Torus bundles and 2-forms on the universal family of Riemann surfaces

Abstract

Abstract. We revisit three results due to Morita expressing certain natural integral cohomology classes on the universal family of Riemann surfaces Cg , coming from the parallel symplectic form on the universal jacobian, in terms of the Euler class e and the Miller-Morita-Mumford class e 1 . Our discussion will be on the level of the natural 2-forms representing the relevant cohomology classes, and involves a comparison with other natural 2-forms representing e, e 1 induced by the Arakelov metric on the relative tangent bundle of Cg over Mg . A secondary object called ag occurs, which was discovered and studied by Kawazumi around 2008. We present alternative proofs of Kawazumi's (unpublished) results on the second variation of ag on Mg . Also we review some results that were previously obtained on the invariant ag , with a focus on its connection with Faltings's delta-invariant and Hain-Reed's beta-invariant.

New explicit formulas for Faltings’ delta-invariant

“…as forms on M g , where Z ⊆ M g is the vanishing locus of ∆ g . Comparing this with the forms (5.1), (5.2) and (5.19), we notice that each of the mentioned formulas for hyperelliptic Riemann surfaces implies 3e A 1 = (2 − 2g) π 2 h 3 , which is not true in general on M g , see also [26,Section 10]. 5.5.…”

“…For ϕ(X) and δ(X) we have an expression for the resulting forms in terms of the canonical forms e A 1 , π 2 h 3 and ω Hdg (see Section 5.1) on M g by de Jong [26]. To calculate the application of ∂∂ to the integral in (1.1), we apply the Laplace operator on the universal abelian variety with level 2 structure to log θ , and we pull back the integral to the (g + 1)-th power of the universal Riemann surface with level 2 structure X g → M g [2].…”

“…To calculate the application of ∂∂ to the integral in (1.1), we apply the Laplace operator on the universal abelian variety with level 2 structure to log θ , and we pull back the integral to the (g + 1)-th power of the universal Riemann surface with level 2 structure X g → M g [2]. The pullback can be expressed in terms of Deligne pairings by a result due to de Jong [26,Proposition 6.3]. The main difficulty is to express the first Chern form of the (g + 1)-th power in the sense of Deligne pairings of the line bundle in terms of the forms e A 1 , π 2 h 3 and e A .…”

Point-like limit of the hyperelliptic Zhang–Kawazumi invariant

“…[24,Corollary 1.2] or [33, Proposition 2.5.1]. Several expressions for the Levi form of ϕ are derived in [16] [24].…”

“…In [23] [24] the invariant ϕ arises in the context of a study of the first extended Johnson homomorphism [26] on the mapping class group of a pointed compact connected oriented topological surface. The results from (the unfortunately unpublished) [24] were revisited in [16]. The motivation in [33] to study ϕ comes from number theory, where ϕ appears as a local archimedean contribution in a formula that relates the height of the canonical Gross-Schoen cycle on a smooth projective and geometrically connected curve with semistable reduction over a number field with the self-intersection of its admissible relative dualizing sheaf.…”