Spaces of algebraic maps from real projective spaces to toric varieties

摘要

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) 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.