MMP for Generalized Pairs on Kähler 3-folds

TL;DR

The paper extends the minimal model program to generalized pairs on compact Kähler spaces of dimension at most three by introducing b-(1,1) currents $\boldsymbol{\beta}$ to encode generalized boundary data. It develops the necessary analytic framework (b-(1,1) currents, generalized pairs, and generalized models) and then proves key MMP tools in dimension three: the cone theorem, existence of flips, and the existence and structure of log terminal and log canonical models, as well as Mori fiber spaces, both in the absolute and relative settings. It also analyzes the geography of minimal models via polyhedral decompositions and proves that minimal models are connected by flips and flops, with Calabi–Yau-type cases linked by flop sequences. The results extend projective MMP techniques to the Kähler realm, enabling finiteness and connectivity statements for threefold generalized pairs and laying groundwork for potential higher-dimensional generalizations and applications to Calabi–Yau birational geometry in the analytic category.

Abstract

In this article we define generalized pairs $(X, B+\boldsymbolβ)$ where $X$ is an analytic variety and $\boldsymbolβ$ is a b-(1,1) current. We then prove that almost all standard results of the MMP hold in this generality for compact Kähler varieties of dim $X\leq 3$. More specifically, we prove the cone theorem, existence of flips, existence of log terminal models, log canonical models and Mori fiber spaces, the geography of log canonical and log terminal models, etc.

Theorems & Definitions (139)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4