Maps to toric varieties, toric degenerations and integrable systems \`a la Harada--Kaveh

TL;DR

This paper constructs explicit continuous maps from general fibers to special fibers in toric degenerations, extending integrable systems construction and providing a more detailed topological understanding.

Contribution

It introduces a more explicit construction of maps in toric degenerations and extends integrable systems to boundary strata, building on Harada--Kaveh's work.

Findings

Constructed a surjective continuous map in toric degenerations.

Extended integrable systems to boundary strata.

Provided a more explicit topological construction.

Abstract

Given a toric degeneration (a degeneration to a toric variety), over the complex numbers, we construct a surjective continuous map from a general fiber to the special fiber of the degeneration in the classical topology. The construction is a variant of one due to Goresky and MacPherson based on the Thom--Mather theory of stratified spaces. As an application, we recover and extend the construction of integrable systems \`a la Harada--Kaveh in "Integrable systems, toric degenerations and okounkov bodies." Compared to their result, our map is constructed more explicitly and we also construct the integrable systems on the boundary strata. This paper is a part of the authors' research on maps to toric degenerations; we refer the readers to "Toric degenerations and projections," arxiv and "Notes on multi-proj and maps to not-necessarily-normal toric varieties," researchgate for more algebraic approaches.