Questions tagged [analytic-geometry]

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Are the higher direct images of pluricanonical bundles torsion-free?

Suppose $f: X\rightarrow S$ is a projective smooth morphism to a smooth variety $S$. Let $m\geq 2$ be a natural number. It is known that the first higher direct image sheaf $R^1f_*\mathcal{O}_X(mK_X)$ ...
Junpeng Jiao's user avatar
5 votes
0 answers
124 views

Is $\mathbf{C}_p(X)$ self-dual?

Let $X$ be a set. Consider $\mathbf{Q}_p$ and $\mathbf{Z}_p$ as the $p$-adic numbers and $p$-adic integers, respectively. For any finite subset $F \subseteq X$, one can construct the topological ...
Luiz Felipe Garcia's user avatar
2 votes
1 answer
172 views

Deform a divisor from a fiber in a fibration

Suppose $X\rightarrow Z$ is a projective smooth morphism. Let $0\in Z$ be a closed point, $X_0$ the corresponding fiber. Suppose $H^1(X_0,\mathcal{O})=H^2(X_0,\mathcal{O})=0$, then a line bundle $L$ ...
Junpeng Jiao's user avatar
0 votes
1 answer
164 views

The intersection number $C\cdot D=\deg(D_{/C})$

Let $S$ be an algebraic complex surface, and $D=[(U_\alpha,f_{\alpha})]$ is a Cartier divisor over $S$, and let $\cal{O}_S(D)$ be the sheaf associated to $D$. And let $C$ be a complex compact curve in ...
Neo's user avatar
  • 117
6 votes
2 answers
297 views

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 ...
Elliot's user avatar
  • 295
1 vote
0 answers
42 views

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)/...
Doug Liu's user avatar
  • 433
2 votes
0 answers
63 views

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,...
Thomas Kurbach's user avatar
11 votes
0 answers
246 views

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 ...
Ariana's user avatar
  • 111
8 votes
0 answers
462 views

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 ...
Tim Campion's user avatar
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7 votes
2 answers
310 views

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 ...
Brian Lins's user avatar
1 vote
0 answers
39 views

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 $BD$ ...
yusancky's user avatar
4 votes
0 answers
217 views

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 $X^\text{an}$ ...
Nuno Hultberg's user avatar
1 vote
0 answers
177 views

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-...
Sam's user avatar
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1 vote
1 answer
93 views

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 ...
AZZOUZ Tinhinane Amina's user avatar
4 votes
0 answers
170 views

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'...
Emily's user avatar
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3 votes
0 answers
73 views

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 $...
Doug Liu's user avatar
  • 433
4 votes
0 answers
291 views

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 ...
Gabriel's user avatar
  • 933
4 votes
1 answer
478 views

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 ...
Minhyong Kim's user avatar
  • 13.5k
4 votes
0 answers
163 views

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 ...
Marsault Chabat's user avatar
1 vote
0 answers
74 views

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 $(H,\alpha)$ ...
R. González Molina's user avatar
6 votes
1 answer
210 views

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 ...
Tim Campion's user avatar
  • 60.5k
1 vote
0 answers
130 views

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 $...
Gergo Pinter's user avatar
1 vote
1 answer
177 views

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 $k$-...
Dcoles's user avatar
  • 51
4 votes
0 answers
156 views

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, ...
Dmitry Vaintrob's user avatar
3 votes
0 answers
530 views

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-...
Mingchen Xia's user avatar
4 votes
0 answers
744 views

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 ...
Longke Tang 唐珑珂's user avatar
2 votes
1 answer
214 views

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 ...
Ma Joad's user avatar
  • 1,591
5 votes
2 answers
581 views

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,...
nicrot000's user avatar
  • 212
0 votes
0 answers
124 views

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....
Manuel Norman's user avatar
5 votes
1 answer
213 views

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")...
Loïc Teyssier's user avatar
3 votes
1 answer
280 views

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^{...
Z Wu's user avatar
  • 340
6 votes
1 answer
549 views

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, ...
Winnie_XP's user avatar
  • 287
5 votes
0 answers
185 views

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 ...
Fabian Ruoff's user avatar
4 votes
1 answer
257 views

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 ...
Mohan Swaminathan's user avatar
37 votes
2 answers
2k views

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 ...
Bananeen's user avatar
  • 1,180
3 votes
0 answers
92 views

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}...
user45397's user avatar
  • 2,195
5 votes
0 answers
319 views

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 ...
G. Gallego's user avatar
4 votes
0 answers
167 views

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:...
Arkadij's user avatar
  • 914
3 votes
0 answers
101 views

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 ...
Jo Wehler's user avatar
  • 219
2 votes
0 answers
21 views

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 $d_2(x,y)$. ...
SMA.D's user avatar
  • 133
1 vote
0 answers
48 views

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 ...
Stéphane Laurent's user avatar
4 votes
0 answers
63 views

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 ...
Greg Egan's user avatar
  • 2,852
-3 votes
1 answer
202 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 ...
niran90's user avatar
  • 167
5 votes
0 answers
497 views

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{...
RyanChan's user avatar
  • 550
12 votes
2 answers
599 views

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 ...
RyanChan's user avatar
  • 550
7 votes
3 answers
894 views

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 ...
Ronald J. Zallman's user avatar
0 votes
1 answer
367 views

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.
Squid with Black Bean Sauce's user avatar
13 votes
1 answer
702 views

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 ...
Mohan Swaminathan's user avatar
2 votes
1 answer
151 views

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 ...
nowhere's user avatar
  • 21
5 votes
1 answer
604 views

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 ...
user111's user avatar
  • 3,751