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Complex analytic geometry and analytic-geometric categories

Authors

Ya’acov Peterzil, Sergei Starchenko
Type: Article
Publication Date: 2009-01-01
Citations: 39
DOI: https://doi.org/10.1515/crelle.2009.002

Abstract

The notion of an analytic-geometric category was introduced by v. d. Dries and Miller in [Lou van den Dries and Chris Miller, Geometric categories and o-minimal structures, Duke Math. J. 84 (1996), no. 2, 497–540.]. It is a category of subsets of real analytic manifolds which extends the category of subanalytic sets. This paper discusses connections between the subanalytic category, or more generally analytic-geometric categories, and complex analytic geometry. The questions are of the following nature: We start with a subset A of a complex analytic manifold M and assume that A is an object of an analytic-geometric category (by viewing M as a real analytic manifold of double dimension). We then formulate conditions under which A, its closure or its image under a holomorphic map is a complex analytic set.

Locations

  • Journal für die reine und angewandte Mathematik (Crelles Journal)
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Subanalytic sets and complex analytic geometry

2004-01-01
Ya’acov Peterzil , Sergei Starchenko
We consider a subanalytic subset A of a complex analytic manifold M (when M is viewed as a real manifold) and formulate conditions under which A is a complex analytic … We consider a subanalytic subset A of a complex analytic manifold M (when M is viewed as a real manifold) and formulate conditions under which A is a complex analytic subset of M.

Towards a model theory of almost complex manifolds

2017-01-01
Michael Wan
Author(s): Wan, Michael | Advisor(s): Scanlon, Thomas | Abstract: We develop notions of complex analytic of almost complex manifolds, modelled after complex analytic subsets of complex manifolds. Basic analytic-geometric results … Author(s): Wan, Michael | Advisor(s): Scanlon, Thomas | Abstract: We develop notions of complex analytic of almost complex manifolds, modelled after complex analytic subsets of complex manifolds. Basic analytic-geometric results are presented, including an identity principle for almost complex maps, and a proof that the singular locus of an almost complex analytic set is itself an equational almost complex analytic set, under certain conditions.This work is partly motivated by geometric model theory. B. Zilber observed that a compact complex manifold, equipped with the logico-topological structure given by its complex analytic subsets, satisfies the axioms for a so-called Zariski geometry, kick-starting a fruitful model-theoretic study of complex manifolds. Our results point towards a natural generalization of Zilber’s theorem to almost complex manifolds, using our notions of almost complex analytic subset. We include a discussion of progress towards this goal.Our development draws inspiration from Y. Peterzil and S. Starchenko’s theory of nonstandard complex analytic geometry. We work primarily in the real analytic setting.

A class of complex-analytic manifolds

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Hans Samelson

Complex-analytic manifolds

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Complex-analytic manifolds

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Beno Eckmann

Complex-analytic manifolds

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Beno Eckmann

K-analytic versus ccm-analytic sets in nonstandard compact complex manifolds

2008-01-01
Rahim Moosa , Sergei Starchenko
It is shown that in an elementary extension of a compact complex manifold $M$, the $K$-analytic sets (where $K$ is the algebraic closure of the underlying real closed field) agree … It is shown that in an elementary extension of a compact complex manifold $M$, the $K$-analytic sets (where $K$ is the algebraic closure of the underlying real closed field) agree with the ccm-analytic sets if and only if $M$ is essentially saturated. In
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Tame Complex Analysis and <font>o</font>-minimality

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Ya’acov Peterzil , Sergei Starchenko

Complex manifolds and analytic spaces

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On the comparison of different notions of geometric categories

2002-01-01
Joerg Schuermann
We explain our notion of a Nash geometric category, which allows an easy comparison between the following different axiomatic notions of geometric categories: o-minimal structures on the real field, analytic … We explain our notion of a Nash geometric category, which allows an easy comparison between the following different axiomatic notions of geometric categories: o-minimal structures on the real field, analytic geometric categories and X-sets (as defined by van den Dries, Miller and Shiota).

Derived complex analytic geometry II: square-zero extensions

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Mauro Porta
We continue the explorations of derived \canal geometry started in [DAG-IX] and in http://arxiv.org/abs/1506.09042. We describe the category of $\mathcal O_X$-modules over a derived complex analytic space $X$ as the … We continue the explorations of derived \canal geometry started in [DAG-IX] and in http://arxiv.org/abs/1506.09042. We describe the category of $\mathcal O_X$-modules over a derived complex analytic space $X$ as the stabilization of a suitable category of analytic algebras over $\mathcal O_X$. Finally, we apply this description to introduce the notion of analytic square-zero extension and prove a fundamental structure theorem for them.

Complex Analytic Geometry

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Gerd Fischer
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Complex Differential Geometry

2002-01-17
Fangyang Zheng
The theory of complex manifolds overlaps with several branches of mathematics, including differential geometry, algebraic geometry, several complex variables, global analysis, topology, algebraic number theory, and mathematical physics. Complex manifolds … The theory of complex manifolds overlaps with several branches of mathematics, including differential geometry, algebraic geometry, several complex variables, global analysis, topology, algebraic number theory, and mathematical physics. Complex manifolds provide a rich class of geometric objects, for example the (common) zero locus of any generic set of complex polynomials is always a complex manifold. Yet complex manifolds behave differently than generic smooth manifolds; they are more coherent and fragile. The rich yet restrictive character of complex manifolds makes them a special and interesting object of study. This book is a self-contained graduate textbook that discusses the differential geometric aspects of complex manifolds. The first part contains standard materials from general topology, differentiable manifolds, and basic Riemannian geometry. The second part discusses complex manifolds and analytic varieties, sheaves and holomorphic vector bundles, and gives a brief account of the surface classification theory, providing readers with some concrete examples of complex manifolds.

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J. Winkelmann
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On globally defined semianalytic sets

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Francesca Acquistapace , Fabrizio Broglia , José F. Fernando
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 … 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 $\leq\dim(X)-2$.

