The Topology of the Space of Holomorphic Maps with Bounded Multiplicity

Abstract

For a complex (quasi-) projective variety X ⊆ ℂP_N_ with π2(X) = ℤ and an integer d ≥ 0, let Hol_d_∗(_S_2, X) denote the space consisting of all basepoint preserving d holomorphic maps f from _S_2 to X with degree d. We study the topology of certain subspaces of Hol_d_∗ (_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.