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Maps to toric varieties, toric degenerations and integrable systems à la Harada--Kaveh

Takuya Murata, Lara Bossinger

TL;DR

This work constructs a topological collapse map from a general fiber to the special fiber of a toric degeneration using Thom–Mather stratified-space theory, enabling a detailed link between algebraic degeneration data and topological collapse. It proves the existence of a continuous surjection $\varphi: X \to W$ with stratum-wise étale behavior over torus orbits, thereby enabling a stratified integrable-system framework on $X$ by composing with the toric moment map on $W$. By integrating Weinstein’s deformation theory, the authors extend Harada–KAVEH’s integrable-system construction to boundary strata, providing a cohesive mechanism to translate toric-degeneration data into symplectic and Poisson-geometric structures on singular spaces. Additionally, the paper develops multi-parameter Rees-algebra techniques to realize and study toric degenerations, and proposes a stratified, orbit-based description of maps to toric varieties, along with speculative extensions to real-closed settings and broader degeneration scenarios. The results advance the synthesis of algebraic, topological, and symplectic methods in the study of toric degenerations and their associated integrable systems, with potential implications for Newton–Okounkov bodies and stratified Hamiltonian dynamics.

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 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.

Maps to toric varieties, toric degenerations and integrable systems à la Harada--Kaveh

TL;DR

This work constructs a topological collapse map from a general fiber to the special fiber of a toric degeneration using Thom–Mather stratified-space theory, enabling a detailed link between algebraic degeneration data and topological collapse. It proves the existence of a continuous surjection with stratum-wise étale behavior over torus orbits, thereby enabling a stratified integrable-system framework on by composing with the toric moment map on . By integrating Weinstein’s deformation theory, the authors extend Harada–KAVEH’s integrable-system construction to boundary strata, providing a cohesive mechanism to translate toric-degeneration data into symplectic and Poisson-geometric structures on singular spaces. Additionally, the paper develops multi-parameter Rees-algebra techniques to realize and study toric degenerations, and proposes a stratified, orbit-based description of maps to toric varieties, along with speculative extensions to real-closed settings and broader degeneration scenarios. The results advance the synthesis of algebraic, topological, and symplectic methods in the study of toric degenerations and their associated integrable systems, with potential implications for Newton–Okounkov bodies and stratified Hamiltonian dynamics.

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 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.
Paper Structure (12 sections, 27 theorems, 74 equations)

This paper contains 12 sections, 27 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Toric degenerations
  3. Symplectic consideration
  4. Possible alternatives
  5. Further topics
  6. Acknowledgements
  7. Rees algebras in several parameters
  8. Collapsing map and its restriction
  9. Moment maps and integrable systems
  10. Moment map
  11. Construction of integrable systems
  12. Torus embedding description

Key Result

Theorem 1

Suppose we are given a toric degeneration $\mathfrak{X}$ from $X$ (a general fiber) to $W$ (the special fiber) over $\mathbb{C}$ that is embedded in a complex smooth variety as a proper family. Then there exists a continuous surjective map (not a morphism of varieties) such that, for each torus orbit $O \subset W$, $\varphi^{-1}(O)$ has the structure of a reducible variety so that $\varphi : \var

Theorems & Definitions (72)

  • Theorem : Theorem \ref{['thm:etale map to a special fiber']}
  • Theorem : Harada--Kaveh type Theorem \ref{['thm:improved Harada-Kaveh']}
  • Remark 1
  • Remark 2
  • Lemma 1.1: graded Nakayama
  • proof
  • Lemma 1.2
  • proof
  • Corollary 1.3
  • Definition 1.4: associated ring
  • ...and 62 more