# Questions tagged [analytic-geometry]

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### Properties of shapes defined by locus of points with a function of distances

Different shapes such as hyperbola and ellipse can be defined as a locus of points. For example, if we denote distance to points $P_1$ and $P_2$ from any arbitrary point as $d_1(x,y)$ and $d_2(x,y)$. ...
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### Associativity property of the gyrobarycenter

I'm using Ungar's terminology and notations. In the open unit ball of $\mathbb{R}^n$, let $GB(A_1, \ldots, A_N; m_1, \ldots, m_N)$ be the gyrobarycenter of the points $(A_1, \ldots, A_N)$ with ...
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### Is there a classification of higher-degree generalisations of confocal conic sections?

The 1-parameter families of ellipses and hyperbolas with a given pair of points in the plane as their foci yield “orthogonal double-foliations” of the plane. That is, once the foci are specified, any ...
175 views

### Conformal map from a 7-sided polyhedron to a square pyramid

I have a right-angled square pyramid, $A$, whose height and base-length are $l$. By 'right-angled', I mean that the apex of $A$ lies vertically above one of the vertices in its base. Now supposed I ...
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### Integrals of real analytic functions

Let $A\subset \mathbb{R}^{n+m}$ be a compact subanalytic subset. Let $F\colon A\to \mathbb{R}$ be a function which is a restriction to $A$ of a real analytic function defined in a neighborhood of $A$. ...
138 views

### Can an analytic variety extend along a codimension 2 subvariety?

Let $X$ be a smooth, connected, complex analytic variety, and $Y\subset X$ a closed, analytic subvariety of codimension at least 2. Now let $V\subset X\backslash Y$ be a closed, analytic subvariety. ...
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### Is the topology generated by the complements of analytic subsets strictly coarser than the Euclidean topology in dimensions $\geq 2$?

Let $\mathbb{K}$ be either $\mathbb{R}$ or $\mathbb{C}$ and let $N\geq 2$. Similarly to the construction of the Zariski topology, take the collection of zero sets of $\mathbb{K}$-analytic functions to ...
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### Equivalence of the term “Divisor”

Throughout my university education, I have studied some theory of Riemann surfaces, focusing particularly on Miranda's Algebraic curves and Riemann surfaces. My current studies however are in the ...
116 views

### Bounded holomorphic functions on hypersurfaces of $\Bbb C^n$

Is it true that every bounded holomorphic functions on a smooth analytic hypersurface $X$ of $\Bbb C^n$ is constant? Remark that if $X$ is algebraic, the answer is yes. Otherwise can you provide ...
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### Pseudo-effective divisor which is not nef in any birational model

Let $X$ be a smooth complex projective algebraic variety and let $D$ be a $\mathbb{Q}$-Cartier pseudo-effective divisor on $X$. Lets say that $D$ is birationally nef if there exists a birational ...
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### Subadditivity of multiplier ideals with a pluriharmonic function

I would like to have a reference for the following two facts (if true): Let $D$ be a nef and big divisor on an algebraic variety $X$ and $h$ a Hermitian metric with minimal singularities on $D$, ...
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### Unboundedness of number of solutions of intersection of bivariate polynomial with graph of function from an o-minimal structure

I am trying to understand a construction sketched in the paper by Gwozdziewicz, Kurdyka and Parusinski in the Proceedings of the AMS 1999 (paper here) and I'd like to request some help. The ...
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### Sheaves of functions on open semi-algebraic sets

Let $U\subsetneq\mathbb{R}^n$ be an open semi-algebraic set which is not Zariski open (the case I have in mind is that of an open ball $B(0,1)$). A function $f:U\rightarrow \mathbb{R}$ is called (1) ...
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### Are continuous rational functions arc-analytic?

Let $X\subseteq\mathbb{R}^n$ be a smooth semi-algebraic set (for simplicity we can assume $X=B(0,r)$ is a small ball around the origin). A function $f:X\rightarrow \mathbb{R}$ is called a continuous ...
734 views

### Galois descent for schemes over fields

Let $K\subset L$ be a finite galois extension of fields (the case I have in mind is $K=\mathbb{R},L=\mathbb{C}$). Given a scheme $X$ over $K$ by pulling back to $L$ we get a scheme $Y=X\times _K L$ ...
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### Necessity of cohomological flatness for the Picard functor

Let $f:X\rightarrow S$ be a proper, flat morphism of complex analytic spaces and let $Pic_{X/S}(T)=H^0(T,R^1 {f_T}_*(\mathcal{O}^*_{X_T}))$ be the relative Picard functor. Here $X_T= X\times_S T$. ...
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### Understanding canonical angles between two subspaces

I am trying to understand Wedin Theorem on the perturbations of the Singular Vectors of a matrix, and a key element for this theorem is the matrix of the canonical angles between two subspaces; I am ...
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### Easiest proof for showing finite etale (analytic) quotients of algebraic varieties are algebraic

