Isomonodromic tau function on the space of admissible covers

TL;DR

This paper investigates the asymptotic behavior and divisor of the isomonodromic tau function on the space of admissible covers, providing explicit formulas relating it to Hodge and boundary classes.

Contribution

It offers a detailed analysis of the tau function's boundary behavior and derives explicit formulas for its divisor in the context of admissible covers.

Findings

Explicit formula for the pullback of the Hodge class

Asymptotic analysis of the tau function near boundary

Computation of the divisor of the tau function

Abstract

The isomonodromic tau function of the Fuchsian differential equations associated to Frobenius structures on Hurwitz spaces can be viewed as a section of a line bundle on the space of admissible covers. We study the asymptotic behavior of the tau function near the boundary of this space and compute its divisor. This yields an explicit formula for the pullback of the Hodge class to the space of admissible covers in terms of the classes of compactification divisors.