Questions tagged [analytic-geometry]
The analytic-geometry tag has no usage guidance.
108
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Is there a notion of a complex/analytic diffeological space?
I have a bit of a general question. This seems like something you can do, but I can't seem to find much reference for this.. Perhaps something like this already exists in a different guise. But, is ...
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Pullback of coherent sheaves on Stein manifolds
Let $i:X\to Y$ be a closed embedding of Stein spaces, $G$ be a coherent $O_Y$-module. Set $I=\ker(i^*:O(Y)\to O(X))$. Then $I$ is an ideal of the ring $O(Y)$. Is that true that $\Gamma(X,i^*G)=G(Y)/...
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Differentiable functions on analytic varieties
Let $\iota\colon X\to \Omega\subseteq \mathbb{C}^n$ be a complex analytic variety $X$ in an open subset $\Omega$ of $\mathbb{C}^n$. If $N$ is a smooth manifold and $h\colon M\to X$ is a continuous map,...
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Detecting topology change of tubular neighbourhoods via smoothness of volume function
Let $M$ be an embedded closed manifold in $\mathbb R^n$, define $M_r=\{x\in\mathbb R^n:d(x,M)<r\}$.
Define $r\in\mathcal S_M$ iff $M_r\subset M_{r+\epsilon}$ is not a homotopy equivalence for all ...
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What lies between algebraic geometry and analytic geometry?
Algebraic geometry and analytic geometry are closely related (witness GAGA). But the latter still seems much "bigger" than the former. I'd like to be able to get from algebraic geometry to ...
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253
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Contractible real analytic varieties
If a real analytic variety $V$ in $\mathbb{R}^n$ is both bounded and contractible, is it true that $V$ must be a single point?
Here a real analytic variety is the set of zeros of a real analytic ...
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Proving Geometric Inequality Using Equation Discriminant
I met this question before:
An acute $\triangle ABC$ (you can imagine $BC$ below) has a point $D$ on side $AC$. The line parallel to BC through $D$ meets $AB$ at $E$, and the parallel line $BD$ ...
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What information does the topology of nonarchimedean Berkovich analytic spaces encode?
Given a finite type scheme $X$ over $\Bbb{C}$ we can associate to it an analytic space $X^\text{an}$. There are then comparison theorems comparing invariants of the topological space $X^\text{an}$ ...
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Contractibility of the quotient of an analytification of a smooth variety by a finite group (if the field is trivially valued)
Let $k$ be a field and $X$ be a smooth irreducible $k$-variety with an action of a finite group $G$. I consider $k$ as a trivially valued field.
It is known from results of Berkovich ("Smooth p-...
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Complete residue field of a point of type 5
Let $(F,|.|)$ be a complete algebraically closed field. Let $x$ be the point of type 5 corresponding to the unit open disc of the adic affine line over $F$. Can we obtain a concrete description of the ...
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Valuations and (semi)norms on ring spectra
Valuations and seminorms on rings play a big role in number theory and analytic geometry, with seminorms being heavily used in Berkovich geometry and valuations featuring heavily in adic geometry.
Let'...
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Intersection of Stein opens admits a Stein neighborhood basis?
Let $X$ be a Stein manifold, $K$ be a compact subset of $X$. Consider the following conditions:
1.$K$ admit an open neighborhood basis in $X$ whose members are Stein;
2.$K=\cap_{j\ge 1}V_j$, where $...
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Can we see the completion of a scheme along a subscheme as an adic space?
Classically, formal schemes were invented to study completions of schemes along closed subschemes. Eventually, people started using them for more arithmetical reasons. (I.e., to study non-archimedean ...
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Origin of 'Analytic' Geometry?
My impression is that the name analytic geometry, which I understand roughly to be geometry in Euclidean space using coordinates, is not used that much anymore. We would probably classify the subject ...
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The notion of border for (complex and non-archimedean) analytic spaces and schemes
Is a manifold with corner an analytic space (just show that $\left[0, +\infty \right)^{n}$ is an analytic space, which seems obvious but maybe I'm wrong...) EDIT: as noted in the comments some complex ...
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Representatives of line bundle cohomology over tori
Let $V^n$ a be a $\mathbb{C}$-vector space. For $U\subset V$ a complete lattice, the holomorphic line bundles over $V/U$ are classified (see e.g. `Abelian varieties', D. Mumford) by data $(H,\alpha)$ ...
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Why are Berkovich spaces locally connected?
A characteristic feature of Berkovich spaces is that they are locally connected (in fact, locally contractible). I'd like to understand the proof. The key ingredient seems to be Corollary 2.2.8 in ...
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Intersection multiplicity via parametrization in general
My question is a generalization of Analytic and algebraic definitions of intersection multiplicity of two complex algebraic curves coincide.
Take two complex space germs $(A, 0)=V(I_A)$ of dimension $...
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Reference request: Gruson's theorem on the tensor product of Banach spaces over a non-Archimedean field
I am looking for a reference for theorem 3.21 of these notes: https://web.math.princeton.edu/~takumim/Berkovich.pdf
The theorem states that if $k$ is a non-Archimedean field and $X$ and $Y$ are $k$-...
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Is the Serre dualizing complex local in the analytic topology?
There is a Serre dualizing complex $S_X\in D^b Coh(X)$ for any scheme $X$ of finite type. For proper schemes, it is characterized as the sheaf defining the right adjoint to derived global sections, ...
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Regularity of fiber integration between complex analytic spaces
Let $f:X\rightarrow Y$ be a flat surjective morphism between reduced complex analytic spaces. Assume that $Y$ is locally irreducible (i.e. unibranch).
We assume that $X$ (resp. $Y$) is pure-...
