Existence of minimal models for varieties of log general type

Abstract

We prove that the canonical ring of a smooth projective variety is finitely generated.

A reduction of canonical stability index of 4 and 5 dimensional projective varieties with large volume

“…We have the following question which has an affirmative answer for n = 3, 4: Question 6.3. -For any n 5, is it true that r n− 1 2…”

“…Thanks are also due to Chenyang Xu who helped us a lot in understanding the new proof of Theorem 2.4 using [1]. Meng Chen thanks Weiping Li for the hospitality during his visit…”

“…-By [1] or [23], the pair (V, X) has a minimal model (V 0 , X 0 ) since, for 0 < ε 1 ε 2 1, one may choose an effective Q-divisor…”

“…By [1] or [23], we know that the canonical ring R(Z, K Z ) is finitely generated over k. One can obtain the morphism π Z : Z −→ Z 0 possibly after a birational modification of Z.…”

“…Take V (1) to be a generic irreducible element of |L 1 | on F . Since |L 1 | is base point free, V (1) is smooth.…”