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.

中文翻译：

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曲线模空间上的同义反复形式环

我们在微分形式水平的标记曲线的模空间上定义并研究了同义反复环的自然系统。我们证明从这些模空间上的自然正规函数获得的某些 2-形式是同义反复的。我们还表明同义反复形式的环总是有限维的。最后，我们将 Kawazumi-Zhang 不变量描述为本质上唯一的曲线模空间上的光滑函数，其 Levi 形式是同义反复形式。