On globally defined semianalytic sets

2015-01-01
Francesca Acquistapace , Fabrizio Broglia , José F. Fernando
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 … 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 $\leq\dim(X)-2$.
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Given a cohesive sheaf $\Cal S$ over a complex Banach manifold $M$, we endow the cohomology groups $H^q(M,\Cal S)$ of $M$ and $H^q(\frak U,\Cal S)$ of open covers $\frak U$ … Given a cohesive sheaf $\Cal S$ over a complex Banach manifold $M$, we endow the cohomology groups $H^q(M,\Cal S)$ of $M$ and $H^q(\frak U,\Cal S)$ of open covers $\frak U$ of $M$ with a locally convex topology. Under certain assumptions we prove that the canonical map $H^q(\frak U,\Cal S)\to H^q(M,\Cal S)$ is an isomorphism of topological vector spaces.
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Analytic cohomology groups of infinite dimensional complex manifolds

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Given a cohesive sheaf $\Cal S$ over a complex Banach manifold $M$, we endow the cohomology groups $H^q(M,\Cal S)$ of $M$ and $H^q(\frak U,\Cal S)$ of open covers $\frak U$ … Given a cohesive sheaf $\Cal S$ over a complex Banach manifold $M$, we endow the cohomology groups $H^q(M,\Cal S)$ of $M$ and $H^q(\frak U,\Cal S)$ of open covers $\frak U$ of $M$ with a locally convex topology. Under certain assumptions we prove that the canonical map $H^q(\frak U,\Cal S)\to H^q(M,\Cal S)$ is an isomorphism of topological vector spaces.

Complex varieties and the analytic topology

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Complex manifolds and analytic varieties

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On the closure of the positive Hodge locus

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Bruno Klingler , Anna Otwinowska
Given a variation of Hodge structures on a quasi-projective base $S$, whose generic Mumford-Tate group is non-product, we prove that the (countable) union of positive components of the Hodge locus … Given a variation of Hodge structures on a quasi-projective base $S$, whose generic Mumford-Tate group is non-product, we prove that the (countable) union of positive components of the Hodge locus is either an algebraic subvariety of $S$, or is Zariski-dense in $S$.

The Ax-Schanuel conjecture for variations of mixed Hodge structures

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Ziyang Gao , Bruno Klingler
We prove in this paper the Ax-Schanuel conjecture for all admissible variations of mixed Hodge structures. We prove in this paper the Ax-Schanuel conjecture for all admissible variations of mixed Hodge structures.
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Mixed Ax-Schanuel for the universal abelian varieties and some applications

2018-01-01
Ziyang Gao
In this paper we prove the mixed Ax-Schanuel theorem for the universal abelian varieties (more generally any mixed Shimura variety of Kuga type), and give some simple applications. In particular … In this paper we prove the mixed Ax-Schanuel theorem for the universal abelian varieties (more generally any mixed Shimura variety of Kuga type), and give some simple applications. In particular we present an application to studying the generic rank of the Betti map.
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Complex analytic geometry in a nonstandard setting

2008-05-22
Ya’acov Peterzil , Sergei Starchenko
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content. A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.

Definability of restricted theta functions and families of abelian varieties

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Ya’acov Peterzil , Sergei Starchenko
We consider some classical maps from the theory of abelian varieties and their moduli spaces, and we prove their definability on restricted domains in the o-minimal structure Ran,exp. In particular, … We consider some classical maps from the theory of abelian varieties and their moduli spaces, and we prove their definability on restricted domains in the o-minimal structure Ran,exp. In particular, we prove that the projective embedding of the moduli space of the principally polarized abelian variety Sp(2g,Z)\Hg is definable in Ran,exp when restricted to Siegel's fundamental set Fg. We also prove the definability on appropriate domains of embeddings of families of abelian varieties into projective spaces.
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Tame Complex Analysis and <font>o</font>-minimality

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Mild manifolds and a non-standard Riemann existence theorem

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Tameness of holomorphic closure dimension in a semialgebraic set

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Big Picard theorem and algebraic hyperbolicity for varieties admitting a variation of Hodge structures

2020-01-13
Ya Deng
In this paper we study three notions of hyperbolicity for varieties admitting complex polarized variation of Hodge structures ($\mathbb{C}$-PVHS for short). In the first part we prove that if a … In this paper we study three notions of hyperbolicity for varieties admitting complex polarized variation of Hodge structures ($\mathbb{C}$-PVHS for short). In the first part we prove that if a quasi-projective manifold $U$ admits a $\mathbb{C}$-PVHS whose period map is quasi-finite, then $U$ is algebraically hyperbolic in the sense of Demailly, and that the generalized big Picard theorem holds for $U$: any holomorphic map $f:\Delta-\{0\}\to U$ from the punctured unit disk to $U$ extends to a holomorphic map of the unit disk $\Delta$ into any projective compactification of $U$. This result generalizes a recent work by Bakker-Brunebarbe-Tsimerman. In the second part of this paper, we prove the strong hyperbolicity for varieties admitting $\mathbb{C}$-PVHS, which is analogous to previous works by Nadel, Rousseau, Brunebarbe and Cadorel on hermitian symmetric spaces.

Holomorphic Lagrangian branes correspond to perverse sheaves

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Xin Jin
Holomorphic Lagrangian branes correspond Holomorphic Lagrangian branes correspond
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Mixed Ax–Schanuel for the universal abelian varieties and some applications

2020-11-01
Ziyang Gao
In this paper we prove the mixed Ax–Schanuel theorem for the universal abelian varieties (more generally any mixed Shimura variety of Kuga type), and give some simple applications. In particular, … In this paper we prove the mixed Ax–Schanuel theorem for the universal abelian varieties (more generally any mixed Shimura variety of Kuga type), and give some simple applications. In particular, we present an application for studying the generic rank of the Betti map.
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Algebraic differential equations from covering maps

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A remark on almost complex manifold with linear connections

