Boundary Expression for Chern Classes of the Hodge Bundle on Spaces of Cyclic Covers

TL;DR

This paper derives explicit formulas for the first and second Chern classes of the Hodge bundle on spaces of cyclic covers, enabling recursive calculations of specific Hodge integrals involving lambda classes.

Contribution

It provides new explicit boundary formulas for Chern classes of the Hodge bundle on cyclic cover moduli spaces, facilitating recursive computations of Hodge integrals.

Findings

Explicit formula for the first Chern class of the Hodge bundle on cyclic covers.

Recursive method for calculating Hodge integrals with lambda classes.

Explicit boundary expression for the second Chern class in the case of Z/2Z covers.

Abstract

We compute an explicit formula for the first Chern class of the Hodge Bundle over the space of admissible cyclic $\mathbb{Z}/3\mathbb{Z}$ covers of $n$-pointed rational stable curves as a linear combination of boundary strata. We then apply this formula to give a recursive formula for calculating certain Hodge integrals containing $\lambda_1$. We also consider covers with a ${\mathbb{Z}}/{2\mathbb{Z}}$ action for which we compute $\lambda_2$ as a linear combination of codimension two boundary strata.