# Questions tagged [analytic-geometry]

The analytic-geometry tag has no usage guidance.

86
questions

**4**

votes

**1**answer

96 views

### Analytically submersed manifold still immersed

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,...

**0**

votes

**0**answers

102 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....

**4**

votes

**1**answer

80 views

### Orbits space of real-analytic planar foliations

Consider a planar foliation (e.g. coming from the trajectories of a 2-dimensionnal vector field). Its orbits space (quotient of $\mathbb R^2$ by the relation "lying on the same trajectory") ...

**3**

votes

**1**answer

176 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^{...

**7**

votes

**1**answer

378 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, ...

**5**

votes

**0**answers

114 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 ...

**4**

votes

**1**answer

163 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 ...

**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 ...

**3**

votes

**0**answers

73 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}...

**5**

votes

**0**answers

186 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 ...

**4**

votes

**0**answers

137 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:...

**3**

votes

**0**answers

93 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 ...

**2**

votes

**0**answers

20 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 $d_2(x,y)$. ...

**1**

vote

**0**answers

40 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 ...

**4**

votes

**0**answers

55 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 ...

**-3**

votes

**1**answer

184 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 ...

**5**

votes

**0**answers

438 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{...

**11**

votes

**2**answers

296 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 ...

**7**

votes

**3**answers

764 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 ...

**0**

votes

**1**answer

288 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.

**12**

votes

**1**answer

477 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 ...

**2**

votes

**1**answer

127 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 ...

**5**

votes

**1**answer

233 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 ...

**4**

votes

**0**answers

112 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. ...

**30**

votes

**3**answers

3k 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 ...

**5**

votes

**1**answer

173 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

**2**answers

759 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

**1**answer

161 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 $Ch(\mathcal{M})$ ...

**12**

votes

**1**answer

643 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} \...

**6**

votes

**0**answers

102 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 $A$.
...

**4**

votes

**1**answer

167 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

**0**answers

99 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 ...

**1**

vote

**1**answer

120 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

**0**answers

394 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

**0**answers

62 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

**0**answers

57 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

**0**answers

84 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

**1**answer

265 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

**0**answers

838 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

**0**answers

163 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 $X_T= X\times_S T$.
...

**3**

votes

**0**answers

288 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

**1**answer

337 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

**0**answers

291 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

**2**answers

357 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

**1**answer

1k 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

**0**answers

248 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

**1**answer

409 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

**0**answers

166 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

**0**answers

169 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

**1**answer

163 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 ...