Relative Nakayama–Zariski Decomposition and Minimal Models of Generalized Pairs

Abstract

We prove some basic properties of the relative Nakayama–Zariski decomposition. We apply them to the study of lc generalized pairs. We prove the existence of log minimal models or Mori fiber spaces for (relative) lc generalized pairs polarized by an ample divisor. This extends a result of Hashizume–Hu to generalized pairs. We also show that, for any lc generalized pair

(X,B+A,\mathbf{M})/Z

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(X,B+A,\textbf{M})/Z$$\end{document}

such that

K_X+B+A+{\mathbf{M}}_X{\sim }_{\mathbb{R},Z}0

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$K_X+B+A+\textbf{M}_X\sim _{{\mathbb {R}},Z}0$$\end{document}

and

B\ge 0,A\ge 0

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$B\ge 0,A\ge 0$$\end{document}

,

(X,B,\mathbf{M})/Z

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(X,B,\textbf{M})/Z$$\end{document}

has either a log minimal model or a Mori fiber space. This is an analogue of a result of Birkar/Hacon–Xu and Hashizume in the category of generalized pairs, and is later shown to be crucial to the proof of the existence of generalized lc flips in full generality.

Keywords

Generalized pairs / Nakayama–Zariski decomposition / Minimal models

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Funding

National Science Foundation(DMS-1801851)

Simons Foundation(256202)

RIGHTS & PERMISSIONS

Peking University