# Differential Geometry tools in Several Complex Variables

I am studying some several complex variables theory.

I have encountered integrals like $$\int_{\partial P(z_0,r)}\frac{f(\zeta_1,\dots,\zeta_n)}{(\zeta_1-z_1)\cdots(\zeta_n-z_n)}\,d\zeta_1\wedge\cdots\wedge d\zeta_n$$ where $P(z_0,r)$ is a polydisc centered in $z_0\in\Bbb C^n$ with muluradius $r=(r_1,\cdots,r_n)$ with $r_j>0$ and $f:\Bbb C^n\to\Bbb C$ is continuous (this integral defines a function of $z=(z_1,\dots,z_n)\in P(z_0,r)$).

Now, I know these kind of integrals are integrals of differential forms on manifolds, but they are new for me.

I took a good book of differential geometry, in order to becoming able to handle these integrals, but learning all the structure relies under integration of differential forms on manifold seems would taking a lot of time. All this theory is really interesting, but I have to read a paper on several complex variables and I cannot "consume" time and energy on things which are not dealing with the work I have to study, since I have deadlines to respect.

Thus my answer is: does a complete study of differential geometry is necessary or can I study somewhere simply the tools I need?

For example, I think studying SCV I will work always with open subsets of $\Bbb C^n$, not with abstract and general differential complex manifold. Are, in this framework, the integration tools needed simpler?

If yes, can you provide some good references?

My goal is understand and work freely with the integrals one find studying SCV.

• I am also of the opinion that you should take a closer look at DG as there is strong, interesting, fundamental interplay between the theories. However, if you simply want a "crash course" in SCV, I would suggest, for example, looking at Chapter 1 of Huybrechts' Complex Geometry. – Bass Jan 31 '18 at 21:28
• I don't want a crash course in SCV, but in DE, only because now I don't have the time to study it properly. However, this book seems useful, thank you – Joe Jan 31 '18 at 23:39

I do not like Huybrecht's book as an introduction, he has a lot of really deep concepts hidden in the background and it is noticeable.

I would recommend either Gunning and Rossi's Analytic Functions of Several Complex Variables -- This integral (The iterated Cauchy integral) is covered at the very beginning.

Or if you want a more analytic approach, I would recommend B.V. Shabat's Introduction to Complex Analysis -- Part II -- Several Complex Variables.

If you need a reference, that is not as easy to learn from, I would suggest Kaup and Kaup's Holomorphic Functions of Several Variables.

A great book on picking up differential geometry but less on the side of SCV is Moroianu's Lecture Notes on K\"ahler Geometry.

Hope this helps.

Personnally I would strongly recommend that you study this abstract differential geometry you mention before studying SCV. This is because SCV is a technical and difficult subject, and sooner or later you will need a solid background in several topics.

Depending on which direction in SCV you are going to go, other topics will probably be required (ring theory, potential theory, covering maps, algebraic geometry come to mind). Not to mention differential geometry that is going to be a lot more advanced than integration of volume forms on manifolds (for instance, fiber bundles, cohomology...)

PS: by the way, there is something wrong with your integral. It is probably either $\int_{\partial P} \frac{f(\zeta_1, \ldots, \zeta_n)}{(z_1-\zeta_1) \ldots (z_n - \zeta_n))} d\zeta_1 \ldots d\zeta_n$ or $\int_{P} \frac{f(\zeta_1, \ldots, \zeta_n)}{(z_1-\zeta_1) \ldots (z_n - \zeta_n))} d\zeta_1 \wedge d\overline{\zeta_1}\wedge \ldots d\zeta_n \wedge d\overline{\zeta_n}$, where $P$ is a polydisk and $\partial P$ is the product of the boundaries of the disks