Multicusps

TL;DR

This paper establishes a decomposition theorem for the source space of a higher version of the reduced Kodaira-Spencer-Mather map for multicusp functions, proving its bijectivity and answering a question in singularity theory.

Contribution

It introduces a direct sum decomposition theorem for the source space of the higher reduced Kodaira-Spencer-Mather map for multicusp functions, demonstrating its bijectivity.

Findings

Proved a direct sum decomposition theorem for the source space.

Showed the bijectivity of the higher reduced Kodaira-Spencer-Mather map.

Provided an affirmative answer to a question posed in singularity theory.

Abstract

For a given multicusp $f=c_{(\theta_0,..., \theta_i)}$ $(1\le i)$, we present a direct sum decomposition theorem of the source space of ${}_i\bar{\omega}f$, where ${}_i\bar{\omega}f$ is a higher version of the reduced Kodaira-Spencer-Mather map $\bar{\omega}f$. As a corollary of our direct sum decomposition theorem, we show that for any $i\in \mathbb{N}$ and any $f=c_{(\theta_0,..., \theta_i)}$, ${}_i\bar{\omega}f$ is bijective. The corollary is an affirmative answer to the question raised by M. A. S. Ruas during the 11th International Workshop on Real and Complex Singularities at the University of S${\tilde {\rm a}}$o Paulo in S${\tilde {\rm a}}$o Carlos (2010).