Journal of the Mathematical Society of Japan

Spaces of algebraic maps from real projective spaces to toric varieties

Andrzej KOZLOWSKI, Masahiro OHNO, and Kohhei YAMAGUCHI

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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 $n_D\to\infty$ as $D\to\infty$. 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.

Article information

J. Math. Soc. Japan, Volume 68, Number 2 (2016), 745-771.

First available in Project Euclid: 15 April 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55R80: Discriminantal varieties, configuration spaces
Secondary: 55P10: Homotopy equivalences 55P35: Loop spaces 14M25: Toric varieties, Newton polyhedra [See also 52B20]

toric variety fan rational polyhedral cone homogenous coordinate primitive element simplicial resolution algebraic map Vassiliev spectral sequence


KOZLOWSKI, Andrzej; OHNO, Masahiro; YAMAGUCHI, Kohhei. Spaces of algebraic maps from real projective spaces to toric varieties. J. Math. Soc. Japan 68 (2016), no. 2, 745--771. doi:10.2969/jmsj/06820745.

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