Simple zero property of some holomorphic functions on the moduli space of tori

TL;DR

This paper proves that certain holomorphic functions on the moduli space of tori have only simple zeros using a conceptual approach involving Painlevé VI equations, leading to insights on degeneracy curves of Green functions.

Contribution

It introduces a novel conceptual proof for the simple zero property of holomorphic functions on the moduli space, avoiding direct derivative computations.

Findings

Holomorphic functions on the moduli space have only simple zeros.

Degeneracy curves of trivial critical points are smooth.

Application to multiple Green functions.

Abstract

We prove that some holomorphic functions on the moduli space of tori have only simple zeros. Instead of computing the derivative with respect to the moduli parameter $\tau$, we introduce a conceptual proof by applying Painlev\'{e} VI\ equation. As an application of this simple zero property, we obtain the smoothness of all the degeneracy curves of trivial critical points for some multiple Green function.