Second variation of Zhang’s λ-invariant on the moduli space of curves

We compute the second variation of the λ-invariant, recently introduced by S. Zhang, on the complex moduli space Mg of curves of genus g ≥ 2, using work of N. Kawazumi. As a result we prove that (8g +4)λ is equal, up to a constant, to the β-invariant introduced some time ago by R. Hain and D. Reed. We deduce some consequences; for example we calculate the λ-invariant for each hyperelliptic curve, expressing it in terms of the Petersson norm of the discriminant modular form. 1. Introduction. Recently, independently S. Zhang (27) and N. Kawazumi (16) introduced a new interesting real-valued function ϕ on the moduli space Mg of complex curves of genus g ≥2. Its value at a curve (X) ∈M g is given as follows. Let H 0 (X, ωX ) be the space of holomorphic differentials on X, equipped with the