Questions tagged [compactifications]
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42 questions with no upvoted or accepted answers
10
votes
0
answers
545
views
When is the one-point compactification well-pointed?
This is a follow up to my previous
question.
Question:
Is there a reasonably natural set of conditions which guarantee that the one-point
compactification $X^+$ of a locally compact Hausdorff ...
- 18.8k
9
votes
0
answers
373
views
Embedding $\beta\mathbb{N}$ into a product of Cantor sets
Let us consider $\beta\mathbb{N}$, the Stone-Čech compactification of the natural numbers (where we do not take $0$ to be a natural number, so the only idempotent elements are nonprincipal ...
- 297
8
votes
0
answers
522
views
Deligne-Mumford moduli spaces and compactification of symmetric matrices
The real Deligne-Mumford moduli space $\bar M_{0,n+1}(\mathbb R)$ of stable genus
zero curves with $n+1$ marked points is a compactification of the space of
configurations of $n$ distinct ordered ...
7
votes
0
answers
499
views
Compactification of the moduli space of Kähler manifolds with negative constant scalar curvatures
Moishezon compactification is very important in the study of the moduli space of varieties which admit canonical metrics. Moishezon showed that any non-projective Moishezon manifold $X$, after a ...
user21574
7
votes
0
answers
491
views
Alterations of regular varieties
Let $X$ be a regular quasi-projective variety over a perfect field $k$. The existence of a "good compactification" of $X$, i.e. a regular projective variety $\bar{X}$ with an embedding $X\...
- 4,450
6
votes
0
answers
75
views
About path-connected components of the Bohr compactification of $\mathbb{R}^d$
Let ${\rm b}(\mathbb{R}^d)$ denote the Bohr compactification of $\mathbb{R}^d$, with $d\in\mathbb{N}$. This is the Pontryagin dual of the group $\mathbb{R}^d_d$, corresponding to $\mathbb{R}^d$ with ...
- 193
6
votes
0
answers
159
views
Completion/Compactification of a Kähler metric on $\mathbb C^2$
Consider $\mathbb{C}^{2}$ equipped with the Kähler form
$$
\omega_{\mu}=\frac{i}{2} \partial \bar{\partial} \log \left(\left(1+|z|^{2}\right)^{\mu}+|w|^{2}\right),
$$
where $\mu$ is a positive real ...
- 81
6
votes
0
answers
243
views
A compactification of the non-negative rationals with the discrete topology
Let $S$ be the set of non-negative rational numbers. (If it makes any difference, feel free to take the non-negative dyadic rationals instead.) Let $B=\ell_\infty(S)$; as a ${\rm C}^*$-algebra this is ...
- 25.8k
5
votes
0
answers
273
views
Intuition for the McGerty-Nevins compactification of quiver varieties
In Section 4 of the paper Kirwan surjectivity for quiver varieties (Inventiones Math. 2018) McGerty and Nevins define a compactification of the moduli space of representations
of the preprojective ...
- 2,964
5
votes
0
answers
294
views
On toroidal compactifications of Hilbert Kuga-Sato varieties
Let $F$ be a totally real field of degree d. There are Hilbert modular varieties over $\mathbb{Q}$ that paramatrize abelian varieties of dimension d with an action of $\mathcal{O}_F$ the ring of ...
- 845
4
votes
0
answers
190
views
Ends of a metric space?
I'm looking for a definition of “ends” of a metric space that is well-defined even for non geodesic or locally finite metric spaces, invariant under quasi-isometries (or more generally coarse ...
- 887
4
votes
0
answers
141
views
Explicit toroidal compactification of Hilbert modular varieties
Hirzebruch's construction of toroidal compactification of Hilbert modular surfaces is explicit, namely one can explicitly choose rational polyhedral cone decomposition in a sort of optimal way using ...
- 1,024
4
votes
0
answers
485
views
Embedding of a smooth variety into a complete smooth variety.
Consider the following fact from algebraic geometry:
Any (complex) smooth algebraic variety can be embedded into a complete smooth variety as a locally closed set.
I know how to prove this fact ...
- 2,639
3
votes
0
answers
107
views
Fulton-MacPherson compactifications (and wonderful compactifications) as relative Proj
Let $X$ be a (smooth complex projective) variety.
The Fulton-MacPherson compactification $X[n]$ is obtained from $X^n$ by blowing up the diagonals in a certain order. Is it possible to write down a (...
