Harmonic volume and its applications

TL;DR

This paper reviews the harmonic volume, a complex analytic invariant extending the classical period, and demonstrates its applications in analyzing the moduli space of Riemann surfaces and algebraic cycles.

Contribution

It introduces the harmonic volume as a computable invariant that captures detailed complex structure information and applies it to study algebraic cycles and the moduli space.

Findings

Harmonic volume extends classical period invariants.

Provides an algorithm to prove nontriviality of algebraic cycles.

Enables quantitative analysis of the moduli space structure.

Abstract

The period is a classical complex analytic invariant for a compact Riemann surface defined by integration of differential 1-forms. It has a strong relationship with the complex structure of the surface. In this chapter, we review another complex analytic invariant called the harmonic volume. It is a natural extension of the period defined using Chen's iterated integrals and captures more detailed information of the complex structure. It is also one of a few explicitly computable examples of complex analytic invariants. As an application, we give an algorithm in proving nontriviality for a class of homologically trivial algebraic cycles obtained from special compact Riemann surfaces. The moduli space of compact Riemann surfaces is the space of all biholomorphism classes of compact Riemann surfaces. The harmonic volume can be regarded as an analytic section of a local system on the moduli space. It enables a quantitative study of the local structure of the moduli space. We explain basic concepts related to the harmonic volume and its applications of the moduli space.