Smash-nilpotence: algebraic, tropical and symplectic

In algebraic geometry, two useful ways of classifying cycles are up to rational equivalence, or up to a weaker equivalence relation known as algebraic equivalence. Using the dictionary of mirror symmetry, these correspond to classifications of Lagrangians in the mirror symplectic manifold: Sheridan—Smith showed that rational equivalence is related to a version of Lagrangian cobordism, and in previous work I introduced an equivalence relation called alg-cobordism as a Lagrangian counterpart to algebraic equivalence.

In the early 90’s, Voevodsky and Voisin independently proved the following surprising Theorem: if an algebraic cycle Z in an algebraic variety Y is algebraically trivial, then for k sufficiently large, Z^k is rationally trivial in Y^k. I will present joint work with Danil Kozevnikov, in which we translate their Theorem into the symplectic realm: if a Lagrangian L is alg-nullcobordant, then for k sufficiently large, L^k is null-cobordant.