I know that there exist many questions on this site on complex analysis books but my question is more specific than that. I am looking for recommendations for a concise complex analysis book but with a view towards algebraic geometry/ complex manifolds. I have studied smooth manifold theory before and it would be interesting to see it combined with complex analysis in the complex setting (all manifolds that I'd studied before were real manifolds). Any recommendations for such a book?
[Math] Complex analysis book for Algebraic Geometers
algebraic-geometrycomplex-analysisreference-request
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I'm not a big fan of Lang's complex analysis book-I consider it the weakest by far of all his textbooks. But since that's what you're using, you're really asking for a recommendation for an advanced course on complex analysis.
The most intensive and yet readable textbook I know on the subject is Complex Analysis in One Variable by University of Chicago master Raghavan Narasimhan and Yves Nievergelt. It is complex analysis for the serious analyst, from rapid coverage of the basics of analytic functions, covering spaces and Runge's theorem through the basics of functions of several complex variables and the elements of complex manifolds. This is unquestionably a graduate level text-it requires a good working knowledge of both real analysis at the level of Rudin or Pugh,a basic knowledge of abstract and linear algebra and topology. In short,it's a serious book for advanced students. I think you'll find it very helpful.
Another possible good text is Function Theory on Planar Domains by Steven Fisher, which covers a second course in complex analysis in a much more geometric manner, focusing on the Dirichlet problem and Hardy spaces.If you're interested in the geometric aspects of function theory, this will be a better choice for you. Best of all,it's in Dover and really cheap!
Those are the 2 best ones I know for a second course.Hope it helped.
The topics that you want to study use mostly the very essential ideas from Linear Algebra. Yes, over $\mathbb{R}$ and $\mathbb{C}$ is all you need. Since you seem to be a theoretically minded person with interest in geometry related subjects, I would recommend Gelfand's "Lectures on Linear Algebra". Strang's textbook is excellent, but probably not the style you are looking for.
My area of research is differential geometry (moving surfaces) and my favorite subjects to teach are Linear Algebra and Tensor Analysis. My advice would be to not treat Linear Algebra merely a means to studying the topics that you listed. Linear Algebra is the language and the framework for dealing with all branches of applied mathematics where physics, algebra and geometry meet. I would therefore recommend to study Linear Algebra thoroughly and, in fact, to give it several passes following a few different sources.
In case you want to learn my personal perspective on Linear Algebra and Differential Geometry, check out the Linear Algebra course on Lemma (http://lem.ma) and my Tensor Calculus videos on YouTube.
When Niels Henrik Abel was asked how he acquired his expertise, we replied "By studying the masters and not their pupils." Just one of my reactions to your desire to read a modern textbook.
Best Answer
An extremely good but shamefully underrated book is Łojasiewicz's Introduction to Complex Analytic Geometry.
The author, a renowned mathematician who discovered a very important inequality pertaining to analytic sets, managed to write a book with the contradictory qualities of giving very detailed explanations and proving quite advanced results.
Indeed the book starts with the definition of a ring (!) and goes on to prove a sophisticated version (due to Thom and Martinet) of the Weierstrass preparation theorem, the Puiseux theorem, Remmert's proper mapping theorem, Chevalley's theorem on images of constructible sets, the Cartan-Oka theorem on the coherence of the ideal sheaf of an analytic subset of $\mathbb C^n$, Chow's theorem on the algebraicity of analytic subsets of $\mathbb P^n$ (=the mother of all GAGA-type theorems!) , and much more.
The book is, alas, out of print but I suppose that many libraries have a copy.
A phonetic link
The undersigned happens to know that some patronyms are considered difficult to pronounce, so here is a link with the correct pronunciation of our author's name.