Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [analytic-geometry]

The tag has no usage guidance.

20
votes
2answers
1k views

Complex analytic vs algebraic geometry

This is more of a philosophical or historical question, and I can be totally wrong in what I am about to write next. It looks to me, that complex-analytic geometry has lost its relative positions ...
2
votes
1answer
107 views

Regular sequence from prime ideal

Let $I$ be a prime ideal in $\mathbb{C}\{x_1, \ldots, x_n\}_0$ (the localization at the maximal ideal that defines $0$) and suppose that the height of $I$ is $h$. Then, there is a standard trick to ...
2
votes
2answers
398 views

Conformal mappings that preserve angles and areas but not perimeters?

Conformal mappings from $U$ to $V$, both subsets of $\mathbb{C}$, locally preserve angles. But, in general, such mappings neither preserve areas nor preserve perimeters. Q. Are there examples of ...
6
votes
1answer
123 views

Additivity of characteristic cycle of holonomic D-module

Let $\mathcal{M}$ be a holonomic D-module on a complex analytic (or alternatively, algebraic) manifold $X$. One can attach to it (using a good filtration) a characteristic cycle $Ch(\mathcal{M})$ ...
10
votes
1answer
359 views

Geometric interpretation of algebraic tangent cone

Suppose $(A,\mathfrak m)$ is a Neotherian local $k$-algebra with residue field $k$. Then, we define (the coordinate ring of) its algebraic tangent cone to be the $k$-algebra $A_c = \sum_{i\ge 0} \...
5
votes
0answers
89 views

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 $A$. ...
3
votes
1answer
113 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. ...
1
vote
0answers
94 views

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 ...
3
votes
1answer
284 views

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 ...
1
vote
1answer
99 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 ...
6
votes
0answers
251 views

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 ...
5
votes
0answers
51 views

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 $D$, ...
2
votes
0answers
48 views

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 ...
2
votes
0answers
71 views

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) ...
5
votes
1answer
221 views

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 ...
8
votes
0answers
513 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 $Y=X\times _K L$ ...
1
vote
0answers
111 views

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 $X_T= X\times_S T$. ...
2
votes
0answers
160 views

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 ...
7
votes
1answer
286 views

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 ...
2
votes
0answers
173 views

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 ...
2
votes
2answers
241 views

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 ...
18
votes
1answer
992 views

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 ...
2
votes
0answers
181 views

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 ...
10
votes
1answer
366 views

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 ...
1
vote
0answers
160 views

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 $X$...
3
votes
0answers
143 views

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 $X$.
0
votes
1answer
133 views

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 ...
10
votes
1answer
295 views

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^...
4
votes
1answer
254 views

GCD in polynomial vs. formal power series rings

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,...
6
votes
1answer
143 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 ...
7
votes
1answer
275 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 ...
1
vote
0answers
115 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 ...
2
votes
1answer
253 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?
1
vote
0answers
68 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 ...
7
votes
0answers
124 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 ...
3
votes
1answer
235 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 $P$...
0
votes
0answers
72 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\}...
2
votes
1answer
190 views

Reference to parabola lemma

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 $P$ ...
1
vote
0answers
108 views

Pascal and Brianchon's theorems generalized for hyperbolic paraboloid

I know that giving a general version of these two theorems for quadrics can be quite tricky, but if we restrict ourselves to a verssion that holds for the hyperbolic paraboloid only it should be ...
11
votes
3answers
560 views

Why does inconstructibility of $\sqrt[3]{2}$ imply impossibility of cube doubling? [closed]

In this question "constructing" and "doubling" is meant in the compass-and-straightedge sense. On my desk I have five Basic Algebra texts treating constructability in the plane $\mathbb{C}$ or $\...
14
votes
1answer
409 views

Groups and pregeometries

Definition. For an infinite structure $\mathcal{A}$ and $cl : P(dom(\mathcal{A})) \longrightarrow P(dom(\mathcal{A}))$ , we say that $(\mathcal{A}, cl)$ is a structure carrying an $\omega$-homogeneous ...
3
votes
0answers
149 views

Universal covering space of a Zariski open subset of projective space

Let $U$ be a Zariski open subset of $\mathbb P^n_{\mathbb C}$. Assume $U$ is the complement of some divisors. Have the possible universal covering spaces of $U$ been classified? Do we know when the ...
4
votes
1answer
404 views

Simple maps: Flat versus locally trivial

In deformation of complex analytic spaces, one usually considers an analytic proper simple surjective map $\varpi: \mathscr{M} \twoheadrightarrow \mathscr{P}$ as an analytic family. However the term ...
0
votes
1answer
118 views

Can an analytic set admit such a foliation?

I confess to be not an expert of analytic geometry, but I have come across the following problem, for which I need an help from experts in this specific field. I was wondering myself if it is ...
3
votes
1answer
224 views

Tangent cone and embedded components

Is it possible for a reduced, equidimensional germ of complex analytic singularity to have a tangent cone with embedded components but without multiple irreducible components? If it is, how can you ...
2
votes
2answers
1k views

supporting facts to fujita conjecture

I came across the Fujita conjecture which is perhaps very widely known. I want to know what are the supporting facts to the truth of the conjecture. http://en.wikipedia.org/wiki/Fujita_conjecture
2
votes
0answers
272 views

Asymptotics vs Puiseux series

Define asymptotic as a class of sequences {$ x_i$},$_{i\in\mathbb N}$ modulo equivalence {$x_i$}={$y_i$} if $\lim_{i\to\infty} (x_i/y_i)=c\in\mathbb R,c\ne 0$. More, we define $X= \{x_i\} \lt Y= \{ ...
26
votes
2answers
831 views

On Determinants of Laplacians on Riemann Surfaces

History of the Formula: In their famous paper "On Determinants of Laplacians on Riemann Surfaces" (1986), D'Hoker and Phong computed the determinant of the Laplacian $\Delta_n^+$ on the space $T^n$ of ...
2
votes
3answers
185 views

How to tell if a second-order curve goes below the $x$ axis?

Suppose we have a second-order curve in general form: (1) $a_{11}x^{2}+2a_{12}xy+a_{22}y^{2}+2a_{13}x+2a_{23}y+a_{33}=0$. I'd like to know if there is a simple condition that ensures that the curve ...
0
votes
0answers
167 views

Asymptotes of hyperbolic sections of a given cone

A book I'm reading (Companion to Concrete Math Vol. I by Melzak) mentions, "...any ellipse occurs as a plane section of any given cone. This is not the case with hyperbolas: for a fixed cone only ...