A vanishing theorem in Siefring's intersection theory

TL;DR

This paper studies the asymptotic behavior of punctured pseudoholomorphic curves and shows that, under generic conditions, the additional contributions to intersection numbers and singularity indices vanish, simplifying their analysis.

Contribution

It provides a stratification of the moduli space to characterize when asymptotic contributions vanish and proves these contributions disappear under generic perturbations.

Findings

Asymptotic contributions to intersection numbers vanish generically.

Stratification of the moduli space describes convergence rates.

Simplification of intersection theory for pseudoholomorphic curves.

Abstract

In 2009, R. Siefring introduced a homotopy-invariant generalized intersection number and singularity index for punctured pseudoholomorphic curves, by adding contributions from curve's asymptotic behavior to the standard intersection number and singularity index. In this article, we provide a stratification of the moduli space that describes the rate of asymptotic convergence of the pseudoholomorphic curves. Using this stratification, we provide a more intricate characterization of the curves for which these added contribution to the intersection number and singularity index vanishes. In doing so, we prove that the asymptotic contribution to intersection number and singularity index vanishes under generic perturbations. This means that in generic situations we only need to consider the usual intersections of the curves.