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October 16, 2025

Algebraic Lagrangian cobordisms, flux and the Lagrangian Ceresa cycle

Key Points

The Lagrangian Ceresa cycle is proven to be non-torsion in its algebraic Lagrangian cobordism group, supporting deep interactions between symplectic geometry and algebraic cycles.
Tropical curves are shown to have a corresponding Lagrangian lift that remains non-torsion, indicating the richness of the relationship between different geometric objects.
Developments include the introduction of tropical and symplectic flux, establishing morphisms that connect Griffiths and Lagrangian cobordism groups.
This research signals significant advancements in understanding the algebraic equivalences and the role of cycles in higher-dimensional geometries.

Abstract

Abstract We introduce an equivalence relation for Lagrangians in a symplectic manifold known as algebraic Lagrangian cobordism , which is meant to mirror algebraic equivalence of cycles. From this, we prove a symplectic, mirror-symmetric analogue of the statement “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 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.

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