Log canonical pairs over varieties with maximal Albanese dimension

TL;DR

This paper proves that for log canonical pairs over varieties with maximal Albanese dimension, relative abundance of the canonical divisor implies its abundance, supporting the subadditivity of Kodaira dimensions and discussing related conjectures.

Contribution

It establishes the abundance of $K_X+B$ under certain conditions over varieties with maximal Albanese dimension, advancing the understanding of the log Iitaka conjecture.

Findings

Proves abundance of $K_X+B$ when relatively abundant over $Z$.

Establishes subadditivity of Kodaira dimensions for these pairs.

Discusses variants and implications for the log Iitaka conjecture.

Abstract

Let $(X,B)$ be a log canonical pair over a normal variety $Z$ with maximal Albanese dimension. If $K_X+B$ is relatively abundant over $Z$ (for example, $K_X+B$ is relatively big over $Z$), then we prove that $K_X+B$ is abundant. In particular, the subadditvity of Kodaira dimensions $\kappa(K_X+B) \geq \kappa(K_F+B_F)+ \kappa(Z)$ holds, where $F$ is a general fiber, $K_F+B_F= (K_X+B)|_F$, and $\kappa(Z)$ means the Kodaira dimension of a smooth model of $Z$. We discuss several variants of this result in Section 4. We also give a remark on the log Iitaka conjecture for log canonical pairs in Section 5.