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On toric foliated pairs

Osamu Fujino, Hiroshi Sato

TL;DR

This paper extends the theory of extremal rays, Fujita's freeness, and Kodaira vanishing to log canonical toric foliated pairs on (not necessarily $\mathbb{Q}$-factorial) toric varieties, using toric Mori theory. It proves a sharp bound $l_{(\mathscr F, \Delta)}(R) \le r+1$ for lengths along extremal rays and characterizes the equality case as a $\mathbb{P}^r$-bundle contraction with $\mathscr F=\mathscr T_{X/Y}$ and $\sum \Delta<1$, advancing the cone theorem for toric foliated pairs. The results yield Fujita-type basepoint-freeness and very ampleness statements, as well as a Kodaira vanishing theorem for these pairs, even in the non-$\mathbb{Q}$-factorial setting. An appendix provides a self-contained toric proof that toric $\mathbb P^r$-bundles arise as $\mathbb P_Y(\mathcal E)$ with toric vector bundles, clarifying the geometry of the bundles involved.

Abstract

We discuss lengths of extremal rational curves, Fujita's freeness, and the Kodaira vanishing theorem for log canonical toric foliated pairs.

On toric foliated pairs

TL;DR

This paper extends the theory of extremal rays, Fujita's freeness, and Kodaira vanishing to log canonical toric foliated pairs on (not necessarily -factorial) toric varieties, using toric Mori theory. It proves a sharp bound for lengths along extremal rays and characterizes the equality case as a -bundle contraction with and , advancing the cone theorem for toric foliated pairs. The results yield Fujita-type basepoint-freeness and very ampleness statements, as well as a Kodaira vanishing theorem for these pairs, even in the non--factorial setting. An appendix provides a self-contained toric proof that toric -bundles arise as with toric vector bundles, clarifying the geometry of the bundles involved.

Abstract

We discuss lengths of extremal rational curves, Fujita's freeness, and the Kodaira vanishing theorem for log canonical toric foliated pairs.

Paper Structure

This paper contains 6 sections, 13 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lemmas on projective bundles
  4. Proof of Theorem \ref{['a-thm1.1']}
  5. Proofs of Corollary \ref{['a-cor1.2']}, Theorems \ref{['a-thm1.3']}, \ref{['a-thm1.4']}, \ref{['a-thm1.5']}, and \ref{['a-thm1.6']}
  6. Appendix: Toric projective bundles

Key Result

Theorem 1.1

Let $X$ be a projective (not necessarily $\mathbb Q$-factorial) toric variety and let $(\mathscr F, \Delta)$ be a log canonical toric foliated pair on $X$ with ${\operatorname{rank}} \mathscr F=r$. Then holds for every extremal ray $R$ of the Kleiman--Mori cone $\overline{{\operatorname{NE}}}(X)= {\operatorname{NE}}(X)$. Moreover, if $l_{(\mathscr F, \Delta)}(R)>r$ holds for some extremal ray $R$

Theorems & Definitions (36)

  • Theorem 1.1: Lengths of extremal rational curves for toric foliated pairs
  • Corollary 1.2: Cone theorem for toric foliated pairs
  • Theorem 1.3: Fujita's freeness for toric foliated pairs
  • Theorem 1.4: Fujita's freeness for toric foliations
  • Theorem 1.5: Fujita's very ampleness for toric foliations
  • Theorem 1.6: Kodaira's vanishing theorem for toric foliated pairs
  • Definition 2.1: Foliations and toric foliations
  • Theorem 2.2: see pang
  • Remark 2.3
  • Remark 2.4: see chang-chen
  • ...and 26 more