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

For every smooth complex projective variety W of dimension d and nonnegative Kodaira dimension, we show the existence of a universal constant m depending only on d and two natural invariants of the very general fibres of an Iitaka fibration of W such that the pluricanonical system |mK W | defines an Iitaka fibration. This is a consequence of a more general result on polarized adjoint divisors. In order to prove these results we develop a generalized theory of pairs, singularities, log canonical thresholds, adjunction, etc.

In this paper, we study the linear systems | − mKX | on Fano varieties X with klt singularities. In a given dimension d, we prove | − mKX| is non-empty and contains an element with "good singularities" for some natural number m depending only on d; if in addition X is ǫ-lc for some ǫ > 0, then we show that we can choose m depending only on d and ǫ so that | − mKX | defines a birational map. Further, we prove Shokurov's conjecture on boundedness of complements, and show that certain classes of Fano varieties form bounded families.

Let (X/Z, B + A) be a Q-factorial dlt pair where B, A ≥ 0 are Q-divisors and K X + B + A ∼ Q 0/Z. We prove that any LMMP/Z on K X + B with scaling of an ample/Z divisor terminates with a good log minimal model or a Mori fibre space. We show that a more general statement follows from the ACC for lc thresholds. An immediate corollary of these results is that log flips exist for log canonical pairs.

We will prove the following results for 3-fold pairs (X, B) over an algebraically closed field k of characteristic p > 5: log flips exist for Qfactorial dlt pairs (X, B); log minimal models exist for projective klt pairs (X, B) with pseudo-effective K X + B; the log canonical ring R(K X + B) is finitely generated for projective klt pairs (X, B) when K X + B is a big Qdivisor; semi-ampleness holds for a nef and big Q-divisor D if D − (K X + B) is nef and big and (X, B) is projective klt; Q-factorial dlt models exist for lc pairs (X, B); terminal models exist for klt pairs (X, B); ACC holds for lc thresholds; etc. Contents 1 45 12. Non-big log divisors: proof of 1.11 45 References 47Existence of flips and minimal models for 3-folds in char p 5 our results on flips and minimal models.Dlt and terminal models. The next two results are standard consequences of the LMMP (more precisely, of special termination). They are proved in Section 7.Theorem 1.6. Let (X, B) be an lc pair of dimension 3 over k of char p > 5. Then (X, B) has a (crepant) Q-factorial dlt model. In particular, if (X, B) is klt, then X has a Q-factorialization by a small morphism.The theorem was proved in [13, Theorem 6.1] for pairs with standard coefficients.Theorem 1.7. Let (X, B) be a klt pair of dimension 3 over k of char p > 5. Then (X, B) has a (crepant) Q-factorial terminal model.The theorem was proved in [13, Theorem 6.1] for pairs with standard coefficients and canonical singularities.The connectedness principle with applications to semi-ampleness. The next result concerns the Kollár-Shokurov connectedness principle. In characteristic 0, the surface case was proved by Shokurov by taking a resolution and then calculating intersection numbers [26, Lemma 5.7] but the higher dimensional case was proved by Kollár by deriving it from the Kawamata-Viehweg vanishing theorem [20, Theorem 17.4].Theorem 1.8. Let (X, B) be a projective Q-factorial pair of dimension 3 over k of char p > 5. Let f : X → Z be a birational contraction such that −(K X +B) is ample/Z. Then for any closed point z ∈ Z, the non-klt locus of (X, B) is connected in any neighborhood of the fibre X z .The theorem is proved in Section 9. To prove it we use the LMMP rather than vanishing theorems. When dim X = 2, the theorem holds in a stronger form (see 9.3).We will use the connectedness principle on surfaces to prove some semiampleness results on surfaces and 3-folds. Here is one of them: Theorem 1.9. Let (X, B + A) be a projective Q-factorial dlt pair of dimension 3 over k of char p > 5. Assume that A, B ≥ 0 are Q-divisors such that A is ample andNote that if one could show that ⌊B⌋ is semi-lc, then the result would follow from Tanaka [29]. In order to show that ⌊B⌋ is semi-lc one needs to check that it satisfies the Serre condition S 2 . In characteristic 0 this is a consequence of Kawamata-Viehweg vanishing (see Kollár [20, Corollary 17.5]). The S 2 condition can be used to glue sections on the various irreducible components of ⌊B⌋. To prove the above semi-ampleness we instead use a ...