Log canonical pairs over varieties with maximal Albanese dimension

Abstract

Let (X,B)(X,B) be a log canonical pair over a normal variety ZZ with maximal Albanese dimension. If KX+BK_X+B is relatively abundant over ZZ (for example, KX+BK_X+B is relatively big over ZZ), then we prove that KX+BK_X+B is abundant. In particular, the subadditvity of Kodaira dimensions κ(KX+B)κ(KF+BF)+κ(Z)\kappa(K_X+B) \geq \kappa(K_F+B_F)+ \kappa(Z) holds, where FF is a general fiber, KF+BF=(KX+B)FK_F+B_F= (K_X+B)|_F, and κ(Z)\kappa(Z) means the Kodaira dimension of a smooth model of ZZ. 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.Comment: 24 pages. Some typos fixe

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