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Kailash Chandra Petwal , Vineet Bhatt , Sunil Kumar +1 more
The complex and almost complex manifolds are enormous and very fruitful fields for differential geometry. J.A. Schouten and D. Van Dantzig were the first to try to apply the finding … The complex and almost complex manifolds are enormous and very fruitful fields for differential geometry. J.A. Schouten and D. Van Dantzig were the first to try to apply the finding in differential geometry of spaces with Riemannian metric and affine connection to the situation of complex structure spaces. C. Ehresmann defined an almost complex space as an even-dimensional differentiable manifold containing a tensor field with a square root of minus unity. The present paper intended to study, some fundamental properties with linear connections of an almost complex manifold. If almost complex structure be converted from a complex structure, then the various integrable and completely integrable condition has been investigated. Furthermore, the symmetric affine connections of almost complex manifold have also been investigated.
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Big Picard theorems and algebraic hyperbolicity for varieties admitting a variation of Hodge structures

2023-04-24
Ya Deng
In this paper, we study various hyperbolicity properties for a quasi-compact K\"ahler manifold $U$ which admits a complex polarized variation of Hodge structures so that each fiber of the period … In this paper, we study various hyperbolicity properties for a quasi-compact K\"ahler manifold $U$ which admits a complex polarized variation of Hodge structures so that each fiber of the period map is zero-dimensional. In the first part, we prove that $U$ is algebraically hyperbolic and that the generalized big Picard theorem holds for $U$. In the second part, we prove that there is a finite \'etale cover $\tilde{U}$ of $U$ from a quasi-projective manifold $\tilde{U}$ such that any projective compactification $X$ of $\tilde{U}$ is Picard hyperbolic modulo the boundary $X-\tilde{U}$, and any irreducible subvariety of $X$ not contained in $X-\tilde{U}$ is of general type. This result coarsely incorporates previous works by Nadel, Rousseau, Brunebarbe and Cadorel on the hyperbolicity of compactifications of quotients of bounded symmetric domains by torsion-free lattices.
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Big Picard theorems and algebraic hyperbolicity for varieties admitting a variation of Hodge structures

2020-01-01
Ya Deng
In this paper, we study various hyperbolicity properties for a quasi-compact K\"ahler manifold $U$ which admits a complex polarized variation of Hodge structures so that each fiber of the period … In this paper, we study various hyperbolicity properties for a quasi-compact K\"ahler manifold $U$ which admits a complex polarized variation of Hodge structures so that each fiber of the period map is zero-dimensional. In the first part, we prove that $U$ is algebraically hyperbolic and that the generalized big Picard theorem holds for $U$. In the second part, we prove that there is a finite \'etale cover $\tilde{U}$ of $U$ from a quasi-projective manifold $\tilde{U}$ such that any projective compactification $X$ of $\tilde{U}$ is Picard hyperbolic modulo the boundary $X-\tilde{U}$, and any irreducible subvariety of $X$ not contained in $X-\tilde{U}$ is of general type. This result coarsely incorporates previous works by Nadel, Rousseau, Brunebarbe and Cadorel on the hyperbolicity of compactifications of quotients of bounded symmetric domains by torsion-free lattices.
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Tame topology of arithmetic quotients and algebraicity of Hodge loci

2018-01-01
Bruno Klingler
We prove that the uniformizing map of any arithmetic quotient, as well as the period map associated to any pure polarized $\mathbb{Z}$-variation of Hodge structure $\mathbb{V}$ on a smooth complex … We prove that the uniformizing map of any arithmetic quotient, as well as the period map associated to any pure polarized $\mathbb{Z}$-variation of Hodge structure $\mathbb{V}$ on a smooth complex quasi-projective variety $S$, are topologically tame. As an easy corollary of these results and of Peterzil-Starchenko's o-minimal GAGA theorem we obtain that the Hodge locus of $(S, \mathbb{V})$ is a countable union of algebraic subvarieties of $S$ (a result originally due to Cattani-Deligne-Kaplan).

Some Recent Results on Holomorphic Isometries of the Complex Unit Ball into Bounded Symmetric Domains and Related Problems

2018-01-01
Ngaiming Mok

On the closure of the Hodge locus of positive period dimension

2021-03-29
Bruno Klingler , A. Otwinowska
Abstract Given $${{\mathbb {V}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>V</mml:mi> </mml:math> a polarizable variation of $${{\mathbb {Z}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Z</mml:mi> </mml:math> -Hodge structures on a smooth connected complex quasi-projective variety S , the … Abstract Given $${{\mathbb {V}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>V</mml:mi> </mml:math> a polarizable variation of $${{\mathbb {Z}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Z</mml:mi> </mml:math> -Hodge structures on a smooth connected complex quasi-projective variety S , the Hodge locus for $${{\mathbb {V}}}^\otimes $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>V</mml:mi> </mml:mrow> <mml:mo>⊗</mml:mo> </mml:msup> </mml:math> is the set of closed points s of S where the fiber $${{\mathbb {V}}}_s$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>V</mml:mi> <mml:mi>s</mml:mi> </mml:msub> </mml:math> has more Hodge tensors than the very general one. A classical result of Cattani, Deligne and Kaplan states that the Hodge locus for $${{\mathbb {V}}}^\otimes $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>V</mml:mi> </mml:mrow> <mml:mo>⊗</mml:mo> </mml:msup> </mml:math> is a countable union of closed irreducible algebraic subvarieties of S , called the special subvarieties of S for $${{\mathbb {V}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>V</mml:mi> </mml:math> . Under the assumption that the adjoint group of the generic Mumford–Tate group of $${{\mathbb {V}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>V</mml:mi> </mml:math> is simple we prove that the union of the special subvarieties for $${{\mathbb {V}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>V</mml:mi> </mml:math> whose image under the period map is not a point is either a closed algebraic subvariety of S or is Zariski-dense in S . This implies for instance the following typical intersection statement: given a Hodge-generic closed irreducible algebraic subvariety S of the moduli space $${{\mathcal {A}}}_g$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>g</mml:mi> </mml:msub> </mml:math> of principally polarized Abelian varieties of dimension g , the union of the positive dimensional irreducible components of the intersection of S with the strict special subvarieties of $${{\mathcal {A}}}_g$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>g</mml:mi> </mml:msub> </mml:math> is either a closed algebraic subvariety of S or is Zariski-dense in S .
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Special subvarieties of non-arithmetic ball quotients and Hodge theory

