The Topology of the Space of Holomorphic Maps with Bounded Multiplicity

Abstract

For a complex (quasi-) projective variety $X\subseteq C{P}^N$ with ${\pi }_2(X)=Z$ and an integer $d\ge 0$, let ${\mathrm{Hol}}_d^{\ast }(S^2,X)$ denote the space consisting of all basepoint preserving holomorphic maps $f$ from $S^2$ to $X$ with degree $d$. We study the topology of certain subspaces of ${\mathrm{Hol}}_d^{\ast }(S^2,X)$ defined using the concept of multiplicity of roots, and we d show that the Atiyah–Jones–Segal type theorem ([1], [11]) holds for these subspaces if $X$ is belong to a certain family of complex quasi-projective varieties.