By Francesco Guaraldo
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Extra resources for Topics on Real Analytic Spaces
Let (X, OX) = E :R. s'. be a complex analytic space and a be an antiinvolution on it. Let x be a point of X such that x =a(x). 1 ) a the diagram p:R -------+0 (W,0:R 4p lw) j" j :R (V,Oxlv) following --------+ p:R I (W, 0:R4P W) is com- 35 Proof. 8) which deX Let us consider, as in (o:P,O pl. 7, the sections sIt = I We have sl' s1 E r(U, 0xlu) every I = 1, ... ,p. 13), I I be s~' for = the mor- = 1, ... 8); by p is an embedding. There exists a neighbourhood W ii) of 0 in o:2 p , with W = bedding.
Fp = f~ + i f~ . f". Finally. let 1 1 P P be the local model for real analytic spaces defined by the ideal IR. is 'P : (M. OM) another -+ (N. ON) model for is analytic an associate. functorially. to -+ R (N. ON ). complex analytic morphism. ) From 1. 1. 3 it follows that for every complex analy- tic space (X. OX) there exists a s;ructure space on X. ~£_~E~£~ of (X. OX). It is straightforward to check that the construction of underlying the real category of analytic spaces is a covariant functor from complex analytic spaces to the category of real analytic spaces.
Let f 1 , ... ,f p holomorphic functions lin a ne i ghbourhood U of a. X is the complexification of x We may suppose for every x E U. Then, x' since f I ® ~, the real and imaginary parts of f , ... 2,i)). 9. 8 it follows that, if X is a coherent real analytic variety, then all local models of X are coherent. Therefore in the sequel we shall speak of germs of coherent analytic varieties without specifying where they are realized. 10. Let Y be a coherent real analytic variety and X be a subvariety of Y.
Topics on Real Analytic Spaces by Francesco Guaraldo