Computations of λ-classes via strata of differentials

We introduce a new set of relations in the tautological Chow rings of the moduli space of stable curves of genus g. These relations are obtained by computing the Poincaré-dual class of empty loci in the Hodge bundle in terms of the standard generators of the tautological rings. In particular, we use these relations to obtain a new expression for the Chern classes of the Hodge bundle. We prove that the \((g-i)\)th Chern class of the Hodge bundle, can be expressed as a linear combination of tautological classes constructed from stable graphs with at most i loops. In particular, the top Chern class can be expressed with trees. This property was expected as a consequence of the DR/DZ equivalence conjecture by Buryak–Guéré–Rossi.