2022-11-23
Gregorio Baldi , Emmanuel Ullmo
Let $\Gamma \subset \mathrm{PU}(1,n)$ be a lattice and $S_\Gamma$ be the associated ball quotient. We prove that, if $S_\Gamma$ contains infinitely many maximal complex totally geodesic subvarieties, then $\Gamma$ is … Let $\Gamma \subset \mathrm{PU}(1,n)$ be a lattice and $S_\Gamma$ be the associated ball quotient. We prove that, if $S_\Gamma$ contains infinitely many maximal complex totally geodesic subvarieties, then $\Gamma$ is arithmetic. We also prove an Ax--Schanuel Conjecture for $S_\Gamma$, similar to the one recently proven by Mok, Pila and Tsimerman. One of the main ingredients in the proofs is to realise $S_\Gamma$ inside a period domain for polarised integral variations of Hodge structure and interpret totally geodesic subvarieties as unlikely intersections.
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Global complexification of real analytic globally subanalytic functions

2016-04-15
Tobias Kaiser

The Ax–Schanuel conjecture for variations of mixed Hodge structures

2023-04-17
Ziyang Gao , Bruno Klingler
Abstract We prove in this paper, the Ax–Schanuel conjecture for all admissible variations of mixed Hodge structures. Abstract We prove in this paper, the Ax–Schanuel conjecture for all admissible variations of mixed Hodge structures.
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Existence and density of typical Hodge loci

2024-07-15
Nazim Khelifa , David Urbanik

Algebraic differential equations from covering maps

2014-08-22
Thomas Scanlon
Let $Y$ be a complex algebraic variety, $G \curvearrowright Y$ an action of an algebraic group on $Y$, $U \subseteq Y({\mathbb C})$ a complex submanifold, $\Gamma < G({\mathbb C})$ a … Let $Y$ be a complex algebraic variety, $G \curvearrowright Y$ an action of an algebraic group on $Y$, $U \subseteq Y({\mathbb C})$ a complex submanifold, $\Gamma < G({\mathbb C})$ a discrete, Zariski dense subgroup of $G({\mathbb C})$ which preserves $U$, and $\pi:U \to X({\mathbb C})$ an analytic covering map of the complex algebraic variety $X$ expressing $X({\mathbb C})$ as $\Gamma \backslash U$. We note that the theory of elimination of imaginaries in differentially closed fields produces a generalized Schwarzian derivative $\widetilde{\chi}:Y \to Z$ (where $Z$ is some algebraic variety) expressing the quotient of $Y$ by the action of the constant points of $G$. Under the additional hypothesis that the restriction of $\pi$ to some set containing a fundamental domain is definable in an o-minimal expansion of the real field, we show as a consequence of the Peterzil-Starchenko o-minimal GAGA theorem that the \emph{prima facie} differentially analytic relation $\chi := \widetilde{\chi} \circ \pi^{-1}$ is a well-defined, differential constructible function. The function $\chi$ nearly inverts $\pi$ in the sense that for any differential field $K$ of meromorphic functions, if $a, b \in X(K)$ then $\chi(a) = \chi(b)$ if and only if after suitable restriction there is some $\gamma \in G({\mathbb C})$ with $\pi(\gamma \cdot \pi^{-1}(a)) = b$.

Effective transcendental Zilber-Pink for variations of Hodge structures

2021-05-12
Jonathan Pila , Thomas Scanlon
We prove function field versions of the Zilber-Pink conjectures for varieties supporting a variation of Hodge structures. A form of these results for Shimura varieties in the context of unlikely … We prove function field versions of the Zilber-Pink conjectures for varieties supporting a variation of Hodge structures. A form of these results for Shimura varieties in the context of unlikely intersections is the following. Let $S$ be a connected pure Shimura variety with a fixed quasiprojective embedding. We show that there is an explicitly computable function $B$ of two natural number arguments so that for any field extension $K$ of the complex numbers and Hodge generic irreducible proper subvariety $X \subsetneq S_K$, the set of nonconstant points in the intersection of $X$ with the union of all special subvarieties of $X$ of dimension less than the codimension of $X$ in $S$ is contained in a proper subvariety of $X$ of degree bounded by $B(\operatorname{deg}(X),\dim(X))$. Our techniques are differential algebraic and rely on Ax-Schanuel functional transcendence theorems. We use these results to show that the differential equations associated with Shimura varieties give new examples of minimal, and sometimes, strongly minimal, types with trivial forking geometry but non-$\aleph_0$-categorical induced structure.

Rational Points of Definable Sets and Results of Andre-Oort-Manin-Mumford type

2009-03-05
Jonathan Pila
Journal Article Rational Points of Definable Sets and Results of André–Oort–Manin–Mumford type Get access Jonathan Pila Jonathan Pila School of Mathematics, University of Bristol, Bristol BS8 1TW, UK Correspondence to … Journal Article Rational Points of Definable Sets and Results of André–Oort–Manin–Mumford type Get access Jonathan Pila Jonathan Pila School of Mathematics, University of Bristol, Bristol BS8 1TW, UK Correspondence to be sent to: [email protected] Search for other works by this author on: Oxford Academic Google Scholar International Mathematics Research Notices, Volume 2009, Issue 13, 2009, Pages 2476–2507, https://doi.org/10.1093/imrn/rnp022 Published: 05 March 2009 Article history Received: 24 October 2008 Revision received: 28 January 2009 Accepted: 30 January 2009 Published: 05 March 2009

