Regular Compactifications Research Articles

Constructive strong regularity and the extension property of a compactification

In contexts in which the principle of dependent choice may not be available, as toposes or Constructive Set Theory, standard locale theoretic results related to complete regularity may fail to hold. To resolve this difficulty, B. Banaschewski and A. Pultr introduced strongly regular locales. Unfortunately, Banaschewski and Pultr's notion relies on non-constructive set existence principles that hinder its use in Constructive Set Theory. In this article, a fully constructive formulation of strong regularity for locales is introduced by replacing non-constructive set existence with coinductive set definitions, and exploiting the Relation Reflection Scheme. As an application, every strongly regular locale L is proved to have a compact regular compactification. The construction of this compactification is then used to derive the main result of this article: a characterization of locale compactifications (and thus, classically, of the compactifications of a space) in terms of their ability of extending continuous functions with compact regular codomains. Finally, an open problem related to the existence of the compact regular reflection of a locale is presented.

K‐theory of regular compactification bundles

AbstractLet G be a split connected reductive algebraic group. Let be a ‐torsor over a smooth base scheme and X be a regular compactification of G. We describe the Grothendieck ring of the associated fibre bundle , as an algebra over the Grothendieck ring of a canonical toric bundle over a flag bundle over . These are relative versions of the corresponding results on the Grothendieck ring of X in the case when is a point, and generalize the classical results on the Grothendieck rings of projective bundles, toric bundles and flag bundles.

Perverse obstructions to flat regular compactifications

Suppose \(\pi :W\rightarrow S\) is a smooth, surjective, proper morphism to a variety S contained as a Zariski open subset in a smooth, complex variety \({\bar{S}}\). The goal of this note is to consider the question of when \(\pi \) admits a regular, flat compactification. In other words, when does there exists a flat, proper morphism \({\bar{\pi }}:\overline{W}\rightarrow {\bar{S}}\) extending \(\pi \) with \(\overline{W}\) regular? One interesting recent example of this occurs in the preprint of Laza et al. (Acta Mathematica. arXiv:1602.05534, 2016) where \(\pi \) is a family of abelian fivefolds over a Zariski open subset S of \({\bar{S}}=\mathbb {P}^5\). In that paper, the authors construct \(\overline{W}\) using the theory of compactified Prym varieties and show that it is a holomorphic symplectic manifold (deformation equivalent to O’Grady’s 10-dimensional example). In this note I observe that non-vanishing of the local intersection cohomology of \(R^1\pi _*\mathbb {Q}\) in degree at least 2 provides an obstruction to finding a \({\bar{\pi }}\). Moreover, non-vanishing in degree 1 provides an obstruction to finding a \({\bar{\pi }}\) with irreducible fibers. Then I observe that, in some cases of interest, results of Brylinski (Asterisque 140–141:3–134, 251, 1986) Beilinson (Regulators 571:19–23, 2012) and Schnell (Math Ann 354(2):727–763, 2012) can be used to compute the intersection cohomology. I also give examples involving cubic fourfolds motivated by Laza et al. (Acta Mathematica. arXiv:1602.05534, 2016) and ask a question about palindromicity of hyperplane sections.

Equivariant -theory of regular compactifications: further developments

We describe the -equivariant -ring of , where is a factorial covering of a connected complex reductive algebraic group , and is a regular compactification of . Furthermore, using the description of , we describe the ordinary -ring as a free module (whose rank is equal to the cardinality of the Weyl group) over the -ring of a toric bundle over whose fibre is equal to the toric variety associated with a smooth subdivision of the positive Weyl chamber. This generalizes our previous work on the wonderful compactification (see [1]). We also give an explicit presentation of and as algebras over and respectively, where is the wonderful compactification of the adjoint semisimple group . In the case when is a regular compactification of , we give a geometric interpretation of these presentations in terms of the equivariant and ordinary Grothendieck rings of a canonical toric bundle over .

