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Francesca Acquistapace

Publications and source records attributed to Francesca Acquistapace.

6 recordsLinked to original sources
arXiv2017

Normalization of complex analytic spaces from a global viewpoint

In this work we study some algebraic and topological properties of the ring ${\mathcal O}(X^\nu)$ of global analytic functions of the normalization $(X^\nu,{\mathcal O}_{X^\nu})$ of a reduced complex analytic space $(X,{\mathcal O}_X)$. If $(X,{\mathcal O}_X)$ is a Stein space, we characterize ${\mathcal O}(X^\nu)$ in terms of the (topological) completion of the integral closure $\overline{{\mathcal O}(X)}^\nu$ of the ring ${\mathcal O}(X)$ of global holomorphic functions on $X$ (inside its total ring of fractions) with respect to the usual Fr\'echet topology of $\overline{{\mathcal O}(X)}^\nu$. This shows that not only the Stein space $(X,{\mathcal O}_X)$ but also its normalization is completely determined by the ring ${\mathcal O}(X)$ of global analytic functions on $X$. This result was already proved in 1988 by Hayes-Pourcin when $(X,{\mathcal O}_X)$ is an irreducible Stein space whereas in this paper we afford the general case. We also analyze the real underlying structures $(X^{\mathbb R},{\mathcal O}_X^{\mathbb R})$ and $(X^{\nu\,{\mathbb R}},{\mathcal O}_{X^\nu}^{\mathbb R})$ of a reduced complex analytic space $(X,{\mathcal O}_X)$ and its normalization $(X^\nu,{\mathcal O}_{X^\nu})$. We prove that the complexification of $(X^{\nu\,{\mathbb R}},{\mathcal O}_{X^\nu}^{\mathbb R})$ provides the normalization of the complexification of $(X^{\mathbb R},{\mathcal O}_X^{\mathbb R})$ if and only if $(X^{\mathbb R},{\mathcal O}_X^{\mathbb R})$ 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.

math.AG↗
arXiv2015

On globally defined semianalytic sets

In this work we present the concept of $C$-semianalytic subset of a real analytic manifold and more generally of a real analytic space. $C$-semianalytic sets can be understood as the natural generalization to the semianalytic setting of global analytic sets introduced by Cartan ($C$-analytic sets for short). More precisely $S$ is a $C$-semianalytic subset of a real analytic space $(X,{\mathcal O}_X)$ if each point of $X$ has a neighborhood $U$ such that $S\cap U$ is a finite boolean combinations of global analytic equalities and strict inequalities on $X$. By means of paracompactness $C$-semianalytic sets are the locally finite unions of finite boolean combinations of global analytic equalities and strict inequalities on $X$. The family of $C$-semianalytic sets is closed under the same operations as the family of semianalytic sets: locally finite unions and intersections, complement, closure, interior, connected components, inverse images under analytic maps, sets of points of dimension $k$, etc. although they are defined involving only global analytic functions. In addition, we characterize subanalytic sets as the images under proper analytic maps of $C$-semianalytic sets. We prove also that the image of a $C$-semianalytic set $S$ under a proper holomorphic map between Stein spaces is again a $C$-semianalytic set. The previous result allows us to understand better the structure of the set $N(X)$ of points of non-coherence of a $C$-analytic subset $X$ of a real analytic manifold $M$. We provide a global geometric-topological description of $N(X)$ inspired by the corresponding local one for analytic sets due to Tancredi-Tognoli (1980), which requires complex analytic normalization. As a consequence it holds that $N(X)$ is a $C$-semianalytic set of dimension $\leq\dim(X)-2$.

math.AG↗
arXiv2014

On the Nullstellensätze for Stein spaces and $C$-analytic sets

In this work we prove the real Nullstellensatz for the ring ${\mathcal O}(X)$ of analytic functions on a $C$-analytic set $X\subset{\mathbb R}^n$ in terms of the saturation of Łojasiewicz's radical in ${\mathcal O}(X)$: The ideal ${\mathcal I}({\mathcal Z}({\mathfrak a}))$ of the zero-set ${\mathcal Z}({\mathfrak a})$ of an ideal ${\mathfrak a}$ of ${\mathcal O}(X)$ coincides with the saturation $\widetilde{\sqrt[\textŁ]{\mathfrak a}}$ of Łojasiewicz's radical $\sqrt[\textŁ]{\mathfrak a}$. If ${\mathcal Z}({\mathfrak a})$ has `good properties' concerning Hilbert's 17th Problem, then ${\mathcal I}({\mathcal Z}({\mathfrak a}))=\widetilde{\sqrt[\mathsf{r}]{\mathfrak a}}$ where $\sqrt[\mathsf{r}]{\mathfrak a}$ stands for the real radical of ${\mathfrak a}$. The same holds if we replace $\sqrt[\mathsf{r}]{\mathfrak a}$ with the real-analytic radical $\sqrt[\mathsf{ra}]{\mathfrak a}$ of ${\mathfrak a}$, which is a natural generalisation of the real radical ideal in the $C$-analytic setting. We revisit the classical results concerning (Hilbert's) Nullstellensatz in the framework of (complex) Stein spaces. Let ${\mathfrak a}$ be a saturated ideal of ${\mathcal O}({\mathbb R}^n)$ and $Y_{{\mathbb R}^n}$ the germ of the support of the coherent sheaf that extends ${\mathfrak a}{\mathcal O}_{{\mathbb R}^n}$ to a suitable complex open neighbourhood of ${\mathbb R}^n$. We study the relationship between a normal primary decomposition of ${\mathfrak a}$ and the decomposition of $Y_{{\mathbb R}^n}$ as the union of its irreducible components. If ${\mathfrak a}:={\mathfrak p}$ is prime, then ${\mathcal I}({\mathcal Z}({\mathfrak p}))={\mathfrak p}$ if and only if the (complex) dimension of $Y_{{\mathbb R}^n}$ coincides with the (real) dimension of ${\mathcal Z}({\mathfrak p})$.

math.AG↗
arXiv2012

A Nullstellensatz for Łojasiewicz ideals

For an ideal of smooth functions that is either Łojasiewicz or weakly Łojasiewicz, we give a complete characterization of the ideal of functions vanishing on its variety in terms of the global Łojasiewicz radical and Whitney closure. We also prove that the Łojasiewicz radical of such an ideal is analytic-like in the sense that its saturation equals its Whitney closure. This allows us to recover in a different way Nullstellensatz results due to Bochnak and Adkins-Leahy and answer positively a modification of the Nullstellensatz conjecture due to Bochnak.

math.AG↗