Questions tagged [analytic-geometry]

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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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4 votes
0 answers
113 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, ...
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0 votes
0 answers
53 views

An equivalent characterization of conical surfaces

Consider an $n-1$ dimensional surface in $\mathbb{R}^n$. If the tangent plane at any point of this surface always passes through the origin, can we show that the surface must be a conical surface? I ...
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3 votes
0 answers
261 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-...
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3 votes
0 answers
324 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 ...
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2 votes
1 answer
158 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 ...
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0 answers
45 views

"Pushforward" of a universal curve by a map (Riemann surfaces)?

For a Riemann surface $X$ with a Beltrami form $\mu \in M(X),$ ($M(X)$ is the space of Beltrami forms on the Riemann surface), we can define the Riemann surface $X_\mu.$ In John H. Hubbard's ...
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  • 1,457
5 votes
2 answers
368 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,...
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110 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....
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5 votes
1 answer
154 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")...
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3 votes
1 answer
215 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^{...
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7 votes
1 answer
439 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, ...
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5 votes
0 answers
138 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 ...
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4 votes
1 answer
199 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 ...
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33 votes
2 answers
1k 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 ...
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3 votes
0 answers
83 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}...
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5 votes
0 answers
233 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 ...
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4 votes
0 answers
146 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:...
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3 votes
0 answers
96 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 ...
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2 votes
0 answers
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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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1 vote
0 answers
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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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4 votes
0 answers
59 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 ...
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-3 votes
1 answer
190 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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5 votes
0 answers
470 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{...
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11 votes
2 answers
388 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 ...
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  • 532
7 votes
3 answers
803 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 ...
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0 votes
1 answer
307 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.
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13 votes
1 answer
565 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 ...
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2 votes
1 answer
132 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 ...
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5 votes
1 answer
314 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 ...
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4 votes
0 answers
135 views

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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31 votes
3 answers
4k 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 ...
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5 votes
1 answer
212 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 ...
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2 votes
2 answers
895 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 ...
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6 votes
1 answer
170 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 $Ch(\mathcal{M})$ ...
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  • 19k
13 votes
1 answer
883 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} \...
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6 votes
0 answers
109 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 $A$. ...
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  • 19k
4 votes
1 answer
210 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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1 vote
0 answers
102 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 ...
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1 vote
1 answer
126 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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6 votes
0 answers
458 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 ...
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5 votes
0 answers
68 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 $D$, ...
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2 votes
0 answers
58 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 ...
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2 votes
0 answers
88 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) ...
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5 votes
1 answer
269 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 ...
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8 votes
0 answers
989 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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1 vote
0 answers
182 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 $X_T= X\times_S T$. ...
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3 votes
0 answers
353 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 ...
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8 votes
1 answer
373 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 ...
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2 votes
0 answers
345 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 ...
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