A Theta lift representation for the Kawazumi-Zhang and Faltings invariants of genus-two Riemann surfaces

Abstract

The Kawazumi-Zhang invariant $φ$ for compact genus-two Riemann surfaces was recently shown to be a eigenmode of the Laplacian on the Siegel upper half-plane, away from the separating degeneration divisor. Using this fact and the known behavior of $φ$ in the non-separating degeneration limit, it is shown that $φ$ is equal to the Theta lift of the unique (up to normalization) weak Jacobi form of weight $-2$. This identification provides the complete Fourier-Jacobi expansion of $φ$ near the non-separating node, gives full control on the asymptotics of $φ$ in the various degeneration limits, and provides a efficient numerical procedure to evaluate $φ$ to arbitrary accuracy. It also reveals a mock-type holomorphic Siegel modular form of weight $-2$ underlying $φ$. From the general relation between the Faltings invariant, the Kawazumi-Zhang invariant and the discriminant for hyperelliptic Riemann surfaces, a Theta lift representation for the Faltings invariant in genus two readily follows.

16 pages; v2: many improvements: the main conjecture is now a theorem, numerical checks and applications are performed, connections to Gromov-Witten invariants are discussed, various clarifications throughout, 3 extra pages, 10 extra references; v3: cosmetic changes, added details on the proof of (78), one new reference; v4: journal version