The topology of spaces of rational curves on certain toric variaties

(Joint with K. Yamaguchi)

The space of holomorphic maps from $ S^2$ to a complex algebraic variety $X$ arises is several areas of geometry and physics. In a seminal paper  Segal showed that in the the case $X=\CP^n$ the inclusion of the space of degree $d$ holomorphic maps in the space of continuous mappings (i.e. double loops on $\CP^n$) induces an isomorphism on homotopy groups up to dimension $n(d)$, where $n(d)\to \infty$ as $d\to \infty$. Segal conjectured that analogous results held for other varieties $X$ and even when $S^2$ is replaced by higher dimensional complex algebraic varieties, such as $\CP^k$. In the following years many such generalisations and extensions  have been proved. In particular, Martin Guest  showed that Segal like result hold for rational curves on a compact toric variety. Recently Mostovoy and Munguia-Villanueva  using the methods due to V. Vassiliev, extended Guest's results to maps from $\CP^k$ (for any $k\ge 1$ ) to compact toric varieties and obtained a much better bound for $n(d)$. In this talk I will consider a simpler case, where $X$ is a certain non-compact toric variety, about which results of the above kind were first proved 20 years ago in [M.A. Guest, A. Kozlowski and K. Yamaguchi].  I will show that using a mixture  of Segal's original method and Vassiliev's approach one can obtain a a much better bound than the one given in [GKY], which can be shown to coincide with the one obtained by Mostovoy and Munguia-Villanueva for compact toric varieties.