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On toric and toroidal foliations

Chih-Wei Chang, Yen-An Chen

TL;DR

This work develops toric descriptions for foliations on toric varieties, introducing toroidal foliations and extended complexes to capture non-toric perturbations. It proves that the toric foliated minimal model program preserves key singularities such as non-dicritical and F-dlt, using discrepancy data $a(E,\mathcal{F},D)$ and support functions $\phi_{(\mathcal F,D)}$ to control birational steps. A cone theorem for log canonical toric foliated pairs is established, showing the negative part of the cone is generated by torus-invariant curves tangent to ${\mathcal F_W}$. Collectively, these results extend FMMP to toric foliations of arbitrary rank and dimension and clarify how singularities relate under toroidal/toric operations within the extended complex framework.

Abstract

In this paper, we provide toric descriptions for various foliation singularities on toric varieties, especially for non-dicritical singularities and F-dlt singularities. We then show that the toric foliated minimal model program works by demonstrating that non-dicritical singularities and F-dlt singularities are preserved.

On toric and toroidal foliations

TL;DR

This work develops toric descriptions for foliations on toric varieties, introducing toroidal foliations and extended complexes to capture non-toric perturbations. It proves that the toric foliated minimal model program preserves key singularities such as non-dicritical and F-dlt, using discrepancy data and support functions to control birational steps. A cone theorem for log canonical toric foliated pairs is established, showing the negative part of the cone is generated by torus-invariant curves tangent to . Collectively, these results extend FMMP to toric foliations of arbitrary rank and dimension and clarify how singularities relate under toroidal/toric operations within the extended complex framework.

Abstract

In this paper, we provide toric descriptions for various foliation singularities on toric varieties, especially for non-dicritical singularities and F-dlt singularities. We then show that the toric foliated minimal model program works by demonstrating that non-dicritical singularities and F-dlt singularities are preserved.
Paper Structure (26 sections, 55 theorems, 43 equations)

This paper contains 26 sections, 55 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basics on foliations
  4. Basics on toric varieties
  5. Toric foliations
  6. Local generators
  7. Singular locus of a toric foliation
  8. Properties
  9. Toroidal foliations and extended complexes
  10. Toric and toroidal foliated pairs
  11. Conical complexes
  12. Extended complexes associated with strict toroidal foliations
  13. Non-dicritical singularities and condition (dagger)
  14. Singularities of toric and toroidal foliated pairs
  15. Support functions
  16. ...and 11 more sections

Key Result

Theorem 1.1

Let ${\mathcal{F}}={\mathcal{F}}_W$ be a toric foliation on a toric variety $X_\Sigma$ of a fan $\Sigma$ in $N\otimes{\mathbb R}$ where $W\subseteq N\otimes{\mathbb C}$ is a complex vector subspace. Then ${\mathcal{F}}_W$ is non-dicritical if and only if $(\Sigma,W)$ satisfies the condition $(\dagge

Theorems & Definitions (136)

  • Theorem 1.1: cf. Theorem \ref{['ND_toroidal']}
  • Proposition 1.3: $=$ Proposition \ref{['lc']}
  • Proposition 1.4: $=$ Proposition \ref{['can_term_toric']}
  • Proposition 1.5: $=$ Proposition \ref{['Fdlt_prop']}
  • Theorem 1.6
  • Theorem 1.7: Propositions \ref{['div_contr']}, \ref{['MFS']}, and \ref{['flip']})
  • Theorem 1.8: $=$ Theorem \ref{['conethm']}, Cone Theorem
  • Definition 2.1: Singular locus
  • Definition 2.2: Invariance
  • Proposition 2.3
  • ...and 126 more