- 318
3
votes
0
answers
64
views
Algebraic characterisation of the end space of a proper geodesic space in terms of non-continuous functions
$\DeclareMathOperator\Bf{B_\mathrm{f}}\DeclareMathOperator\Bc{B_\mathrm{c}}\DeclareMathOperator\Cf{C_\mathrm{f}}\DeclareMathOperator\Cd{C_\mathrm{d}}\DeclareMathOperator\Cc{C_\mathrm{c}}$Based on a ...
3
votes
0
answers
204
views
Smooth toric compactification of $\mathbb C^n$
By a compactification $(X,Y)$ of $\mathbb C^n$, we mean an irreducible compact complex space $X$ and a closed analytic subspace $Y\subset X$ such that $X\setminus Y$ is biholomorphic to $\mathbb C^n$. ...
- 2,789
3
votes
0
answers
250
views
Functoriality for compactifications of locally symmetric spaces
Let $X$ be a symmetric space associated to an algebraic group $G$ defined over $\mathbb{Q}$ and $G(\mathbb{R})$ acts on $X$ from the left. Let $\Gamma \subset G(\mathbb{Q})$ be an arithmetic subgroup ...
- 443
3
votes
0
answers
195
views
Universal closure of schemes à la Nagata
Nagata compactification theorem is the following fundamental result:
Let $S$ be a qcqs scheme. Let $X$ be a separated $S$-scheme of finite type. Then there exists a proper $S$-scheme $\overline{X}$ ...
user143077
3
votes
0
answers
442
views
Equivariant form of Nagata's compactification theorem?
Given a finite group $G$ acting on an algebraic variety $X$ (let's say over $\mathbb C$, if that helps), is there always a proper variety $\bar X$ with a $G$ action such that $X \to \bar X$ is a $G$-...
- 3,017
2
votes
0
answers
102
views
What is the meaning of universal family of Fulton Macpherson configuration space?
Fulton and Macpherson suggests the way to compactify the set of $n$-labelled distinct point on variety in their paper, "A Compactification of Configuration Spaces"
In this paper, the process ...
- 285
2
votes
0
answers
152
views
Line bundles on toric varieties associated to Weyl chamber
I am interested in studying toric varieties associated to the fan of Weyl chambers. General information would be best but I am also interested in the specific case of the Weyl chamber of $\mathfrak{sl}...
- 41
2
votes
0
answers
218
views
Borel-Weil-Bott theorem for wonderful compactification in characteristic p
Are there any known results for a Borel-Weil-Bott theorem for the wonderful compactifications over characteristic $p$ (i.e., theorems that classify the cohomologies of all line bundles on a wonderful ...
- 41
2
votes
0
answers
93
views
Extending Kahler metric across a divisor
Let $(X,\omega)$ be a complete noncompact Kahler manifold of finite volume. Suppose $X$ is can be compactified to a compact projective manifold $M$ so that $D=M-X$ is a divisor of simple normal ...
2
votes
0
answers
159
views
toroidal compactifications of modulis spaces of ppav's
Are the modular toroidal compactifications of ppav's (second Voronoi) defined by Alexeev without self-intersections? i.e. are the irreducible component of the boundary divisor normal? If not, can one ...
- 357
2
votes
0
answers
191
views
degenerate abelian surfaces
I am wondering if the family of degenerate abelian surfaces constructed by K. Hulek, C. Kahn and S.H. Weintraub in "Moduli spaces of Abelian Surfaces: Compactification, Degenarations, and Theta ...
- 357
2
votes
0
answers
574
views
tangent bundle of the toric variety of the wonderful compactification.
Let G be a adjoint group over $k$,algebraically closed of caracteristic zero.
Let $\overline{G}$ be its wonderful compactification.
I denote by $\overline{T}$ the closure of the torus $T$ in $\...
- 3,472
1
vote
0
answers
101
views
When is the "Gelfand Remainder" compact?
Suppose we have a noncompact Hausdorff space $S$ and a Banach algebra $A \subset C^*(S,\mathbb R)$ of the space of real-valued bounded functions on $S$. For niceness let's assume $A$ separates the ...
- 1,955
1
vote
0
answers
114
views
Homeomorphism between interiors of simplex and permutohedron
The $n$-dimensional permutohedron $P_n$ is a polytope whose facets (i.e.\ codimension $1$ faces) are in 1-to-1 correspondence with all faces (of codimension${}\geq 1$) of the $n$-simplex $\Delta_n$, ...
- 1,804
1
vote
0
answers
263
views
Implicit function theorem and compactification of algebraic curve
Let $C$ be a singular curve defined over a local field $K$.
