Algebraic Lagrangian cobordisms, flux and the Lagrangian Ceresa cycle

TL;DR

This paper introduces algebraic Lagrangian cobordism as an equivalence relation for Lagrangians, demonstrating a non-torsion property of the Lagrangian Ceresa cycle in a symplectic mirror-symmetric context, with new notions of tropical and symplectic flux.

Contribution

It defines algebraic Lagrangian cobordism and proves a mirror-symmetric analogue of a classical algebraic cycle non-torsion result, introducing tropical and symplectic flux concepts.

Findings

Lagrangian Ceresa cycle is non-torsion in its cobordism group.

Develops tropical and symplectic flux morphisms.

Establishes a mirror-symmetric analogue of a classical algebraic cycle result.

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.