Spaces of rational maps and the Stone–Weierstrass theorem

Mostovoy, Jacob

doi:10.1016/j.top.2005.08.003

Cited by 33 publications

(44 citation statements)

References 12 publications

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“…The strategy of the proof is exactly the same as in [18] and [19] and mirrors other applications of Vassiliev's method. While it contains no essential novelty as compared to the case of a projective space [19], we find it necessary to go through the whole proof in some detail, since the case of a general smooth toric variety involves a number of additional features.…”

Section: Toric Varietiesmentioning

confidence: 96%

“…The principal tool in the proof of this statement is the Stone-Weierstrass Theorem for vector bundles (see, for instance, [18]): The Stone-Weierstrass Theorem has the following consequence:…”

Section: The Stone-weierstrass Theoremmentioning

confidence: 99%

“…The purpose of this note is to show that the proof given in [18,19] extends to the spaces of maps from CP m to a smooth compact toric variety. We shall use the homogeneous coordinates for a toric variety X and the description of the morphisms CP m → X given by Cox in [5] and [6]: a morphism CP m → X is given in these coordinates by a collection of homogeneous polynomials with certain properties.…”

Section: Introductionmentioning

confidence: 97%

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Spaces of Morphisms From a Projective Space to a Toric Variety

Mostovoy¹,

Munguia-Villanueva²

2014

Revista Colombiana de Matemáticas

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In this paper we study the space of morphisms from a complex projective space to a compact smooth toric variety X. It is shown that the first author's stability theorem for the spaces of rational maps from CP m to CP n extends to the spaces of continuous morphisms from CP m to X essentially, with the same proof. In the case of curves, our result improves the known bounds for the stabilization dimension.2010 Mathematics Subject Classification. Primary: 58D15; Secondary: 32Q55.

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Section: Toric Varietiesmentioning

confidence: 96%

Section: The Stone-weierstrass Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

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Spaces of Morphisms From a Projective Space to a Toric Variety

Mostovoy¹,

Munguia-Villanueva²

2014

Revista Colombiana de Matemáticas

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“…In this section, we summarize the definitions of the non-degenerate simplicial resolution and the associated truncated simplicial resolutions ( [19], [24]). Definition 5.1.…”

Section: Simplicial Resolutionsmentioning

confidence: 99%

The homotopy type of spaces of rational curves on a toric variety

Kozłowski¹,

Yamaguchi²

2018

Topology and its Applications

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Spaces of holomorphic maps from the Riemann sphere to various complex manifolds (holomorphic curves ) have played an important role in several area of mathematics. In a seminal paper G. Segal investigated the homotopy type of holomorphic curves on complex projective spaces and M. Guest on compact smooth toric varieties.. Recently Mostovoy and Villanueva, obtained a far reaching generalisation of these results, and in particular (for holomorphic curves) improved the stability dimension obtained by Guest. In this paper, we generalize their result to holomorphic curves, on certain non-compact smooth toric varieties.In [17], the present authors studied this problem of the case m = 1 for a certain family of non-compact smooth toric subvarieties X I of CP n , and showed that the result of Mostovoy-Villanueva [21] can be extended to this case (with the same stability dimension).In this paper, we shall prove that this result can be further extended to non-compact smooth toric varieties X which satisfy certain two conditions (see the conditions (1.12.1) and (1.12.2)). These conditions are satisfied for a wide range of smooth toric varieties (including all compact ones).In fact, we will do better and show that under the certain condition "homology equivalence"can be replaced by "homotopy equivalence"(up to the same dimension).The broad outline of our argument is analogous to Segal's seminal paper [23] (a brief sketch of such an argument is given in [10, §5]). Namely, for a smooth toric variety X, we first prove that there is a homotopy equivalence between certain limits of spaces Hol * D (S 2 , X) of holomorphic maps, stabilized with respect to a suitably defined degree D, and the double loop space Map * (S 2 , X) = Ω 2 X. We can refer to this as the stable result. The method used to prove it is a generalization of the scanning map technique used by Segal in [23]. In particular, we describe a generalization to the case of toric varieties of a fibration sequence that plays the key role in Segal's argument (see Proposition 3.4).Note that in [21] a quite different stabilization is used, which is based on the Stone-Weierstrass theorem. This stabilization has the advantage that it can used in the case of holomorphic maps from CP m to a compact toric variety X for any m ≥ 1. However, the usefulness of the Stone-Weierstrass theorem is based on the fact that two holomorphic maps that are 'uniformly close', with respect to some metric, are actually homotopic. This is true when the metric on X is complete (e.g. when X is compact), but not for general X. We are, therefore, unsure if our results can be extended to the case Hol * (CP m , X), for m > 1. Even if this is possible, we believe that our generalization of Segal's argument to the case of toric varieties is of some independent interest.The second part of the paper is concerned with establishing "homology stability dimensions" for the inclusion map from Hol * D (S 2 , X) to the double loop space Ω 2 D X of maps of degree D. These stability dimensions depend both on the degree D ...

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“…In this section, we summarize the definitions of the non-degenerate simplicial resolution and the associated truncated resolutions ( [2], [14], [17], [18], [22]).…”

Section: Simplicial Resolutionsmentioning

confidence: 99%

Spaces of algebraic maps from real projective spaces to toric varieties

Kozłowski¹,

Ohno²,

Yamaguchi³

2016

J. Math. Soc. Japan

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Abstract. The problem of approximating the infinite dimensional space of all continuous maps from an algebraic variety X to an algebraic variety Y by finite dimensional spaces of algebraic maps arises in several areas of geometry and mathematical physics. An often considered formulation of the problem (sometimes called the Atiyah-Jones problem after [1]) is to determine a (preferably optimal) integer n D such that the inclusion from this finite dimensional algebraic space into the corresponding infinite dimensional one induces isomorphisms of homology (or homotopy) groups through dimension n D , where D denotes a tuple of integers called the "degree"of the algebraic maps and n D → ∞ as D → ∞. In this paper we investigate this problem in the case when X is a real projective space and Y is a smooth compact toric variety. Introduction.Let X and Y be manifolds with some additional structure S, e.g. holomorphic, symplectic, real algebraic etc. Let S(X, Y ) denote the space of base-point preserving continuous maps f : X → Y preserving the structure S and let Map * (X, Y ) be the space of corresponding continuous maps. The relation between the topology of the spaces S(X, Y ) and Map * (X, Y ) has long been an object of study in several areas of topology and geometry (e.g. In [2] and [14] the case where the structure S is that of a real algebraic variety was considered, and integers n D were found, such that the natural projection map

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Spaces of rational maps and the Stone–Weierstrass theorem

Abstract: It is shown that Segal's theorem on the spaces of rational maps from CP 1 to CP n can be extended to the spaces of continuous rational maps from CP m to CP n for any m n. The tools are the Stone-Weierstrass theorem and Vassiliev's machinery of simplicial resolutions.

Cited by 33 publications

References 12 publications

Spaces of Morphisms From a Projective Space to a Toric Variety

Spaces of Morphisms From a Projective Space to a Toric Variety

The homotopy type of spaces of rational curves on a toric variety

Spaces of algebraic maps from real projective spaces to toric varieties

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