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An attempt to define partial properness and compactification for some maps between analytic spaces
The paper Étale cohomology of diamonds defines partial properness and compactification for maps between v-sheaves, and in particular for perfectoid spaces and rigid-analytic spaces. Recently when ...
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Two definitions of Teichmüller space: relative isotopy or not?
The definition of Teichmüller space on wikipedia via marked Riemann surfaces say that two markings are equivalent if the map $fg^{-1}$ is isotopic to a holomorphic diffeomorphism.
The definition on ...
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When is a real-analytic variety a union of non-singular subvarieties?
I have asked this before on MSE, but received no answer yet.
Say I have a set in $\mathbb{R}^n$ defined to be the zero set of an analytic function $F:\mathbb{R}^n\to\mathbb{R}^k$, $k<n$. Everywhere,...
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On the number of integral points of analytic curves
Consider a curve over some number field $\mathbb{K}$. By Falting's Theorem, if the genus $g$ is greater than $1$, the curve has only finitely many integral points. Moreover, as shown in Bilu, Y. et al....
- 1,155
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Orbits space of real-analytic planar foliations
Consider a foliation of $\mathbb{R}^2$, say coming from the trajectories of a vector field $X$. Its orbit space (the quotient of $\mathbb{R}^2$ by the relation "lying on the same trajectory")...
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How to show analytification functor commutes with forgetful functor?
Let $k$ be a field complete with respect to a non-trivial non-archimedean
absolute value (so that rigid $k$-space makes sense). Denote $K$ a finite field extension of $k$.
Denote $X\rightsquigarrow X^{...
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Intersection theory in analytic geometry
This might be a weird/stupid question, but it came to me a couple of times, and I would like to get an answer for that.
In some papers I read, constantly the authors define some analytic subspaces, ...
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Berkovich Integration on algebraic curves
Berkovich developed a theory of integrating one-forms on his analytic spaces in his book "Integration of One-forms on $P$-adic analytic spaces". As this book is difficult to digest for me, I ...
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English reference for Douady/Grauert construction of versal deformations of compact complex spaces
I'm trying to learn about the deformation theory of compact complex spaces. I'm familiar with the case of compact complex manifolds from the paper "On the Locally Complete Families of Complex ...
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Residues in several complex variables
I am trying to educate myself about the basics of the theory of residues in several complex variables. As is usually written in the introduction in the textbooks on the topic, the situation is much ...
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Decomposability and analytification of coherent sheaves
Let $X$ be an affine (algebraic) complex variety and $f:Y \to X$ be a finite morphism. Given any coherent sheaf $\mathcal{F}$ on $X$, we denote by $\mathcal{F}^{an}$ the analytification of $\mathcal{F}...
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GAGA for vector bundles over Riemann surfaces
Serre’s GAGA theorem gives an equivalence of categories between algebraic and analytic coherent sheaves over a complex projective variety. The proof relies on the finiteness of the cohomologies of ...
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Sheaf of smooth functions and restriction to a divisor
My question is targeted towards a very particular detail in my research that I am trying to understand. I will therefore break it down into some more general questions.
Let $X$ be a smooth variety, $i:...
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Isomorphism between two families of curves over the Teichmueller space
In his construction of the Teichmueller space of curves of genus $\geq 2$ Grothendieck states in Corollaire 2.4 that the map $$\underline{Isom}_S(X,Y) \xrightarrow{} S$$ is finite. The map represents ...
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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 ...
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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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How to compute the volume of a region transformed by a matrix?
This is a rewrite of the OP's question to emphasize what I think are the research level issues here.
Let $\mathscr{R}$ be a bounded convex body in $\mathbb{R}^n$ and let $H : \mathbb{R}^n \to \mathbb{...
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The $r$-dimensional volume of the Minkowski sum of $n$ ($n\geq r$) line sets
Let $n$ line sets be $\mathcal{S}_i=\{a\mathbf{h}_i:0 \le a \le 1\}$, for $1 \le i \le n$, where $\{\mathbf{h}_1,\cdots,\mathbf{h}_n\}$ is a vector group of rank $r$ in the $r$-dimensional Euclidean ...
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Norms as Points in $C(X)$
$\newcommand\abs[1]{\lvert{#1}\rvert}$Let $X$ be a compact hausdorff space, and put $C(X)$ for the $\mathbb{R}$-algebra of continuous maps from $X$ to $\mathbb{R}$.
For each point $x$, there is a ...
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Reference request: Oldest books on analytic geometry with unsolved exercises?
Per the title, what are some of the oldest books on analytic geometry out there with unsolved exercises? Maybe there are some hidden gems from before the 20th century out there.
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Cotangent Complex in Analytic Category
I am looking for a reference which develops the theory of the cotangent complex for complex analytic spaces. I need this to justify some computations I did assuming some formal properties which hold ...
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The minimal volume of the intersection of two $\mathscr{l}_1$-ball in high dimension
We define
$$B_1(r,c) = \{x\in\mathbb{R}^d : \Vert{}x-c\Vert_1 \le r\}$$
Now for arbitrary constant $r \ge s > 0$, given constant $\epsilon \in (r-s,r+s)$, considering $v \in \mathbb{R}^d$ such ...
- 21
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Polynomials (or analytic functions) vanishing on a real algebraic set
I have seen the following result stated several times in the literature, without proof:
Let $\mathbb{K}$ be $\mathbb{R}$ or $\mathbb{C}$, and assume $P\in\mathbb{K}[X_{1},\ldots,X_{n}]$ is an ...
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Localization of multiplicity in algebraic geometry
first a disclaimer: I am not an expert in alg. geometry so please don't shoot. Suppose X is a closed subscheme (not nec. reduced, and $dim >0$) of a smooth (projective if you want) variety Y. ...
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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 ...
- 1,180
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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 ...
- 1,354
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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 ...
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