Tame topology of arithmetic quotients and algebraicity of Hodge loci

2020-07-08
Benjamin Bakker , Bruno Klingler , Jacob Tsimerman
In this paper we prove the following results: <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> We show that any arithmetic … In this paper we prove the following results: <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> We show that any arithmetic quotient of a homogeneous space admits a natural real semi-algebraic structure for which its Hecke correspondences are semi-algebraic. A particularly important example is given by Hodge varieties, which parametrize pure polarized integral Hodge structures. <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> We prove that the period map associated to any pure polarized variation of integral Hodge structures <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper V"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">V</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {V}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on a smooth complex quasi-projective variety <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is definable with respect to an o-minimal structure on the relevant Hodge variety induced by the above semi-algebraic structure. <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mn>3</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">3)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> As a corollary of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and of Peterzil-Starchenko’s o-minimal Chow theorem we recover that the Hodge locus of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper S comma double-struck upper V right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>S</mml:mi> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">V</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(S, \mathbb {V})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a countable union of algebraic subvarieties of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, a result originally due to Cattani-Deligne-Kaplan. Our approach simplifies the proof of Cattani-Deligne-Kaplan, as it does not use the full power of the difficult multivariable <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S upper L 2"> <mml:semantics> <mml:mrow> <mml:mi>S</mml:mi> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">SL_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-orbit theorem of Cattani-Kaplan-Schmid.
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ENLARGED MIXED SHIMURA VARIETIES, BI-ALGEBRAIC SYSTEM AND SOME AX TYPE TRANSCENDENTAL RESULTS

2019-01-01
Ziyang Gao
We develop a theory of enlarged mixed Shimura varieties, putting the universal vectorial bi-extension defined by Coleman into this framework to study some functional transcendental results of Ax type. We … We develop a theory of enlarged mixed Shimura varieties, putting the universal vectorial bi-extension defined by Coleman into this framework to study some functional transcendental results of Ax type. We study their bi-algebraic systems, formulate the Ax-Schanuel conjecture and explain its relation with the logarithmic Ax theorem and the Ax-Lindemann theorem which we shall prove. All these bi-algebraic and transcendental results extend their counterparts for mixed Shimura varieties. In the end we briefly discuss the André–Oort and Zilber–Pink type problems for enlarged mixed Shimura varieties.
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Piecewise Weierstrass preparation and division for o-minimal holomorphic functions

2016-10-17
Tobias Kaiser
Given an o-minimal structure expanding the field of reals, we show a piecewise Weierstrass preparation theorem and a piecewise Weierstrass division theorem for definable holomorphic functions. In the semialgebraic setting … Given an o-minimal structure expanding the field of reals, we show a piecewise Weierstrass preparation theorem and a piecewise Weierstrass division theorem for definable holomorphic functions. In the semialgebraic setting and for the structure of globally subanalytic sets and functions we obtain the corresponding results for definable real analytic functions. As an application we show a definable global Nullstellensatz for principal ideals.
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Ax–Schanuel for Shimura Varieties

2022-05-26
Jonathan Pila
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content. A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
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Volumes of definable sets in o-minimal expansions and affine GAGA theorems

2023-01-01
Patrick Brosnan
In this mostly expository note, I give a quick proof of the definable Chow theorem of Peterzil and Starchenko using the Bishop-Stoll theorem and a volume estimate for definable sets … In this mostly expository note, I give a quick proof of the definable Chow theorem of Peterzil and Starchenko using the Bishop-Stoll theorem and a volume estimate for definable sets due to Nguyen and Valette. The volume estimate says that any <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d"> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding="application/x-tex">d</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-dimensional definable subset of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S subset-of-or-equal-to double-struck upper R Superscript n"> <mml:semantics> <mml:mrow> <mml:mi>S</mml:mi> <mml:mo>⊆<!-- ⊆ --></mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">S\subseteq \mathbb {R}^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in an o-minimal expansion of the ordered field of real numbers satisfies the inequality <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper H Superscript d Baseline left-parenthesis StartSet x element-of upper S colon double-vertical-bar x double-vertical-bar greater-than r EndSet right-parenthesis less-than-or-equal-to upper C r Superscript d"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">H</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mi>x</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mi>S</mml:mi> <mml:mo>:</mml:mo> <mml:mo fence="false" stretchy="false">‖<!-- ‖ --></mml:mo> <mml:mi>x</mml:mi> <mml:mo fence="false" stretchy="false">‖<!-- ‖ --></mml:mo> <mml:mo>&gt;</mml:mo> <mml:mi>r</mml:mi> <mml:mo fence="false" stretchy="false">}</mml:mo> <mml:mo stretchy="false">)</mml:mo> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mi>C</mml:mi> <mml:msup> <mml:mi>r</mml:mi> <mml:mi>d</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {H}^d(\{x\in S:\lVert x\rVert &gt;r\})\leq Cr^d</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper H Superscript d"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">H</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\mathcal {H}^d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> denotes the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d"> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding="application/x-tex">d</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-dimensional Hausdorff measure on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Superscript n"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {R}^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C"> <mml:semantics> <mml:mi>C</mml:mi> <mml:annotation encoding="application/x-tex">C</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a constant depending on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Since this note is intended to be helpful to algebraic geometers not versed in o-minimal structures and definable sets, I review these notions and also prove the main volume estimate from scratch.
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Unstable structures definable in o-minimal theories

2010-03-09
Assaf Hasson , Alf Onshuus
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Ax-Lindemann-Weierstrass with derivatives and the genus 0 Fuchsian groups

2020-11-01
Guy Casale , James Freitag , Joel Nagloo
We prove the Ax-Lindemann-Weierstrass theorem with derivatives for the uniformizing functions of genus zero Fuchsian groups of the first kind. Our proof relies on differential Galois theory, monodromy of linear … We prove the Ax-Lindemann-Weierstrass theorem with derivatives for the uniformizing functions of genus zero Fuchsian groups of the first kind. Our proof relies on differential Galois theory, monodromy of linear differential equations, the study of algebraic and Liouvillian solutions, differential algebraic work of Nishioka towards the Painlevé irreducibility of certain Schwarzian equations, and considerable machinery from the model theory of differentially closed fields. Our techniques allow for certain generalizations of the Ax-Lindemann-Weierstrass theorem that have interesting consequences. In particular, we apply our results to give a complete proof of an assertion of Painlevé (1895). We also answer certain cases of the André-Pink conjecture, namely, in the case of orbits of commensurators of Fuchsian groups.
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Ax–Schanuel for the j-function

