Second variation of Zhang's lambda-invariant on the moduli space of curves

TL;DR

This paper computes the second variation of Zhang's lambda-invariant on the moduli space of curves, revealing its relation to Hain and Reed's beta-invariant and providing explicit calculations for hyperelliptic curves.

Contribution

It establishes a precise relationship between the lambda-invariant and the beta-invariant, and computes the lambda-invariant explicitly for hyperelliptic curves.

Findings

(8g+4)λ equals the β-invariant up to a constant.

Lambda-invariant for hyperelliptic curves expressed via Petersson norm.

Second variation of λ-invariant computed using Kawazumi's work.

Abstract

We compute the second variation of the \lambda-invariant, recently introduced by S. Zhang, on the complex moduli space M_g of curves of genus g>1, using work of N. Kawazumi. As a result we prove that (8g+4)\lambda is equal, up to a constant, to the \beta-invariant introduced some time ago by R. Hain and D. Reed. We deduce some consequences; for example we calculate the \lambda-invariant for each hyperelliptic curve, expressing it in terms of the Petersson norm of the discriminant modular form.