Johnson's homomorphisms and the Arakelov-Green function

Abstract

Let $\pi :{\mathbb{C}}_g\to {\mathbb{M}}_g$ be the universal family of compact Riemann surfaces of genus $g\ge 1$. We introduce a real-valued function on the moduli space ${\mathbb{M}}_g$ and compute the first and the second variations of the function. As a consequence we relate the Chern form of the relative tangent bundle $T_{{\mathbb{C}}_g/{\mathbb{M}}_g}$ induced by the Arakelov-Green function with differential forms on ${\mathbb{C}}_g$ induced by a flat connection whose holonomy gives Johnson's homomorphisms on the mapping class group.

References in corpus (1)

• Harmonic Magnus Expansion on the Universal Family of Riemann Surfaces

Cited by in corpus (23)

• On the modular structure of the genus-one Type II superstring low energy expansion
• Matching the $D^6{\mathcal{R}}^4$ interaction at two-loops
• Higher genus modular graph functions, string invariants, and their exact asymptotics
• Eisenstein series and automorphic representations
• $D^6{R}^4$ amplitudes in various dimensions
• A Theta lift representation for the Kawazumi-Zhang and Faltings invariants of genus-two Riemann surfaces
• Infrared divergences and harmonic anomalies in the two-loop superstring effective action
• New explicit formulas for Faltings' delta-invariant
• SL(2,Z)-invariance and D-instanton contributions to the $D^6{R}^4$ interaction
• 1/8-BPS Couplings and Exceptional Automorphic Functions
• Asymptotics of the $D^8{R}^4$ genus-two string invariant
• Hierarchy of Modular Graph Identities
• Identities among higher genus modular graph tensors
• Second variation of Zhang's lambda-invariant on the moduli space of curves
• Degeneration of Riemann theta functions and of the Zhang-Kawazumi invariant with applications to a uniform Bogomolov conjecture
• Jumps in the height of the Ceresa cycle
• Integrating simple genus two string invariants over moduli space
• On a question of Ekedahl and Serre
• On the height of Gross-Schoen cycles in genus three
• Frobenius' theta function and Arakelov invariants in genus three
• Rings of tautological forms on moduli spaces of curves
• Asymptotic behavior of the Kawazumi-Zhang invariant for degenerating Riemann surfaces
• On Faltings' Delta-Invariant of Hyperelliptic Riemann Surfaces