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Questions tagged [compactifications]

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23
votes
5answers
2k views

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 [...
5
votes
1answer
217 views

Is each compactification of $\mathbb N$ soft?

Definition. A compactification $c\mathbb N$ of the countable discrete space $\mathbb N$ is defined to be soft if for any disjoint sets $A,B\subset\mathbb N\subset c\mathbb N$ with $\bar A\cap\bar B\ne\...
4
votes
2answers
394 views

Bohr compactification as a topological compactification

Let $G$ be a locally compact Hausdorff group. Denote its Bohr compactification by $bG$. Despite group structure, $G$ has several (Hausdorff) compactifications that, in a sense, the smallest one is ...
9
votes
2answers
439 views

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 ...
7
votes
2answers
1k views

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 ...
4
votes
2answers
588 views

Non-uniqueness of smooth compactification

Let $U$ be a smooth quasi-projective variety. Does there always exist a smooth compactification of $U$? If not always when can we have smooth compactification? In particular, suppose $X$ is a ...
7
votes
0answers
416 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 ...
2
votes
1answer
108 views

Locally toric resolutions of compactifications

Suppose $U$ is a smooth, open $n$-dimensional variety over $\mathbb{C}.$ Say $X, X'$ are two proper normal-crossings compactifications of $U$. Call a map $m: X'\to X$ a modification if it is an ...