Holomorphic sections of line bundles on the jet spaces of the Riemann sphere

TL;DR

This paper studies the structure of holomorphic sections of line bundles on the jet spaces of smooth maps from the circle to the Riemann sphere, revealing new algebraic geometric properties of these infinite-dimensional spaces.

Contribution

It identifies a specific class of holomorphic sections on line bundles over jet spaces of maps from the circle to the Riemann sphere, advancing understanding of their complex geometry.

Findings

Characterization of holomorphic sections on jet spaces

Identification of a class of such sections

Insights into the algebraic structure of jet spaces

Abstract

Fix a point $t_0$ in the circle $S^1$. The space $J^k(t_0, \mathbb{P}^1)$ of $k$-jets at $t_0$ of $C^{\infty}$ maps from $S^1$ to the Riemann sphere $\mathbb{P}^1$ is a $k+1$ dimensional complex algebraic manifold. We identify a class of holomorphic sections of line bundles on $J^k(t_0, \mathbb{P}^1)$.