On quasi-Albanese morphisms for log canonical Calabi-Yau pairs

TL;DR

Addresses the quasi-Albanese morphism for log canonical Calabi-Yau pairs, introducing a framework based on differential forms, lc-triple theory, and MMP techniques to study the map $X^0\\rightarrow G$. The main result shows that for a log smooth pair $(X,D_X)$ with $D_X$ reduced and $K_X+D_X\\sim_{\\mathbb{Q}}0$, the quasi-Albanese morphism $\\alpha:X^0\\to G$ is surjective in codimension one, flat in codimension one, and semistable in codimension one. The paper develops subadditivity and positivity tools for logarithmic Kodaira dimensions, constructs a good minimal model over a compactification $\\mathbb{P}_A$ to derive a canonical bundle formula, and derives consequences such as crepant birationality and connectedness criteria for $D_X$. It also discusses orbifold fundamental groups, proving virtual nilpotence in the standard coefficient case and outlining a reduction strategy to the reduced divisor and the $q(X,D_X)=0$ case via polycyclicity. Together, these results advance understanding of quasi-Albanese maps in the MMP context and provide new avenues for studying fundamental groups of lc pairs.

Abstract

We study the quasi-Albanese morphisms for log canonical Calabi-Yau pairs.

Theorems & Definitions (32)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4: greb2011differential
• Proposition 5
• proof
• Lemma 6
• proof
• Theorem 7: Quasi-Albanese morphism iitaka1976logarithmic, see also fujino2024quasi
• Remark 9