Let $\tilde{C}$ be its smooth compactification(maybe this is not normalization).
Why $\tilde{C}(K)\neq \emptyset$ implies ${C}(K)\neq \...
- 1,531
1
vote
0
answers
96
views
Are the irreducible components appearing in the resolution of singularities of a Hilbert modular surface defined over $\mathbb{Q}$?
It seems to me that this is claimed in van der Geer's "Hilbert modular surfaces" on p. 245 at the beginning of XI.2 (without justification).
My current state of belief/knowledge:
The ...
- 93
1
vote
0
answers
143
views
End space of non-compact 2-manifolds described with proper rays
I am wondering if the classification of noncompact surfaces given by Ian Richards can be stated with proper rays instead of nested sequences of connected open subsets with compact boundary. I asked ...
1
vote
0
answers
166
views
Subspaces of compact spaces and quotients of Hausdorff spaces
Let $\operatorname{Top}$ be the class of topological spaces. Furthermore, let $\mathcal{U}\subset\operatorname{Top}$ and $\mathcal{V}\subset\operatorname{Top}$ classes satisfying the following ...
- 353
1
vote
0
answers
501
views
The Deligne-Mumford Compactification for Closed Surfaces
I am reading this note on super-Riemann surfaces. In the second paragraph of section 7.4.1 (page 87), there is a statement that I am trying to understand:
The compactified moduli space of closed ...
- 989
1
vote
0
answers
275
views
Siegel domains and the Baily-Borel compactification of $\mathcal{A}_2$
Consider the connected, almost simple, algebraic group $Sp_4$ over $\mathbb{Q}$ (embedded canonically in $GL_4$). For the following facts, I refer the reader to Murnaghan, Linear Algebraic Groups, ...
- 11
1
vote
0
answers
68
views
Extending a model to a given compactification of its generic fiber
Let $R$ be a discrete valuation ring and $K$ its field of fraction. Let $X$ be a proper $K$-variety, $U$ a dense open and consider an $R$-model $\mathcal{U}$ of $U$.
Can we embed $\mathcal{U}$ in a ...
- 11
1
vote
0
answers
101
views
How can the interersection number of $2$ $D6$ branes wrapping around a CY manifold be derived?
For two intersecting $D6$ branes $a$ and $b$ wrapped around a $6$ dimensional torus $T^6 = T^2 \times T^2 \times T^2$ specified by
$$
\textrm{D6-brane a:}\, (l_1^a,l_2^a,l_3^a)
$$
$$
\textrm{D6-...
- 418
1
vote
0
answers
116
views
How to find number of points at infinity of a Riemann surface
Let $X \subset \mathbb C^2$ be a Riemann surface with boundary $\partial X \subset \mathbb C^2$ and without compact components. Let $\bar X = X \cup \{p_1,\ldots,p_N\} \subseteq \mathbb CP^2$ be its ...
- 1,329
0
votes
0
answers
59
views
Spectral analysis of Dirac operators coupled to gauge potential on $\mathbb{R}^n$
Dirac operators on compact manifolds seem to have been studied well, such as in this book and also this one.
However, I cannot easily find comprehensive treatment of Dirac operators coupled to gauge ...
- 3,477
0
votes
0
answers
188
views
Behavior of subtree of $\mathbb{Z}^2$ embedded in $\mathbb{C}$ under compactification of the latter to the riemann sphere
I consider a countable subtree $T$ of the integer lattice isomorphic to $\mathbb{Z}^2$ with directed edges. It shall be embedded in $\mathbb{C}$ where the edge $(u,v)$ points from $u$ to $v$ if and ...
- 91
0
votes
0
answers
153
views
Ergodic action on product spaces
Let $(X_1 \times X_2,d\mu)$ be a measure space with $X_2$ compact. Suppose that we have a continuous (diagonal) action of a topological group $G$ on $X=X_1 \times X_2$. I know that the action of $G$ ...
- 59
0
votes
0
answers
162
views
A ``1-soft'' improvement of the Parovichenko theorem
This is a ``1-soft'' modification of this problem. We start with the necessary definitions.
Definition 1. A compactification $c\mathbb N$ of the discrete space $\mathbb N$ is called 1-soft if for any ...
- 41.8k
0
votes
0
answers
93
views
Can we express separability of a ray-remainder in terms of the function algebra?
Let $X = [0, 1)$ be a ray and $C(X)$ the algebra of bounded continuous real functions. The spectrum of $C(X)$ is the Stone-Cech compactification $\beta [0,1) $ of the ray. It's easy to see the ...
- 1,955