Realizability criteria in tropical geometry from symplectic geometry

Abstract: The realizability problem asks if a given tropical subvariety is the tropicalization of some algebraic subvariety. Realizability is already an interesting question for curves in $\mathbb {R}^3$, where Mikhalkin exhibited a tropical curve of genus 1 which is non-realizable. In recent independent work, Mak-Ruddat, Matessi, Mikhalkin, and I show that for many examples of tropcial subvarieties in $\mathbb {R}^n$ there exists a Lagrangian lift. This is a Lagrangian submanifold of $(\mathbb {C}^*)^n$ whose image under the moment map approximates a given tropical subvariety. In particular, every smooth tropical curve in $\mathbb {R}^n$ can be lifted to a Lagrangian submanifold (in contrast to the algebraic setting!)

In this talk, I'll discuss what it means to be a Lagrangian lift of a tropical curve. We will then look at what symplectic conditions on the resulting Lagrangian detect realizability of the underlying tropical curve. As an application, we will prove that every tropical curve in a tropical abelian surface has a rigid-analytic realization.

algebraic geometrycombinatorics

Audience: researchers in the topic

(LAGARTOS) Latin American Real and Tropical Geometry Seminar

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