A definability result for compact complex spaces

2004 ◽  
Vol 69 (1) ◽  
pp. 241-254 ◽  
Author(s):  
Dale Radin
Keyword(s):  
Algebraic Geometry ◽  
Complex Space ◽  
Analogous Result ◽  
Order Structure ◽  
Holomorphic Map ◽  
Complex Spaces ◽  
Compact Complex Space ◽  
Image Position

AbstractA compact complex space X is viewed as a 1-st order structure by taking predicates for analytic subsets of X, X x X, … as basic relations. Let f: X → Y be a proper surjective holomorphic map between complex spaces and set Xy ≔ f−1(y). We show that the setis analytically constructible, i.e.. is a definable set when X and Y are compact complex spaces and f: X → Y is a holomorphic map. The analogous result in the context of algebraic geometry gives rise to the definability of Morley degree.

scholarly journals On The Holomorphic Automorphism Groups of Complex Spaces

1968 ◽  
Vol 33 ◽  
pp. 85-106 ◽  
Author(s):  
Hirotaka Fujimoto
Keyword(s):  
Lie Group ◽  
Complex Space ◽  
Automorphism Groups ◽  
Analogous Result ◽  
Holomorphic Mapping ◽  
Compact Complex Manifold ◽  
Holomorphic Automorphism ◽  
Complex Spaces ◽  
Weaker Conditions ◽  
Holomorphic Automorphism Groups

For a complex space X we consider the group Aut (X) of all automorphisms of X, where an automorphism means a holomorphic automorphism, i.e. an injective holomorphic mapping of X onto X itself with the holomorphic inverse. In 1935, H. Cartan showed that Aut (X) has a structure of a real Lie group if X is a bounded domain in CN([7]) and, in 1946, S. Bochner and D. Montgomery got the analogous result for a compact complex manifold X ([2] and [3]). Afterwards, the latter was generalized by R.C. Gunning ([11]) and H. Kerner ([16]), and the former by W. Kaup ([14]), to complex spaces. The purpose of this paper is to generalize these results to the case of complex spaces with weaker conditions. For brevity, we restrict ourselves to the study of σ-compact irreducible complex spaces only.

scholarly journals The polarization constant of finite dimensional complex spaces is one

2021 ◽  
pp. 1-19
Author(s):  
VERÓNICA DIMANT ◽  
DANIEL GALICER ◽  
JORGE TOMÁS RODRÍGUEZ
Keyword(s):  
Banach Space ◽  
Analytic Functions ◽  
Homogeneous Polynomial ◽  
Complex Space ◽  
Necessary Conditions ◽  
Best Constant ◽  
The Real ◽  
Complex Spaces ◽  
Finite Dimensional ◽  
Image Position

Abstract The polarization constant of a Banach space X is defined as \[{\text{c}}(X){\text{ }}{\text{ }}\mathop {\lim }\limits_{k \to \infty } {\text{ }}\sup {\text{c}}{(k,X)^{\frac{1}{k}}},\] where \[{\text{c}}(k,X)\] stands for the best constant \[C > 0\] such that \[\mathop P\limits^ \vee \leqslant CP\] for every k-homogeneous polynomial \[P \in \mathcal{P}{(^k}X)\] . We show that if X is a finite dimensional complex space then \[{\text{c}}(X) = 1\] . We derive some consequences of this fact regarding the convergence of analytic functions on such spaces. The result is no longer true in the real setting. Here we relate this constant with the so-called Bochnak’s complexification procedure. We also study some other properties connected with polarization. Namely, we provide necessary conditions related to the geometry of X for \[c(2,X) = 1\] to hold. Additionally we link polarization constants with certain estimates of the nuclear norm of the product of polynomials.

Discreteness For the Set of Complex Structures On a Real Variety

2003 ◽  
Vol 46 (3) ◽  
pp. 321-322
Author(s):  
E. Ballico
Keyword(s):  
Complex Space ◽  
Complex Conjugate ◽  
Complex Structures ◽  
Complex Spaces ◽  
Isomorphism Classes ◽  
Real Variety ◽  
Compact Complex Space

AbstractLet X, Y be reduced and irreducible compact complex spaces and S the set of all isomorphism classes of reduced and irreducible compact complex spaces W such that X × Y ≅ X × W. Here we prove that S is at most countable. We apply this result to show that for every reduced and irreducible compact complex space X the set S(X) of all complex reduced compact complex spaces W with X × Xσ ≅ W × Wσ (where Aσ denotes the complex conjugate of any variety A) is at most countable.

scholarly journals A NEW DP-MINIMAL EXPANSION OF THE INTEGERS

2019 ◽  
Vol 84 (02) ◽  
pp. 632-663 ◽  
Author(s):  
ERAN ALOUF ◽  
CHRISTIAN D’ELBÉE
Keyword(s):  
Analogous Result ◽  
Abelian Groups ◽  
Quantifier Elimination ◽  
Alternative Proof ◽  
Order Structure ◽  
First Order ◽  
Adic Valuation ◽  
Image Position ◽  
Natural Expansion

AbstractWe consider the structure $({\Bbb Z}, + ,0,|_{p_1 } , \ldots ,|_{p_n } )$, where $x|_p y$ means $v_p \left( x \right) \leqslant v_p \left( y \right)$ and vp is the p-adic valuation. We prove that this structure has quantifier elimination in a natural expansion of the language of abelian groups, and that it has dp-rank n. In addition, we prove that a first order structure with universe ${\Bbb Z}$ which is an expansion of $({\Bbb Z}, + ,0)$ and a reduct of $({\Bbb Z}, + ,0,|_p )$ must be interdefinable with one of them. We also give an alternative proof for Conant’s analogous result about $({\Bbb Z}, + ,0, < )$.