Let $X$ be an algebraic variety over $\mathbb C$. Let $X^{an}\to Y$ be a finite etale morphism with $Y$ a complex analytic space. I read somewhere that $Y$ algebraizes, ie, $Y=V^{an}$ for some ...
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### Descent for complex-analytic spaces

I'm basically interested in knowing the difference between complex spaces and schemes when studying stacks. I'd like to use stacks to study moduli problems in complex analytic geometry. Citing a ...
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### Nontrivial Analytic Varieties

In short, I'd like to know the following: Is there an irreducible analytic variety that has to be defined by at least two distinct sets of holomorphic functions? Are there two irreducible analytic ...
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### When do real analytic functions form a coherent sheaf?

It is known that, in general, the sheaf of real analytic functions on a real analytic manifold is not coherent. However, there are some examples, where we have coherence: for example, if $X$ is a ...
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### Representability of relative Hilbert and Picard functors over analytic spaces

Let $f:X \to S$ be a morphism of complex analytic spaces. Then, just like in the case of schemes, we can define the relative Hilbert and Picard functors. For instance, if $\text{An}_{/S}$ denotes de ...
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### Does GAGA hold over other topological fields?

If k is a non-discrete topological field, we can define an analytic space over k just like complex analytic spaces over $\mathbb{C}$. If you replace "complex analytic space" and "complex algebraic ...
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### Is there an analytic criterion for quasi-compactness of a scheme?

Let $X$ be a locally finite type scheme over $\mathbb C$. I'm looking for the analogue of the notion "finite type" for $X^{an}$ and an SGA 1 Exp. XII type of criterion which says that The scheme $X$...
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### Does constructible and analytically open imply Zariski open

Let $U$ be a constructible subset of a complex algebraic variety. Is the following statement true? If $U$ is open in the analytic topology, then $U$ is open in the Zariski topology on $X$.
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### Some questions related to the unitary operators

A unitary operator is a surjective linear operator between complex inner product spaces, which preserves the inner product. What is the name of the analogue for the real case? Orthogonal operator ...
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### Is there a divisor in $\mathbb P^2$ such that all analytic maps into its complement algebraize?

Is there a closed subscheme $D$ in $\mathbb P^2_{\mathbb C}$ pure of codimension one such that, for all algebraic varieties $X$ over $\mathbb C$, any analytic map  \phi: X(\mathbb C) \to \mathbb P^...
I'm having problems finding an appropriate reference for this question. Given two elements $f, g \in \mathbb{C}[x_1, \dots, x_n]$, consider their greatest common divisor, $\gcd_{\mathbb{C}[x_1, \dots,... 1answer 161 views ### why use Crofton's formula in order to prove log-analyticity of volume? In their interesting paper "Integration des fonctions sous-analytiques et volumes des sous-ensembles sous-analytiques" Lion and Rollin show that the volume of a fibre$Y_x$of a globally subanalytic ... 1answer 338 views ### Connectivity of complements of Stein opens Let$Y$be an affine open subset of a locally noetherian scheme$X$. Then,$X \setminus Y$has pure codimension one [EGAIV$_4$, Cor. 21.12.7]. Moreover, if$X$is proper and of finite type over a ... 0answers 121 views ### Analytification of Poisson structures on an affine variety It is well known that one can transfer every affine variety$X$over$\mathbb{C}$into an analytic space$X^{an}$in a natural way. This process is called the analytification. My question is that does ... 1answer 319 views ### Conic sections in high dimensions Can every$n$-dimensional ellipsoid be obtained as a (spherical) conic section? This is false for generic quadrics but seems true for ellipsoid. Does anybody have any references? 0answers 76 views ### Dimensions of fibers of analytic map I must admit that I know nothing about p-adic geometry, so the following question may be completely trivial. Let$V\subset K^n$be an affine algebraic variety. Let$D$be a polydisk, and$F$be an ... 0answers 135 views ### Finite covers in complex analytic geometry Given a complex manifold or complex analytic space, one has the standard notion of open set. There are two different Grothendieck topologies that one can define using this notion, one where covers ... 1answer 272 views ### about transverse complete intersection There are several questions about transverse complete intersection arising from L. Guth's paper: http://www.ams.org/journals/jams/0000-000-00/S0894-0347-2015-00827-X/home.html We say a polynomial$P$... 0answers 74 views ### Good covering of a (singular) curve in a complex surface Let$W$be a$2$-dimensional complex manifold and$C\subset W$a compact complex curve (possibly singular). I would like to know a reference for the following fact: there exists a collection$\{V_j\}...
I am looking for a previous reference to the following very simple geometric lemma, which I use in my paper [arXiv:1503.03462]: Let $P$ be the parabola $y=x^2$. Let $a,b,c,d$ be four points on $P$ ...