Abstract

One can extract geometric information about an algebraic variety by studying and comparing the different equivalence relations between its subvarieties; these include algebraic, homological, and rational equivalence. An interesting example is the Ceresa cycle associated to a generic curve of genus g>2, which is homologically but not algebraically trivial. One might hope to use insights from mirror symmetry to understand how comparing Lagrangians in a symplectic manifold can exhibit interesting symplectic phenomena. Lagrangian cobordisms are known to be related to rational equivalences of cycles in the mirror, and we introduce algebraic Lagrangian cobordisms mirroring algebraic equivalence. We illustrate with the Lagrangian Ceresa cycle, a Lagrangian in a symplectic 6-torus, that this notion captures non-trivial symplectic geometry.