Tau function and moduli of differentials

TL;DR

This paper studies the tau function on the moduli space of holomorphic differentials, analyzing its boundary behavior and relating it to geometric classes, thereby clarifying the Kontsevich-Zorich formula for Lyapunov exponents.

Contribution

It provides an explicit expression for the Hodge class pullback in terms of tautological and boundary classes, linking tau function asymptotics to the geometry of moduli spaces.

Findings

Asymptotics of tau function near boundary computed

Explicit formula for Hodge class pullback derived

Clarification of Kontsevich-Zorich formula achieved

Abstract

The tau function on the moduli space of generic holomorphic 1-differentials on complex algebraic curves is interpreted as a section of a line bundle on the projectivized Hodge bundle over the moduli space of stable curves. The asymptotics of the tau function near the boundary of the moduli space of 1-differentials is computed, and an explicit expression for the pullback of the Hodge class on the projectivized Hodge bundle in terms of the tautological class and the classes of boundary divisors is derived. This expression is used to clarify the geometric meaning of the Kontsevich-Zorich formula for the sum of the Lyapunov exponents associated with the Teichm\"uller flow on the Hodge bundle.