Pointed harmonic volume and its relation to the extended Johnson homomorphism

TL;DR

This paper explores the pointed harmonic volume, a complex analytic invariant of Riemann surfaces, its explicit computation for certain curves, and its connection to the extended Johnson homomorphism in the mapping class group.

Contribution

It introduces the pointed harmonic volume, computes its value for a specific hyperelliptic curve, and elucidates its relationship with the extended Johnson homomorphism.

Findings

Computed the pointed harmonic volume for a particular hyperelliptic curve.

Established a relationship between harmonic volume and the extended Johnson homomorphism.

Provided an application demonstrating the utility of the pointed harmonic volume.

Abstract

The period for a compact Riemann surface, defined by the integral of differential 1-forms, is a classical complex analytic invariant, strongly related to the complex structure of the surface. In this paper, we treat another complex analytic invariant called the pointed harmonic volume. As a natural extension of the period defined using Chen's iterated integrals, it captures more detailed information of the complex structure. It is also one of a few explicitly computable examples of complex analytic invariants. We obtain its new value for a certain pointed hyperelliptic curve. An application of the pointed harmonic volume is presented. We explain the relationship between the harmonic volume and first extended Johnson homomorphism on the mapping class group of a pointed oriented closed surface.