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By a complex analytic space I mean what Wikipedia calls a “complex analytic variety” (complex analytic spaces need not be Hausdorff). I am trying to learn about the history of this notion.

In H. Grauert, R. Remmert, Coherent Analytic Sheaves, Chapter 1, §1.9, there is an account of the genesis of the concept. I will reproduce here the last paragraph of the section. (Note: this book calls “complex space” to a Hausdorff complex analytic space.)

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The last cite is

[Gr₂] Grauert, H.: Ein Theorem der analytischen Garbentheorie und die Modulräume komplexer Strukturen, Publ. Inst. Hautes Etudes Sci. N° 5, 233-292 (1960).

When one looks it up, in Definition 2, p. 9, one finds an equivalent definition to the aforementioned one from wikipedia (up to Hausdorff condition; where a “bimorphe surjektive Abbildung” means an isomorphism of ringed spaces, see Definition 1 from p. 8).

My questions are:

  • Is [Gr₂] the first published piece of writing in which (Hausdorff) complex analytic spaces were defined? If not, is there a “first one”?

  • Do we know if anyone else suggested the definition to Grauert? As it usually happens with the coming of new mathematical ideas, these were already “floating in the air” before anyone decided to finally write them down on a published text. For this reason, I wonder: do we know about prior appearances of the concept (at meetings, seminars, letters) that predate [Gr₂]?

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Reference: Several Complex Variables IV, Springer, 1990.

In 1951 Behnke and Stein introduced the concept of a branched covering over a domain in $\mathbb{C}^n$ and defined a complex space to be a space whose local model is this branched covering. Soon after this Cartan and Serre introduced a different concept of complex space taking as local model an arbitrary analytic set in a domain in $\mathbb{C}^n$. The relation between these two concepts was finally cleared up in 1958 by Grauert and Remmert who proved that a Behnke-Stein space is a normal complex space in the sense of Cartan-Serre.

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  • $\begingroup$ I am looking at the reference you quoted but I cannot find the exact excerpt in the book. Could you tell me the page/s from where you are quoting? Also, where did “Cartan and Serre introduce a different concept of complex space taking as local model an arbitrary analytic set in a domain in $\mathbb{C}^n$”? According to the excerpt on my post from Grauert-Remmert, this was done by Serre on GAGA. However, the notion defined on GAGA is what we modernly know to be a “reduced complex analytic space,” a concept less general than that of complex analytic space I referred to. $\endgroup$ Apr 10, 2023 at 13:24

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