On the irreducible components of globally defined semianalytic sets

Fernando, José F.

doi:10.1007/s00209-016-1634-9

Cited by 6 publications

(4 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The C-analytic subspaces X i are called the irreducible components of X. For further details see [Fe8,WB]. It is important to point out that M(X) ∼ = i∈I M(X i ).…”

Section: 33mentioning

confidence: 99%

“…Let X = {z ∈ Y : σ(z) = z}.By Corollary 2.3 there exists a ( σ-)invariant analytic curve C ⊂ Y of real dimension 1 without isolated points such that D 0 := C ∩ X is a discrete subset of X and π −1 (X) = X ∪ C. As X has a system of open Stein neighborhoods in Y , by[G, M.Thm.3] also X ∪ C has a system of open Stein neighborhoods in Y . As the irreducible components of Y are its connected components, we may assume (after shrinking Y if necessary) that both X ∪ C and Y are connected[Fe8, Thm.1.2, Prop.5.16].Let f : X → R be a non-zero positive semidefinite analytic function. Shrinking Y if necessary, we may assume that f extends to an invariant holomorphic functionF : Y → C. Consider the holomorphic function F ′ := F • π : Y → C. As σ • π = π • σ, we have that F ′ := F • π is an invariant holomorphic function whose restriction to X is positive semidefinite.…”

mentioning

confidence: 99%

See 1 more Smart Citation

Positive Semidefinite Analytic Functions on Real Analytic Surfaces

Fernando

2021

J Geom Anal

Self Cite

View full text Add to dashboard Cite

show abstract

“…The C-analytic subspaces X i are called the irreducible components of X. For further details see [Fe8,WB]. It is important to point out that M(X) ∼ = i∈I M(X i ).…”

Section: 33mentioning

confidence: 99%

mentioning

confidence: 99%

Positive Semidefinite Analytic Functions on Real Analytic Surfaces

Fernando

2021

J Geom Anal

Self Cite

View full text Add to dashboard Cite

show abstract

“…The section on amenable C-semianalytic sets comes from [Fe2]. Irreducibility and irreducible components are usual concepts in Geometry and Algebra.…”

Section: Bmentioning

confidence: 99%

Topics in global real analytic geometry

Acquistapace¹,

Broglia²,

Fernando³

2021

Preprint

View full text Add to dashboard Cite

“…Thus, we may assume X is irreducible. Consequently, its normalization X ν is connected and both X ν R and X R are irreducible [Fe,§5]. By [N3, IV.…”

Section: Bmentioning

confidence: 99%

Normalization of Complex Analytic Spaces from a Global Viewpoint

Fernando

Broglia

2018

J Geom Anal

Self Cite

View full text Add to dashboard Cite

In this work we study some algebraic and topological properties of the ring O(X ν ) of global analytic functions of the normalization (X ν , OXν ) of a reduced complex analytic space (X, OX). If (X, OX ) is a Stein space, we characterize O(X ν ) in terms of the (topological) completion of the integral closure O(X) ν of the ring O(X) of global holomorphic functions on X (inside its total ring of fractions) with respect to the usual Fréchet topology of O(X) ν . This shows that not only the Stein space (X, OX ) but also its normalization is completely determined by the ring O(X) of global analytic functions on X. This result was already proved in 1988 by Hayes-Pourcin when (X, OX ) is an irreducible Stein space whereas in this paper we afford the general case. We also analyze the real underlying structures (X R , O R X ) and (X ν R , O R X ν ) of a reduced complex analytic space (X, OX ) and its normalization (X ν , OXν ). We prove that the complexification of (is a coherent real analytic space. Roughly speaking, coherence of the real underlying structure is equivalent to the equality of the following two combined operations: (1) normalization + real underlying structure + complexification, and (2) real underlying structure + complexification + normalization. . This article is the fruit of the close collaboration of the authors in the last fifteen years and has been mainly written during a two-months research stay of third author in the Dipartimento di Matematica of the Università di Pisa. Third author would like to thank the department for the invitation and the very pleasant working conditions. Proposition 1.3. The closure Cl(N) is (isomorphic to) the inverse limit lim ←− K⊂X compactIn addition, if {K ℓ } ℓ≥1 is an exhaustion of X by compact sets, then Cl(N)As a straightforward application of Proposition 1.3 (making N = O(X)), we write the ring O(X) as the inverse limit of the rings of fractions S −1 K O(X) where each K ⊂ X is a compact set. Corollary 1.4. The ring O(X) is (isomorphic to) the inverse limit lim ←− K⊂X compactIn addition, if {K ℓ } ℓ≥1 is an exhaustion of X by compact sets, then O(X)We analyze the behavior of the normalization of a reduced complex analytic space (X, O X ) when considering the real underlying structure (X R , O R X ). By 2.E and 2.G there exists a complexification of (X R , O R X ), that is, a complex analytic space (such that X R is the set of fixed points of σ. If (X ν , O X ν , π) is the normalization of (X, O X ) and (X ν R , O R X ν ) is its real analytic structure, we complexify the real analytic morphism π R : X ν R → X R to obtain a complex analytic morphism π R ∼ : X ν R ∼ → X R

show abstract

On the irreducible components of globally defined semianalytic sets

Cited by 6 publications

References 22 publications

Positive Semidefinite Analytic Functions on Real Analytic Surfaces

Positive Semidefinite Analytic Functions on Real Analytic Surfaces

Topics in global real analytic geometry

Normalization of Complex Analytic Spaces from a Global Viewpoint

Contact Info

Product

Resources

About