On spaces of holomorphic maps from two copies of the Riemann sphere to complex Grassmannians

Abstract

This thesis studies the structure of the space $X\sb{k}(f,g)$ of holomorphic maps from $\IP \sp{1} \times \IP \sp1$ to the complex Grassmannian $G(2,s)$ which induce second Chern class k and whose restriction to $\IP \sp{1} \vee \IP \sp1$ is $f \vee g$. As a space of maps from $\IP \sp{1} \times \IP \sp1$ to a variety, $X\sb{k}(f,g)$ exhibits geometry in transcendence degree two, which has been little understood and which is typically much more complicated than the geometry of spaces of holomorphic maps from $\IP \sp1$ to a variety. $X\sb{k}(f,g)$ is also related to the moduli space ${\cal M}\sb{k}$ of SU(2)-instantons on $S\sp4$ with charge k. The basing of the maps in $X\sb{k}(f,g)$ by fixing their restriction to $\IP \sp{1} \vee \IP \sp1$ is motivated from the description of ${\cal M}\sb{k}$ in terms of rank two holomorphic vector bundles on $\IP \sp{1} \times \IP \sp1$ given by Atiyah and Donaldson. The main result of this thesis gives a complete description of some examples of $X\sb{k}(f,g)$ where f is taken in a simple form, but of arbitrary degree, and where g is a generic map of degree one. Specifically, the examples of $X\sb{k}(f,g)$ are shown to be homeomorphic to spaces whose homology and stable homotopy type are known. The main result is proved by analyzing a space which is equivalent to $X\sb{k}(f,g).$ Using adjointness, the elements of $X\sb{k}(f,g)$ can be identified with holomorphic sections of a bundle E(f) over $\IP \sp1$ which is induced by the map f. The sections are based according to the map g. The space of based holomorphic sections of E(f) is understood by proceeding in analogy with spaces of holomorphic sections of line bundles over $\IP \sp1$. Using the manifold structure of the fiber, one finds normal forms for the sections over trivializing neighborhoods for the bundle E(f) and checks the constraint conditions for patching them together over the intersections.