Compactification: Limit Tower Spaces

Convergence approach spaces, defined by E. Lowen and R. Lowen, possess both quantitative and topological properties. These spaces are equipped with a structure which provides information as to whether or not a sequence or filter approximately converges. P. Brock and D. Kent showed that the category of convergence approach spaces with contractions as morphisms is isomorphic to the category of limit tower spaces. It is shown below that every limit tower space has a compactification. Moreover, a characterization of the limit tower spaces which possess a strongly regular compactification is given here. Further, a strongly regular S-compactification of a limit tower space is studied, where S is a limit tower monoid acting on the limit tower space.

Equivariant K-theory of compactifications of algebraic groups

In this paper we describe the G × G-equivariant K-ring of X, where X is a regular compactification of a connected complex reductive algebraic group G. Furthermore, in the case when G is a semisimple group of adjoint type, and X its wonderful compactification, we describe its ordinary K-ring K(X). More precisely, we prove that K(X) is a free module over K(G/B) of rank the cardinality of the Weyl group. We further give an explicit basis of K(X) over K(G/B), and also determine the structure constants with respect to this basis.

On Intersection Indices of Subvarieties in Reductive Groups

In this paper, I give an explicit formula for the intersection indices of the Chern classes (defined in [11]) of an arbitrary reductive group with hypersurfaces. This formula has the following applications. First, it allows to compute explicitly the Euler characteristic of complete intersections in reductive groups thus extending the beautiful result by D.Bernstein and Khovanskii, which holds for a complex torus. Second, for any regular compactification of a reductive group, it computes the intersection indices of the Chern classes of the compactification with hypersurfaces. The formula is similar to the Brion– Kazarnovskii formula for the intersection indices of hypersurfaces in reductive groups. The proof uses an algorithm of De Concini and Procesi for computing such intersection indices. In particular, it is shown that this algorithm produces the Brion–Kazarnovskii formula.

Regular compactifications and Higgs model vortices

We present full numerical solutions to the system of a global string embedded in a six-dimensional space-time. The solutions are regular everywhere and do confine gravity in our four-dimensional world. They depend on the value of the (negative) cosmological constant in the bulk and on the parameters of the Higgs potential, and we perform a systematic study to determine their allowed values. We also comment on the relation of our results with previous studies on the same subject and on their phenomenological viability.

Compactifications and A-compactifications of frames. Proximal frames

The aim of this paper is to give two new descriptions of the ordered set of all (up to equivalence) regular compactifications of a completely regular frame. F and to introduce and study the notion of A-frame as a generalization of the notion of Alexandroff space (known also as zero-set space) (Alexandroff[l], Gordon[15]). A description of the ordered set of all (up to equivalence) A-compactifications of an A-frame by means of an ordered by inclusion set of some distributive lattices (called AP-sublattices) is obtained. It implies that any A-frame has a greatest A-compactification and leads to the descriptions of A new category isomorphic to the category of proximal frames is introduced. A question for compactifications of frames analogous to the R. Chandler's question [8, p. 71] for compactifications of spaces is formulated and solved. Many results of [1], [3], [15], [23], [9], [10] and [11] are generalized.

Cohomology of regular embeddings

We give in this paper an explicit description of the rational cohomology ring of a “complete symmetric variety” or regular compactification of an algebraic symmetric variety. The motivation for this computation comes from the desire to describe an explicit Schubert calculus for these varieties, which have been considered since the work of Chasles and Schubert in several special cases. The technique we use is the result of our attempts to understand better the work of Jurkiewicz and Danilov in the case of torus embeddings. The combinatorial presentation of the cohomology ring of a smooth torus embedding finds a natural explanation and proof using the language of equivariant cohomology. The key point is that the Reisner-Stanley algebra of the torus embedding coincides with its equivariant cohomology ring. We consider a class of spaces, regular embeddings, that contains both “complete symmetric varieties” and smooth torus embeddings. By suitably generalizing the notion of Reisner-Stanley algebra, we are able to compute the rational equivariant cohomology ring of these spaces in an explicit way.

