Normal functions and the height of Gross-Schoen cycles

Abstract

Abstract. We prove a variant of a formula due to S. Zhang relating the Beilinson-Bloch height of the Gross-Schoen cycle on a pointed curve with the self-intersection of its relative dualizing sheaf. In our approach the height of the Gross-Schoen cycle occurs as the degree of a suitable Bloch line bundle. We show that the Chern form of this line bundle is non-negative, and we calculate its class in the Picard group of the moduli space of pointed stable curves of compact type. The basic tools are normal functions and biextensions associated to the cohomology of the universal jacobian.

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

“…For example, in his paper [38] S. Zhang independently introduced the invariant ϕ g = 2π a g and showed the relation of its strict positivity with some important conjectures in arithmetic geometry. In the papers [25], [26], [27], [28] the present author found a number of further properties of a g , including its asymptotics towards the boundary of M g in the Deligne-Mumford compactification, and closed expressions in the case of a hyperelliptic Riemann surface. We will review these results briefly in our final Section 10.…”

Point-like limit of the hyperelliptic Zhang–Kawazumi invariant

“…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. The connection between these two seemingly different approaches was established in [18].…”