Algebraic Lagrangian cobordisms, flux and the Lagrangian Ceresa cycle

TL;DR

The paper develops a symplectic analogue of algebraic cycle equivalence through algebraic Lagrangian cobordism $algCob(M,\omega)$ and proves a non-torsion result for the Lagrangian Ceresa cycle in the Lefschetz framework. By coupling tropical flux in tropical tori with symplectic flux in Lagrangian torus fibrations, it transfers Zharkov’s tropical Ceresa nontriviality to a nontrivial flux obstruction for Lagrangian Ceresa cycles in a polarised symplectic torus $X(J(C))$ associated to genus $3$ tropical curves of type $K4$. A Lefschetz Jacobian $LJ(M)$ is introduced to host flux data, and explicit chain maps relate tropical and symplectic data, yielding a commutative picture that explains the non-torsion phenomenon. The main theorem asserts infinite order of $L_C-L_C^-$ in $algCob^{or}(X(J(C)))_{hom}$ for generic $K4$ data, with corollaries connecting to the ordinary cobordism setting; the work also speculates on higher-genus extensions and connections to hyperelliptic-type curves and Reznikov’s characters. Overall, the results provide a principled bridge between tropical geometry, symplectic topology, and mirror-symmetric invariants of Griffiths-type groups.

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.

Figures (10)

• Figure 1: A tropical curve of type $K4$.
• Figure 2: A tropical curve of type $K4$.
• Figure 3: Bases for $\Gamma_1$ and $\Gamma_2$ represented in $C$.
• Figure 4: A representation of the embedding of the type $K4$ tropical curve in its Jacobian, with the edges colour-coded as in Figure \ref{['K4_basis']}, and the two bases $\{\gamma_1,\gamma_2,\gamma_3\}$ and $\{e_1,e_2,e_3\}$ represented as well. The two black parallelepipeds are two copies of $J(C)$ inside $\Gamma_2\otimes\mathbb{R}\cong\mathbb{R}^3$, and edges with the same symbols $\blacksquare,\blacktriangle$, or $\bigstar$ are identified.
• Figure 5: Gluing cylindrical cobordisms into an algebraic Lagrangian cobordism.

Theorems & Definitions (69)

• Theorem 1.1
• Theorem 1.2: zharkov_tropical_ceresaCeresa
• Remark 1.3
• Theorem 1.4
• Definition 2.1
• Remark 2.2
• Definition 2.3
• Remark 2.4
• Theorem 2.5
• Remark 2.6