2016-06-10
Jonathan Pila , Jacob Tsimerman
In this paper we prove a functional transcendence statement for the j-function which is an analogue of the Ax-Schanuel theorem for the exponential function. It asserts, roughly, that atypical algebraic … In this paper we prove a functional transcendence statement for the j-function which is an analogue of the Ax-Schanuel theorem for the exponential function. It asserts, roughly, that atypical algebraic relations among functions and their compositions with the j-function are governed by modular relations.
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Asymptotic Hodge Theory in String Compactifications and Integrable Systems

2024-08-14
Jeroen Monnee
String theory is a proposal for a description of nature on the smallest length scales, where both quantum mechanics and gravity are expected to play an essential role. A curious … String theory is a proposal for a description of nature on the smallest length scales, where both quantum mechanics and gravity are expected to play an essential role. A curious feature of the theory is that its basic building blocks – one-dimensional strings – behave as if they were moving in nine-dimensional space, as opposed to the three-dimensional space we are familiar with. String theory proposes that the six additional dimensions are curled up into extremely small sizes, in a process called “compactification”. Importantly, what happens in these hidden six dimensions has an enormous influence on the physics we observe at larger length scales, including the strength of the interactions between elementary particles and the value of the cosmological constant. An intriguing aspect of string theory is that there are countless possible ways to compactify these extra dimensions, resulting in a vast "landscape" of potential universes, each with distinct physical properties. In a beautiful interplay of physics and mathematics, the physical characteristics of these universes are intricately linked to the geometric properties of the internal six-dimensional space. The main aim of this thesis is to study the latter using the sophisticated mathematical framework of asymptotic Hodge theory. One of the central results is that one can obtain a good approximation of numerous physical observables by studying the allowed singularities of the six-dimensional internal space. In accordance with some fundamental theorems in the field of algebraic geometry, these can be classified in great generality and reveal intriguing underlying structures. In particular, this abstract approach leads to a comprehensive, algorithmic method for calculating physical observables in string compactifications. In this way, we obtain a very general characterization of universes that can be constructed in string theory and find that they are in fact finite in number. Additionally, based on recent advances in the field of tame geometry it is suggested that this number might be much smaller than previously expected. Finally, we observe that the same mathematical structures applied above in the context of string compactifications arise in a completely different corner of physics: the study of integrable non-linear sigma models. In particular, the same computational methods used to compute physical observables in string theory can be used to find previously unknown solutions for certain classes of such models. Our results open up a new avenue of research, bridging two distinct areas of theoretical physics.
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Exact flux vacua, symmetries, and the structure of the landscape

2025-01-02
Thomas W. Grimm , Damian van de Heisteeg
A bstract Identifying flux vacua in string theory with stabilized complex structure moduli presents a significant challenge, necessitating the minimization of a scalar potential complicated by infinitely many exponential corrections. … A bstract Identifying flux vacua in string theory with stabilized complex structure moduli presents a significant challenge, necessitating the minimization of a scalar potential complicated by infinitely many exponential corrections. In order to obtain exact results we connect three central topics: transcendentality or algebraicity of coupling functions, emergent symmetries, and the distribution of vacua. Beginning with explicit examples, we determine the first exact landscape of flux vacua with a vanishing superpotential within F-theory compactifications on a genuine Calabi-Yau fourfold. We find that along certain symmetry loci in moduli space the generically transcendental vacuum conditions become algebraic and can be described using the periods of a K3 surface. On such loci the vacua become dense when we do not bound the flux tadpole, while imposing the tadpole bound yields a small finite landscape of distinct vacua. Away from these symmetry loci, the transcendentality of the fourfold periods ensures that there are only a finite number of vacua with a vanishing superpotential, even when the tadpole constraint is removed. These observations exemplify the general patterns emerging in the bulk of moduli space that we expose in this work. They are deeply tied to the arithmetic structure underlying flux vacua and generalize the finiteness claims about rational CFTs and rank-two attractors. From a mathematical perspective, our study is linked with the recent landmark results by Baldi, Klingler, and Ullmo about the Hodge locus that arose from connecting tame geometry and Hodge theory.
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Quasianalytic Denjoy-Carleman classes and o-minimality

2003-03-21
Jean-Philippe Rolin , Patrick Speissegger , A. J. Wilkie
We show that the expansion of the real field generated by the functions of a quasianalytic Denjoy-Carleman class is model complete and o-minimal, provided that the class satisfies certain closure … We show that the expansion of the real field generated by the functions of a quasianalytic Denjoy-Carleman class is model complete and o-minimal, provided that the class satisfies certain closure conditions. Some of these structures do not admit analytic cell decomposition, and they show that there is no largest o-minimal expansion of the real field.
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On the douady space of a compact complex space in the category

1982-03-01
Akira Fujiki
Let X be a complex space. Let D x be the Douady space of compact complex subspaces of X [6] and p x : Z x → D x the … Let X be a complex space. Let D x be the Douady space of compact complex subspaces of X [6] and p x : Z x → D x the corresponding universal family of subspaces of X .
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Complex Analytic Sets

1989-07-31
E. M. Chirka
The theory of complex analytic sets is part of the modern geometric theory of functions of several complex variables. Traditionally, the presentation of the foundations of the theory of analytic … The theory of complex analytic sets is part of the modern geometric theory of functions of several complex variables. Traditionally, the presentation of the foundations of the theory of analytic sets is introduced in the algebraic language of ideals in Noetherian rings as, for example, in the books of Herve [23] or Gunning-Rossi [19]. However, the modern methods of this theory, the principal directions and applications, are basically related to geometry and analysis (without regard to the traditional direction which is essentially related to algebraic geometry). Thus, at the beginning of this survey, the geometric construction of the local theory of analytic sets is presented. Its foundations are worked out in detail in the book of Gunning and Rossi [19] via the notion of analytic cover which together with analytic theorems on the removal of singularities leads to the minimum of algebraic apparatus necessary in order to get the theory started.

