# Questions tagged [compactifications]

The compactifications tag has no usage guidance.

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### 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 $\...

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### A product on the square roots of unit matrix

There is a strange product that takes two square roots of unit matrix, say $A$ and $B$, $A^2=I$, $B^2=I$ to a square root again,
$$ A\star B=(A+B)^{-1}(A-B+2I), \qquad (A\star B)^2=I$$
Could anybody ...

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### Positive function with zero Haar integral

If $C$ is a compact (semi-)topological (semi-)group, are there nonzero positive functions having zero Haar integral? In other words: is the Hermitian product associated to the Haar integral degenerate?...

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### Locally compact, 0-dimensional, pseudocompact space

Is a 0-dimensional, locally compact and pseudocompact space $X$ necessarily strongly 0-dimensional? I.e., must $\beta X$ be 0-dimensional?
It is known that a 0-dimensional locally compact space which ...

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### Does locally compact plus pseudocompact imply paracompact?

This one is probably simple, but I don't see it yet.
Is a locally compact, pseudocompact Hausdorff space necessarily paracompact?

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### When is the Freudenthal compactification an ANR?

Let $X$ be a locally compact metric ANR (or, if preferred, a locally compact simplicial complex). If needed, assume that $X$ has finitely many ends or is of finite dimension. My question is:
What ...

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### 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 ...

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### Stone-Čech compactification of $\mathbb R$

Let $\beta X$ - is a Stone-Čech compactification of $X$. $I=(-1,1)$ - is an interval of the real line. Is it true that $\beta \mathbb R\setminus I = \beta(\mathbb R\setminus I)$? In other words, it ...

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### Non-bimeromorphic compactifications

Let $X$ be a (smooth, non-compact) complex space. By a compactification of $X$ we mean a compact complex space $\overline X$ which contains a dense open subset biholomorphic to $X$ (we shall identify ...

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### Ends of topological spaces. Why independent of choice of ascending sequence of compact subsets?

Quoting from http://en.wikipedia.org/wiki/End_(topology):
"Let X be a topological space, and suppose that
K1 ⊂ K2 ⊂ K3 ⊂ · · ·
is an ascending sequence of compact ...

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### Direct construction of the Stone-Čech compactification using ultrafilters?

If $X$ is a set (regarded as a discrete space), its Stone-Čech compactification can be identified with the set of ultrafilters on $X$ with its natural (Stone) topology. If $X$ is a general ...

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### End point compactification for metric spaces

Freundenthal introduced ends of topological spaces and the end point compactification of locally compact topological spaces adding one point for each end of the topological space (see here).
For ...

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### Flatly compactifiable morphisms

Let $f:U \to S$ be a flat morphism. Let us say that $f$ is flatly compactifiable if there exists a proper morphism $\bar{f}:X \to S$ and a closed subscheme $Z \subset X$ such that
1) $U = X \...

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### Compactifications of varieties with small complement

Let $X$ be a smooth variety over an algebraically closed field $k$. If it makes things easier, $X$ may be assumed to be quasi-projective. By Nagata (or quasi-projectivity) there exists a proper ...

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### Reference Request for Drinfeld and Laumon Compactifications

Background
Let $X$ denote a smooth projective curve over $\mathbb{C}$ and let $G$ denote a semi-simple simply connected algebraic group over $\mathbb{C},$ which has associated flag variety $G/B.$
...

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### 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\...

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### Compactification theorem for differentiable manifolds ?

Just parallelling this question, that seemed not to admit an easy answer at all, let's "soft down" the category and ask the same thing in the case of $\mathcal{C}^{\infty}$-differentiable manifolds [...

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### “Transitivity” of the Stone-Cech compactification

Let $\beta \mathbb{N}$ be the Stone-Cech compactification of the natural numbers $\mathbb{N}$, and let $x, y \in \beta \mathbb{N} \setminus \mathbb{N}$ be two non-principal elements of this ...