2022/10/24 by Takuya Murata, Murata, Takuya, Lara Bossinger +1
Mathematics · #14D06 (Primary) #14M25 (Secondary) #37J35 (Secondary) #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.2210.13137
openalex publication_date 2022/10/24 · openalex created_date 2022/10/31 · openalex updated_date 2026/07/28
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 à 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.