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# Pure and Applied Mathematics Quarterly

## Volume 12 (2016)

### Number 4

### Log canonical pairs over varieties with maximal Albanese dimension

Pages: 543 – 571

DOI: http://dx.doi.org/10.4310/PAMQ.2016.v12.n4.a5

#### Author

#### 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 subadditivity of Kodaira dimensions $\kappa (K_X + B) \geq \kappa (K_F + B_F) + \kappa (Z)$ holds, where $F$ is a general fiber, $K_F + B_F = {(K_X + B) \vert}_ F$, 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.

#### 2010 Mathematics Subject Classification

14E30

Received 12 September 2016