Reference requestion for complex analysis with a view towards complex geometry

Question

As a background, I am a beginning graduate student whose background is primarily algebra and algebraic geometry. I have some background in analysis, but it is weaker. In particular I know the basics of measure theory and complex analysis but only at an undergraduate level.

I am really interested in the Riemann-Roch theorem, and am approaching it from an algebraic geometry perspective at the moment, but I thought the best way to learn complex analysis would be to approach from the Riemann surface point of view.

With that in mind, I am looking for a complex analysis/complex geometry book that is fairly self contained in terms of complex analytic background. Since I want to learn some complex analysis as well, I would like something that doesn't shy away from analytic arguments when necessary, but I would like something with a definite geometric flavour.

I feel like the standard answer here is Griffiths and Harris, so perhaps it would be good if people could give me an idea on how this book would serve me for this purpose? I got the impression is was much more algebraic geometry than complex analysis though. The other is Schlag's Complex Analysis and Riemann Surfaces if anyone is able to give me an idea of how that would serve.

Does such a book exist, or an I asking too much? To put it shortly, a self-contained complex analysis and complex geometry which doesn't shy away from complex analytic arguments but has a geometric slant with the view towards proving the Riemann-Roch theorem for complex manifolds.

Answer 1

B. V. Shabat's complex analysis in several variables.

Answer 2

I would recommend Forster's book Lectures on Riemann Surfaces. He first introduces the basics of complex differential geometry and then discusses non-compact and compact Riemann surfaces. It is heavily analytic and was understandable with my undergraduate background (basic courses on algebra, (complex) analysis, and a bit of differential geometry).

Answer 3

Hahn, Liang-shin. Complex numbers and geometry

The purpose of this book is to demonstrate that complex numbers and geometry can be blended together beautifully. This results in easy proofs and natural generalizations of many theorems in plane geometry, such as the Napoleon theorem, the Ptolemy-Euler theorem, the Simson theorem, and the Morley theorem. The book is self-contained-no background in complex numbers is assumed-and can be covered at a leisurely pace in a one-semester course. Many of the chapters can be read independently. Over 100 exercises are included.