Spaces of algebraic maps from real projective spaces to toric varieties
Andrzej Kozlowski, Masahiro Ohno, Kohhei Yamaguchi

TL;DR
This paper studies how well finite-dimensional algebraic maps approximate the space of all continuous maps from real projective spaces to toric varieties, focusing on homology and homotopy isomorphisms as degrees grow.
Contribution
It advances understanding of the approximation problem for algebraic maps into toric varieties, especially in the context of real projective spaces, and explores homological and homotopical properties.
Findings
Identifies conditions for homology isomorphisms up to a certain dimension.
Establishes bounds on the degree for approximation accuracy.
Provides new insights into the topology of algebraic mapping spaces.
Abstract
The problem of approximating the infinite dimensional space of all continuous maps from an algebraic variety to an algebraic variety 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 \cite{AJ}) is to determine a (preferably optimal) integer 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 , where denotes a tuple of integers called the "degree" of the algebraic maps and as . In this paper we investigate this problem in the case when is a real projective space and is a smooth compact toric variety.
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Algebraic Geometry and Number Theory · Commutative Algebra and Its Applications
