Spaces of algebraic maps from real projective spaces to toric varieties

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 $X$ 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 \cite{AJ}) 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.