Topics in global real analytic geometry

TL;DR

This paper explores the development and challenges of real analytic geometry, contrasting it with complex analytic spaces, and discusses the properties and limitations of real analytic sets and spaces.

Contribution

It provides a survey of the historical development, key concepts, and difficulties in formulating real analytic spaces compared to complex ones.

Findings

Complex analytic spaces have well-behaved properties like coherence and Theorems A and B.

Real analytic spaces form a category with less desirable properties, such as lack of coherence.

Historical efforts to define real analytic spaces faced significant obstacles.

Abstract

In the first half of twentieth century the theory of complex analytic functions and of their zerosets was fully developed. The definition of holomorphic function has a local nature. Germs of holomorphic functions form a distinguished subring of the ring of germs of continuous functions. This way came out the notion of analytic space. The definition of a complex analytic set is by local models as in the case of complex manifolds. But while local models for manifolds are open sets of ${\mathbb C}^n$, a local model of an analytic space is the zeroset of finitely many analytic functions on an open set of ${\mathbb C}^n$ together with a sheaf of continuous function to be called holomorphic. Towards the years $50$ of the last century, Cartan, Whitney, Bruhat and others tried to formulate the notion of analytic space over ${\mathbb R}$. Immediately they realize that the real sets verifying a definition similar to the complex one form a cathegory whose elements can get unpleasant behaviour. In particular this cathegory does not get the good properties of complex analytic spaces, as for instance coherence of their structural sheaves and Theorems A and B do not hold in general.