On the Existence of Good Minimal Models for Kähler Varieties with Projective Albanese Map

TL;DR

The paper proves the existence of a good minimal model for a compact Kähler klt pair $(X,B)$ when the Albanese map $a_X:X\to A$ is projective with connected fibers and the general fiber $(F,B_F)$ has a good minimal model. The authors extend projective results (Fujino, Lai, and related work) to the Kähler setting by employing a relative MMP over the Albanese, cyclic covering techniques, and generic vanishing to control fixed parts and nonvanishing phenomena, avoiding reliance on a full cone theorem in the analytic setting. They establish nonnegativity of Kodaira dimension $\kappa(X, K_X+B)$, handle the $\kappa=0$ case via a log-cyclic cover and GV arguments to force semi-ampleness, and treat $\kappa>0$ by invoking finite generation of the canonical ring, the Iitaka fibration, and an MMP with scaling to obtain a global good minimal model. The results advance Abundance-type questions in complex analytic geometry and broaden the applicability of MMP techniques to Kähler varieties, providing a cone-theorem–free route adapted from the projective case.

Abstract

In this article, we establish the existence of a good minimal model for a compact Kähler klt pair $(X, B)$ when the Albanese map of $X$ is a projective morphism and the general fiber of $(X, B)$ has a good minimal model.

Theorems & Definitions (31)

• Theorem 1.1
• Theorem 1.2: lai2011varieties
• Theorem 1.3: das2024existence
• Theorem 2.1: cf. DAS2024109615 and fujino2022minimal
• Theorem 2.2: cf. DAS2024109615 and fujino2022minimal
• Remark 2.3
• Definition 2.4
• Lemma 2.5: das2024transcendentalminimalmodelprogram
• Lemma 2.6
• Proof