The Angehrn-Siu Type Effective Freeness For Quasi-Log Canonical Pairs

We prove the Angehrn-Siu Type effective freeness and effective point separation for quasi-log canonical pairs. As a natural consequence, we obtain that these two results hold for semi-log canonical pairs. One of the main ingredients of our proof is the inversion of adjunction for quasi-log canonical pairs, which is established in this paper

On the log canonical ring of projective plt pairs with the Kodaira dimension two

The log canonical ring of a projective plt pair with the Kodaira dimension two is finitely generated.Comment: 11 pages, v2: revision following the referee's comment

Quasi-log canonical pairs are Du Bois

We prove that every quasi-log canonical pair has only Du Bois singularities. Note that our arguments are free from the minimal model program.Comment: 6 pages, v2: very minor modifications, v3: add some explanations in the proof of Theorem 1.1. arXiv admin note: text overlap with arXiv:1804.1113