The fractional parts of an additive form

1972 ◽  
Vol 72 (2) ◽  
pp. 209-212 ◽  
Author(s):  
R. J. Cook
Keyword(s):  
Quadratic Form ◽  
Analogous Result ◽  
The Difference ◽  
Image Position ◽  
Forms Of Higher Degree ◽  
Additive Form ◽  
Real Quadratic

Heilbronn (6) proved that for every ε ≥ 0 and N ≥ 1 and every real θ there is an integer x such that,where C(ε) depends only on ε and ∥α∥ is the difference between α and the nearest integer, taken positively. Danicic(1) obtained an analogous result for the fractional parts of nkθ, the proof of this is more readily accessible in Davenport(4). Danicic(2) also obtained an estimate for the fractional parts of a real quadratic form in n variables, and in order to extend this result to forms of higher degree it is desirable to first obtain results for additive forms.

scholarly journals TOPOLOGICAL CELL DECOMPOSITION AND DIMENSION THEORY IN P-MINIMAL FIELDS

2017 ◽  
Vol 82 (1) ◽  
pp. 347-358 ◽  
Author(s):  
PABLO CUBIDES KOVACSICS ◽  
LUCK DARNIÈRE ◽  
EVA LEENKNEGT
Keyword(s):  
Algebraic Geometry ◽  
Open Subset ◽  
Cell Decomposition ◽  
Real Algebraic Geometry ◽  
Dimension Theory ◽  
Definable Set ◽  
Image Position ◽  
Definable Function

AbstractThis paper addresses some questions about dimension theory for P-minimal structures. We show that, for any definable set A, the dimension of $\bar A\backslash A$ is strictly smaller than the dimension of A itself, and that A has a decomposition into definable, pure-dimensional components. This is then used to show that the intersection of finitely many definable dense subsets of A is still dense in A. As an application, we obtain that any definable function $f:D \subseteq {K^m} \to {K^n}$ is continuous on a dense, relatively open subset of its domain D, thereby answering a question that was originally posed by Haskell and Macpherson.In order to obtain these results, we show that P-minimal structures admit a type of cell decomposition, using a topological notion of cells inspired by real algebraic geometry.

scholarly journals THE UNITY AND IDENTITY OF DECIDABLE OBJECTS AND DOUBLE-NEGATION SHEAVES

2018 ◽  
Vol 83 (04) ◽  
pp. 1667-1679
Author(s):  
MATÍAS MENNI
Keyword(s):  
Differential Geometry ◽  
Algebraic Geometry ◽  
Algebraic Topology ◽  
Full Subcategory ◽  
Sufficient Conditions ◽  
Double Negation ◽  
Synthetic Differential Geometry ◽  
Simplicial Sets ◽  
Image Position

AbstractLet ${\cal E}$ be a topos, ${\rm{Dec}}\left( {\cal E} \right) \to {\cal E}$ be the full subcategory of decidable objects, and ${{\cal E}_{\neg \,\,\neg }} \to {\cal E}$ be the full subcategory of double-negation sheaves. We give sufficient conditions for the existence of a Unity and Identity ${\cal E} \to {\cal S}$ for the two subcategories of ${\cal E}$ above, making them Adjointly Opposite. Typical examples of such ${\cal E}$ include many ‘gros’ toposes in Algebraic Geometry, simplicial sets and other toposes of ‘combinatorial’ spaces in Algebraic Topology, and certain models of Synthetic Differential Geometry.

scholarly journals A smoothness criterion for complex spaces in terms of differential forms

2020 ◽  
Vol 126 (2) ◽  
pp. 221-228
Author(s):  
Håkan Samuelsson Kalm ◽  
Martin Sera
Keyword(s):  
Complex Space ◽  
Differential Forms ◽  
Complex Spaces ◽  
Smoothness Criterion

For a reduced pure dimensional complex space $X$, we show that if Barlet's recently introduced sheaf $\alpha _X^1$ of holomorphic $1$-forms or the sheaf of germs of weakly holomorphic $1$-forms is locally free, then $X$ is smooth. Moreover, we discuss the connection to Barlet's well-known sheaf $\omega _X^1$.

A Multiple Exponential Sum to Modulus p2

1985 ◽  
Vol 28 (4) ◽  
pp. 394-396 ◽  
Author(s):  
D. R. Heath-Brown
Keyword(s):  
Algebraic Geometry ◽  
Exponential Sums ◽  
Exponential Sum ◽  
Total Degree ◽  
Image Position

AbstractFor suitable polynomials f(x) ∊ ℤ[x] in n variables, of total degree d, it is shown thatThis is, formally, a precise analogue of a theorem of Deligne [1] on exponential sums (mod p). However the proof uses no more than elementary algebraic geometry.

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