Computation of $\lambda$-classes via strata of differentials

TL;DR

This paper introduces new tautological relations in the moduli space of stable curves, derived from the Poincaré-dual classes of empty loci in the Hodge bundle, leading to novel expressions for Chern classes.

Contribution

It provides a new family of tautological relations and expresses Chern classes of the Hodge bundle using stable graphs with limited loops, confirming a conjecture.

Findings

New tautological relations derived from empty loci in the Hodge bundle

Expressed Chern classes using stable graphs with at most i loops

Confirmed the DR/DZ equivalence conjecture for top Chern classes

Abstract

We introduce a new family of tautological relations of the moduli space of stable curves of genus $g$. These relations are obtained by computing the Poincar\'e-dual class of empty loci in the Hodge bundle. We use these relations to obtain a new expression for the Chern classes of the Hodge bundle. We prove that the $(g-i)$th class can be expressed as a linear combination of tautological classes involving only stable graphs with at most $i$ loops. In particular the top Chern class may be expressed with trees. This property was expected as a consequence of the DR/DZ equivalence conjecture by Buryak-Gu\'er\'e-Rossi.