Semilinear and semialgebraic loci of o-minimal sets

2001-10-01
Artur Piękosz
We consider some semilinear (= semiaffine) and semialgebraic loci of o-minimal sets in euclidean spaces. Semilinear loci have good properties. Some of these properties hold for semialgebraic loci when we … We consider some semilinear (= semiaffine) and semialgebraic loci of o-minimal sets in euclidean spaces. Semilinear loci have good properties. Some of these properties hold for semialgebraic loci when we restrict to a smaller class of analytically o-minimal sets.
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Jet spaces in complex analytic geometry: an exposition

2004-01-01
Rahim Moosa
These notes are an exposition and synthesis of various "jet space" constructions in complex analytic geometry. They are written primarily for model-theorists interested in the results of Campana and Fujiki … These notes are an exposition and synthesis of various "jet space" constructions in complex analytic geometry. They are written primarily for model-theorists interested in the results of Campana and Fujiki (whose model-theoretic significance has been observed by Pillay).

Model-theoretic consequences of a theorem of Campana and Fujiki

2002-01-01
Anand Pillay
We give a model-theoretic interpretation of a result by Campana and Fujiki on the algebraicity of certain spaces of cycles on compact complex spaces. The model-theoretic interpretation is in the … We give a model-theoretic interpretation of a result by Campana and Fujiki on the algebraicity of certain spaces of cycles on compact complex spaces. The model-theoretic interpretation is in the language of canonical bases, and says that if $b,c$ are tupl
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Uniformly amenable discrete groups

1980-02-01
Marek Bożejko

Exponentiation is hard to avoid

1994-01-01
Chris Miller
Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper R"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be an O-minimal expansion of the field of real … Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper R"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be an O-minimal expansion of the field of real numbers. If <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper R"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is not polynomially bounded, then the exponential function is definable (without parameters) in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper R"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. If <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper R"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is polynomially bounded, then for every definable function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f colon double-struck upper R right-arrow double-struck upper R"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>:</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mo stretchy="false">→<!-- → --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">f:\mathbb {R} \to \mathbb {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <italic>f</italic> not ultimately identically 0, there exist <italic>c</italic>, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r element-of double-struck upper R comma c not-equals 0"> <mml:semantics> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mo>,</mml:mo> <mml:mi>c</mml:mi> <mml:mo>≠<!-- ≠ --></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">r \in \mathbb {R},c \ne 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x right-arrow from bar x Superscript r Baseline colon left-parenthesis 0 comma plus normal infinity right-parenthesis right-arrow double-struck upper R"> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo stretchy="false">↦<!-- ↦ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>x</mml:mi> <mml:mi>r</mml:mi> </mml:msup> </mml:mrow> <mml:mo>:</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mo>+</mml:mo> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">→<!-- → --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">x \mapsto {x^r}:(0, + \infty ) \to \mathbb {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is definable in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper R"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Underscript x right-arrow plus normal infinity Endscripts f left-parenthesis x right-parenthesis slash x Superscript r Baseline equals c"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:munder> <mml:mo movablelimits="true" form="prefix">lim</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>x</mml:mi> <mml:mo stretchy="false">→<!-- → --></mml:mo> <mml:mo>+</mml:mo> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> </mml:mrow> </mml:munder> </mml:mrow> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>x</mml:mi> <mml:mi>r</mml:mi> </mml:msup> </mml:mrow> <mml:mo>=</mml:mo> <mml:mi>c</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">{\lim _{x \to + \infty }}f(x)/{x^r} = c</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.
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Geometric categories and o-minimal structures

1996-08-01
Lou van den Dries , Chris Miller

Algébricité et compacité dans l'espace des cycles d'un espace analytique complexe

1980-02-01
F. Campana

Infinite differentiability in polynomially bounded o-minimal structures

1995-01-01
Chris Miller
Infinitely differentiable functions definable in a polynomially bounded o-minimal expansion <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal fraktur upper R"> <mml:semantics> <mml:mi mathvariant="normal">ℜ</mml:mi> <mml:annotation encoding="application/x-tex">\Re</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the ordered field … Infinitely differentiable functions definable in a polynomially bounded o-minimal expansion <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal fraktur upper R"> <mml:semantics> <mml:mi mathvariant="normal">ℜ</mml:mi> <mml:annotation encoding="application/x-tex">\Re</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the ordered field of real numbers are shown to have some of the nice properties of real analytic functions. In particular, if a definable function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f colon double-struck upper R Superscript n Baseline right-arrow double-struck upper R"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>:</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:mo stretchy="false">→</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">f:{\mathbb {R}^n} \to \mathbb {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript upper N"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>C</mml:mi> <mml:mi>N</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{C^N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> at <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a element-of double-struck upper R Superscript n"> <mml:semantics> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>∈</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">a \in {\mathbb {R}^n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for all <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N element-of double-struck upper N"> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>∈</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">N</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">N \in \mathbb {N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and all partial derivatives of <italic>f</italic> vanish at <italic>a</italic>, then <italic>f</italic> vanishes identically on some open neighborhood of <italic>a</italic>. Combining this with the Abhyankar-Moh theorem on convergence of power series, it is shown that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal fraktur upper R"> <mml:semantics> <mml:mi mathvariant="normal">ℜ</mml:mi> <mml:annotation encoding="application/x-tex">\Re</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a polynomially bounded o-minimal expansion of the field of real numbers with restricted analytic functions, then all <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript normal infinity"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>C</mml:mi> <mml:mi mathvariant="normal">∞</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{C^\infty }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> functions definable in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal fraktur upper R"> <mml:semantics> <mml:mi mathvariant="normal">ℜ</mml:mi> <mml:annotation encoding="application/x-tex">\Re</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are real analytic, provided that this is true for all definable functions of one variable.
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Expansions of algebraically closed fields in o-minimal structures

2001-09-01
Ya’acov Peterzil , Sergei Starchenko

A theorem of the complement and some new o-minimal structures

1999-12-01
A. J. Wilkie

The Field of Reals with Multisummable Series and the Exponential Function

2000-11-01
Lou van den Dries , Patrick Speissegger
We show that the field of real numbers with multisummable real power series is model complete, o-minimal and polynomially bounded. Further expansion by the exponential function yields again a model … We show that the field of real numbers with multisummable real power series is model complete, o-minimal and polynomially bounded. Further expansion by the exponential function yields again a model complete and o-minimal structure which is exponentially bounded, and in which the Gamma function on the positive real line is definable. 2000 Mathematics Subject Classification: primary 03C10, 32B05, 32B20; secondary, 26E05.

