Rings of tautological forms on moduli spaces of curves

TL;DR

This paper introduces a system of tautological rings of differential forms on moduli spaces of marked curves, demonstrating their finiteness and characterizing the Kawazumi-Zhang invariant within this framework.

Contribution

It defines tautological rings of forms on moduli spaces and proves their finite dimensionality, also characterizing a key invariant as a tautological form.

Findings

Certain 2-forms from normal functions are tautological

Tautological rings of forms are finite dimensional

Kawazumi-Zhang invariant characterized as a tautological Levi form

Abstract

We define and study a natural system of tautological rings on the moduli spaces of marked curves at the level of differential forms. We show that certain 2-forms obtained from the natural normal functions on these moduli spaces are tautological. Also we show that rings of tautological forms are always finite dimensional. Finally we characterize the Kawazumi-Zhang invariant as essentially the only smooth function on the moduli space of curves whose Levi form is a tautological form.