Homological Stability among Moduli Spaces of Holomorphic Curves in Complex Projective Space

TL;DR

This paper investigates the homological stability of moduli spaces of holomorphic curves in complex projective space, using homotopy theory and Gromov-Witten theory concepts to approximate and analyze their topology.

Contribution

It introduces a homotopy theoretic approach to approximate moduli spaces of holomorphic maps, extending Segal's methods to include partial compactifications with stable maps.

Findings

Established homological stability results for these moduli spaces

Connected the topology of holomorphic maps with Gromov-Witten theory

Provided new homotopy theoretic models for moduli space analysis

Abstract

The primary goal of this paper is to find a homotopy theoretic approximation to moduli spaces of holomorphic maps Riemann surfaces into complex projective space. There is a similar treatment of a partial compactification of these moduli spaces of consisting of irreducible stable maps in the sense of Gromov-Witten theory. The arguments follow those from a paper of G. Segal on the topology of the space of rational functions.