Algebraic Lagrangian cobordisms, flux and the Lagrangian Ceresa cycle

Abstract

We introduce an equivalence relation for Lagrangians in a symplectic manifold known as \textit{algebraic Lagrangian cobordism}, which is meant to mirror algebraic equivalence of cycles. From this we prove a symplectic, mirror-symmetric analogue of the statement \enquote{the Ceresa cycle is non-torsion in the Griffiths group of the Jacobian of a generic genus $3$ curve}. Namely, we show that for a family of tropical curves, the \textit{Lagrangian Ceresa cycle}, which is the Lagrangian lift of their tropical Ceresa cycle to the corresponding Lagrangian torus fibration, is non-torsion in its oriented algebraic Lagrangian cobordism group. We proceed by developing the notions of tropical (resp. symplectic) flux, which are morphisms from the tropical Griffiths (resp. algebraic Lagrangian cobordism) groups.