Where can I learn about complex differential forms?

So I'm a 3rd year grad student in number theory/modular forms/algebraic geometry, and I've worked with differential forms from an algebraic point of view without ever knowing what they really are. I'd like to fix this.

I've recently read a section in Bredon's "Geometry and Topology", where he defines differential forms on real manifolds. However, most of the differential forms I work with are complex differential forms, and I understand there are some additional subtleties involved coming from the complex structure. I'd like to understand exactly how complex differential forms compare with real differential forms, and especially how to see them as special cases of real differential forms with additional structure.

Note that I've never taken a course in differential geometry, so if possible I'd like a resource that's as explicit as possible while still using the most general and precise language.

thanks,

• will

Solution 1:

Huybrechts' Complex Geometry: An Introduction has a nice section on them, as does the book by Demailly mentioned in mrf's answer. The only place where I found a careful construction of the exterior algebra on a complex manifold is in the second volume of Kobayashi and Nomizu's Foundations of Differential Geometry; every other source I've seen defines $\bigwedge^{p,q}V$ as $\bigwedge^qV^{1,0}\otimes\bigwedge^qV^{0,1}$, but this does not allow for skew-symmetry among indices of different types.

Solution 2:

Most books on several complex variables has at least a little about complex differential forms.

For something freely available, you could try reading the first chapter in Jean-Pierre Demailly's book Complex Analytic and Differential Geometry. It's really an excellent text that I imagine would be appealing to someone with a background in algebraic geometry who wants to learn some of the analytic approaches to the area.