EXPANSIONS OF ALGEBRAICALLY CLOSED FIELDS II: FUNCTIONS OF SEVERAL VARIABLES

2003-05-01
Ya’acov Peterzil , Sergei Starchenko
Let ℛ be an o-minimal expansion of a real closed field R. We continue here the investigation we began in [11] of differentiability with respect to the algebraically closed field … Let ℛ be an o-minimal expansion of a real closed field R. We continue here the investigation we began in [11] of differentiability with respect to the algebraically closed field [Formula: see text]. We develop the basic theory of such K-differentiability for definable functions of several variables, proving theorems on removable singularities as well as analogues of the Weierstrass preparation and division theorems for definable functions. We consider also definably meromorphic functions and prove that every definable function which is meromorphic on K n is necessarily a rational function. We finally discuss definable analogues of complex analytic manifolds, with possible connections to the model theoretic work on compact complex manifolds, and present two examples of "nonstandard manifolds" in our setting.

Tame Topology and O-minimal Structures

1998-05-07
L. P. D. van den Dries
Following their introduction in the early 1980s o-minimal structures were found to provide an elegant and surprisingly efficient generalization of semialgebraic and subanalytic geometry. These notes give a self-contained treatment … Following their introduction in the early 1980s o-minimal structures were found to provide an elegant and surprisingly efficient generalization of semialgebraic and subanalytic geometry. These notes give a self-contained treatment of the theory of o-minimal structures from a geometric and topological viewpoint, assuming only rudimentary algebra and analysis. The book starts with an introduction and overview of the subject. Later chapters cover the monotonicity theorem, cell decomposition, and the Euler characteristic in the o-minimal setting and show how these notions are easier to handle than in ordinary topology. The remarkable combinatorial property of o-minimal structures, the Vapnik-Chervonenkis property, is also covered. This book should be of interest to model theorists, analytic geometers and topologists.

The Pfaffian closure of an o-minimal structure

1999-03-12
Patrick Speissegger
Abstract Every o-minimal expansion of the real field has an o-minimal expansion in which the solutions to Pfaffian equations with definable C 1 coefficients are definable. Abstract Every o-minimal expansion of the real field has an o-minimal expansion in which the solutions to Pfaffian equations with definable C 1 coefficients are definable.

TAME TOPOLOGY AND O-MINIMAL STRUCTURES (London Mathematical Society Lecture Note Series 248)

2000-05-01
A. J. Wilkie
By Lou van den Dries: 180 pp., £24.95 (US$39.95, LMS Members' price £18.70), isbn 0 521 59838 9 (Cambridge University Press 1998). By Lou van den Dries: 180 pp., £24.95 (US$39.95, LMS Members' price £18.70), isbn 0 521 59838 9 (Cambridge University Press 1998).

Quasi-convex decomposition in o-minimal structures. Application to the gradient conjecture

2019-03-06
Krzysztof Kurdyka , Adam Parusiński
<!-- *** Custom HTML *** --> We show that every subset of $\mathbb{R}^n$ definable in an o-minimal structure can be decomposed into a finite number of definable sets that are … <!-- *** Custom HTML *** --> We show that every subset of $\mathbb{R}^n$ definable in an o-minimal structure can be decomposed into a finite number of definable sets that are quasi-convex i.e. have comparable, up to a constant, the intrinsic distance and the distance induced from the embedding. We apply this result to study the limits of secants of the trajectories of gradient vector field $\nabla f$ of a $C^1$ definable function $f$ defined in an open subset of $\mathbb{R}^n$. We show that if the o-minimal structure is polynomially bounded then the limit of such secants exists, that is an analog of the gradient conjecture of R. Thom holds. Moreover we prove that for $n = 2$ the result is true in any o-minimal structure.
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Exponentiation is Hard to Avoid

1994-09-01
Chris Miller
Let $\mathcal {R}$ be an O-minimal expansion of the field of real numbers. If $\mathcal {R}$ is not polynomially bounded, then the exponential function is definable (without parameters) in $\mathcal … Let $\mathcal {R}$ be an O-minimal expansion of the field of real numbers. If $\mathcal {R}$ is not polynomially bounded, then the exponential function is definable (without parameters) in $\mathcal {R}$. If $\mathcal {R}$ is polynomially bounded, then for every definable function $f:\mathbb {R} \to \mathbb {R}$, f not ultimately identically 0, there exist c, $r \in \mathbb {R},c \ne 0$, such that $x \mapsto {x^r}:(0, + \infty ) \to \mathbb {R}$ is definable in $\mathcal {R}$ and ${\lim _{x \to + \infty }}f(x)/{x^r} = c$.
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Infinite Differentiability in Polynomially Bounded O-Minimal Structures

1995-08-01
Chris Miller
Infinitely differentiable functions definable in a polynomially bounded o-minimal expansion $\Re$ of the ordered field of real numbers are shown to have some of the nice properties of real analytic … Infinitely differentiable functions definable in a polynomially bounded o-minimal expansion $\Re$ of the ordered field of real numbers are shown to have some of the nice properties of real analytic functions. In particular, if a definable function $f:{\mathbb {R}^n} \to \mathbb {R}$ is ${C^N}$ at $a \in {\mathbb {R}^n}$ for all $N \in \mathbb {N}$ and all partial derivatives of f vanish at a, then f vanishes identically on some open neighborhood of a. Combining this with the Abhyankar-Moh theorem on convergence of power series, it is shown that if $\Re$ is a polynomially bounded o-minimal expansion of the field of real numbers with restricted analytic functions, then all ${C^\infty }$ functions definable in $\Re$ are real analytic, provided that this is true for all definable functions of one variable.