Remarks on the Griffiths infinitesimal invariant of algebraic curves
Haohua Deng
TL;DR
This work addresses the Griffiths infinitesimal invariants of two canonically defined normal functions on the genus-$4$ moduli, focusing on the Ceresa cycle and its associated normal function $\nu_c$ in a trigonal genus-$4$ setting. The author extends Collino–Pirola techniques to genus $4$, linking the vanishing of $\delta\nu_c$ for rank-$1$ deformations to Schiffer variations and relating $\delta\nu_{c,C}$ to the canonical embedding; this yields a precise description of the zero locus as a degree-$6$ canonical image in $\mathbb{P}(H^1(C,\mathcal{O}_C))$ for general curves. In a concrete family of trigonal genus-$4$ curves, $\nu_0$ and its invariant vanish identically while $\nu_c$ achieves maximal rank, and the locus where $\nu_c$ is locally constant is at most one-dimensional and organized by a distinguished subbundle of the tangent bundle; this leads to a foliation of the base by algebraic leaves. The paper also develops an o-minimal definability framework for normal functions, showing that leaves are definable and that monodromy arguments (notably $\mathrm{SU}(3,1)$) govern the structure of the rank loci and torsion phenomena. Overall, the results deepen understanding of Ceresa-type normal functions in genus $4$, establish rank and vanishing patterns, and provide tools to detect torsion loci via algebraic monodromy and definable foliations.
Abstract
We study two canonically defined admissible normal functions on the moduli space of smooth genus 4 algebraic curves including the Ceresa normal function. In particular, we study the vanishing criteria for the Griffiths infinitesimal invariants of both normal functions over a specific family of curves.