A class of extension properties that are not simply generated

Let P be a closed-hereditary topological property preserved by products. Call a space P -regular if it is homeomorphic to a subspace of a product of spaces with P . Suppose that each P -regular space possesses a P -regular compactification. It is well-known that each P -regular space X is densely embedded in a unique space γ scP X with P such that if f: X → Y is continuous and Y has P , then f extends continuously to γ scP X. Call P -pseudocompact if γ scP X is compact. Associated with P is another topological property P #, possessing all the properties hypothesized for P above, defined as follows: a P -regular space X has P # if each P -pseudocompact closed subspace of X is compact. It is known that the P -pseudocompact spaces coincide with the P #-pseudocompact spaces, and that P # is the largest closed-hereditary, productive property for which this is the case. In this paper we prove that if P is not the property of being compact and P -regular, then P # is not simply generated; in other words, there does not exist a space E such that the spaces with P # are precisely those spaces homeomorphic to closed subspaces of powers of E.

Topological Extension Properties and Projective Covers

Introduction. All spaces considered in this paper are assumed to be (Hausdorff) completely regular, and all maps are continuous. Let be a topological property of spaces. We shall identify with the class of spaces having . A space having is called a -space, and a subspace of a -space is called a -regular space. The class of -regular spaces is denoted by R(). Following [37], we call a closed hereditary, productive, topological property such that each -regular space has a -regular compactification a topological extension property, or simply, an extension property. In this paper, we restrict our attention to extension properties satisfying the following axioms:(A1) The two-point discrete space has .(A2) If each -regular space of nonmeasurable cardinal has , then = R().

A class of Wallman-type extensions

This paper grew out of an attempt to determine what T2-compactifications, the Wallman compactification, and the one-point compactification have in common. It turns out that in each case the associated nearness is generated by those grills which contain ultraclosed filters and are clans with respect to the associated proximity. Such a nearness will be called a Wallman nearness, and this paper is a study of the properties of Wallman nearnesses and their extensions. A mild covering condition on a proximity guarantees that it contains a Wallman nearness. Each covered proximity contains exactly one Wallman nearness. This sets up a 1-1 correspondence between covered proximities and extensions obtained from Wallman nearnesses. The latter will be called Wallman-type extensions. These can be characterized by the fact that they are covered extensions and satisfy a certain completeness property; namely, the duals of certain clans must converge. This summarizes the first two sections. The last section is a study of compact Wallman-type extensions. A condition on the proximity is obtained which guarantees that the associated Wallman-type extension is compact. The condition states that certain very large grills containing ultraclosed filters must be clans with respect to the given proximity. Such a proximity will be called a compactification proximity. It turns out that compactification proximities give rise to weakly regular compactifications. The paper ends with a study of the relation between weak regularity and Wallman-type extensions.

Completely regular and 𝜔-regular spaces

The completely regular convergence spaces are characterized as those spaces having symmetric compactifications. The ω \omega -regular convergence spaces are those which have regular compactifications.

𝑇-regular-closed convergence spaces

It is known that a convergence space which has a regular compactification is almost identical to a completely regular topological space. It is shown that a less restricitive class of convetgence spaces have T T -regular-closed extensions with the universal property of the Stone-Čech compactification.

Regularity of Richardson's Compactification

In [7] Richardson constructed a Stone-Čech type compactification R(E) of a Hausdorff convergence space E. Two questions arise in this regard. First, when is R(E) homeomorphic to β(E), β(E) the topological Stone-Čech compactification of E, for a Tychonoff topological space E? Second, if E is a regular convergence space, when is R(E) regular? The last question is motivated by the study of regular compactifications in [6]. In section 2 it will be shown that a necessary and sufficient condition in answer to both questions, is that α = cl(α) for each nonconvergent ultrafilter α on E.

On smallest compactification for convergence spaces

In this note we obtain necessary and sufficient conditions for a convergence space to have a smallest Hausdorff compactification and to have a smallest regular compactification.

Regular compactifications of convergence spaces

This note gives a simple characterization for the class of convergence spaces for which regular compactifications exist and shows that each such convergence space has a largest regular compactification.

Completely regular compactifications

Completely regular compactifications