Remarks on very basic slc-trivial fibrations

Abstract

We study very basic slc-trivial fibrations. We show that restricting on any lc center of a very basic slc-trivial fibration, its moduli part is numerically trivial if and only if it is $\mathbb{Q}$-linearly trivial. We then prove that abundance conjecture for very basic slc-trivial fibrations holds true in dimension two when the moduli part is $\mathbb{Q}$-Cartier. As an application, we prove that the log canonical ring of a projective plt pair with Kodaira dimension 3 is finitely generated.

22 pages, any comments are welcome; v2: we correct some typos and modify Section 6 a little bit by referring to suitable conference