A twisted invariant of a compact Riemann surface

TL;DR

This paper introduces a twisted variant of the Kawazumi-Zhang invariant for compact Riemann surfaces, exploring its relation to the first Mumford-Morita-Milller class and the original invariant, enriching the understanding of moduli space geometry.

Contribution

It presents a new twisted invariant of Riemann surfaces and analyzes its connection to key classes on the moduli space, extending previous invariants.

Findings

Definition of the twisted Kawazumi-Zhang invariant

Relation to the first Mumford-Morita-Milller class

Insights into the geometry of the moduli space

Abstract

We introduce a twisted version of the Kawazumi-Zhang invariant $a_g(C) = \varphi(C)$ of a compact Riemann surface $C$ of genus $g \geq 1$, and discuss how it is related to the first Mumford-Morita-Milller class $e_1 = \kappa_1$ on the moduli space of compact Riemann surfaces and the original Kawazumi-Zhang invariant.