Textbooks on Complex Analytic Spaces

Question

I'm looking for textbooks or monographs on complex analytic spaces. I'm aware of Coherent Analytic Sheaves by Grauert/Remmert and Several Complex Variables by Gunning/Rossi, but more references would be appreciated. I would prefer resources that allow nilpotents.

Context: Complex-Analytic techniques play an important role in algebraic geometry. I'm interested in algebraic geometry, so naturally I want to learn it. A particular interest is Berkovich spaces over $\Bbb Z$ à la Poineau, and these contain complex analytic spaces as fibers over archimedean points, the reason being that Berkovich geometry over $\Bbb C$ is complex-analytic geometry. In terms of potentially relevant background, I'm familiar with some algebraic geometry, sheaf cohomology, complex analysis, algebraic topology and some theory of Riemann Surfaces.

Answer 1

You should study Gerd Fischer's Complex Analytic Spaces, Lecture Notes in Mathematics 538.
Fischer got his PhD under Remmert, one of the greatest 20th century complex analysts.
Fischer has written many excellent research articles in complex analytic geometry but is also a great pedagogue who has written many books at various levels.
His books on linear algebra, algebraic curves, etc. have been tremendously successful in Germany in the last decades.
The book I recommend is about general, i.e. non reduced, complex spaces.
Reducedness is never assumed except in a few places where that hypothesis is explicitly mentioned.

Answer 2

What comes to mind is:

1. Algebraic curves and Riemann surfaces by Rick Miranda

2. Complex Geometry by Daniel Huybrechst

3. Principles of Algebraic Geometry by Griffith and Harris.

The first two are an easier read the last one requires knowledge of basic differential geometry namely differential forms. But all three should focus on what you’re looking for and 3. Being more algebraic and less analytic as well as the first one but 2. Is purely analytic.

Answer 3

Despite perhaps the fact that the following references are too "elementary" (in a sense to be precise below) respect to the needs of the Asker, I cannot refrain to advise at least to have a look at the following two masterpieces

[1] Stanisław Łojasiewicz, Introduction to complex analytic geometry, translated from the Polish by Maciej Klimek, (English), Basel-Boston-Berlin: Birkhäuser Veralg, pp. xiv+523 (1991), ISBN 3-7643-1935-6, MR1131081, Zbl 0747.32001.

[2] Hassler Whitney, Complex analytic varieties, (English) Addison-Wesley Series in Mathematics. Reading, Massachussets-Menlo Park, California-London-Don Mills, Ontario: Addison-Wesley Publishing Company. XII, 399 p. (1972), MR0387634, Zbl 0265.32008.

A brief (scant...) description of the approach of these texts.

From the preface of Łojasiewicz [1], p. v:

The subject of this book is analytic geometry understood as the geometry of analytic sets (or, more generally, analytic spaces), i.e., sets described locally by systems of analytic equations.
... Its aim is to familiarize the reader with the basic range of problems, using means as elementary as possible.

Whitney [2] p. vii, is even more straightforward:

The purpose of the present book is to provide a foundation, with reasonably elementary methods, for the theory of analytic varieties and spaces.

These works are textbooks, meaning that their aim is to teach the topic from the scratch, by using only basic prerequisites. Nevertheless they are advanced ones in that their aim is to provide a sound introduction to the field to future researchers.
Regarding their contents, I've arranged below a brief description of what you'll find (and of what you'll not find) there:

• Both books start from a very elementary level. Whitney has an introductory chapter on holomorphic functions and then goes to the very heart of the matter i.e. to analytic varieties. On the other hand, Łojasiewicz starts with tre chapters on preliminaries (A on algebra, B on topology and C on complex analysis) and in chapter 1 describes the rings of germs of holomorphic functions: analytic varieties are the topic of the second chapter.

• Both the books do not use the language of sheaves nor the one of currents (as instead Chirka does in his wonderful monograph cited by Moishe Kohan in his comment above)

• Finally a personal observation: the main reason I like these works is that they make you feel you are understanding what the topic is about, this does not means that they are easy, just that they are masterful expositions.

Well, my two cents.