sections of $K_X$") are differential $n$-forms and can be integrated over submanifolds of $X$ of real dimension $n$.
Unfortunately this is not a super interesting thing to do if you're a complex geometer, because if $n$ is even and you can find a complex submanifold of (real) dimension $n$ in $X$, the integral of anything in $K_X$ over it will be zero. The reason is essentially one of linear algebra, but vaguely speaking to get a nonzero integral over a complex manifold you need the same number of $dz$ and $d\bar z$ in your form, and we only have $dz$'s here.
"But wait," you say, "I integrate holomorphic forms like $f dz$ in the plane over paths all the time and I love it."
That is true, but turns out to be a rather specific feature of complex dimension $1$. There a form like $f dz$ can be integrated over a real path, and closed such paths are the boundaries of open complex submanifolds, so Stokes' theorem applies. By Cauchy's theorem, the real money in integrating holomorphic forms is really in integrating meromorphic forms, that is forms with poles. In higher dimensions the poles of a meromorphic form are no longer isolated and can't be contained inside the boundary of a real hypersurface (and even less inside a lower-dimensional real manifold).
What we can do in higher dimensions is to take two holomorphic $n$-forms $\alpha$ and $\beta$ and try to integrate $\alpha \wedge \overline{\beta}$ over $X$, as that will be a $2n$-form on a (real) $2n$-dimensional manifold. This leads to things like Bergman kernels and Weil-Peterson metrics, which are both active areas of research. There is also some kind of theory of residues in higher dimensions, but it is less developed and important than in complex dimension one.