A signature formula for hyperelliptic broken Lefschetz fibrations

TL;DR

This paper generalizes the concept of local signature from hyperelliptic Lefschetz fibrations to hyperelliptic broken Lefschetz fibrations, providing a formula involving the hyperelliptic mapping class group.

Contribution

It introduces a new formula for the local signature of hyperelliptic broken Lefschetz fibrations, extending Endo's work to a broader class of fibrations.

Findings

Constructed the local signature for hyperelliptic broken Lefschetz fibrations.

Expressed the local signature using Endo's signature and a homomorphism on the hyperelliptic mapping class group.

Provides a computational method for signatures in this generalized setting.

Abstract

A hyperelliptic broken Lefschetz fibration is a generalization of a hyperelliptic Lefschetz fibration. We construct and compute a local signature of hyperelliptic directed broken Lefschetz fibrations by generalizing Endo's local signature of hyperelliptic Lefschetz fibrations. It is described by his local signature and a rational-valued homomorphism on the subgroup of the hyperelliptic mapping class group which preserves a simple closed curve setwise.