New outlook on the Minimal Model Program, I

Abstract

We give a new and self-contained proof of the finite generation of adjoint rings with big boundaries. As a consequence, we show that the canonical ring of a smooth projective variety is finitely generated.Part of this work was written while the second author was a PhD student of A. Corti, who influenced ideas developed here immensely. Part of the paper started as a collaboration with J. M c Kernan. We would like to express our gratitude to both of them for their encouragement, support and continuous inspiration. We thank F. Ambro, C. Hacon, J. Hausen, A.-S. Kaloghiros, A. Lopez, K. Matsuki, and M. Reid for many useful comments. We are particularly grateful to the referees who helped to improve the presentation of the paper considerably. This work supersedes [Laz09], where the results of this paper were first proved without the Minimal Model Program by the second author. Several arguments here follow closely those in [Laz09] and, based on these methods, we obtain a streamlined proof which is almost entirely self-contained. We even prove a lifting statement for adjoint bundles without relying on asymptotic multiplier ideals, assuming only Kawamata-Viehweg vanishing and some elementary arithmetic -this is Theorem 3.4, which slightly generalises the lifting theorem from [HM10].The results presented here were originally proved by extensive use of methods of the Minimal Model Program in [BCHM10, HM10], and an analytic proof of finite generation of the canonical ring for varieties of general type is announced in [Siu08]. By contrast, in this paper we avoid the following tools which are commonly used in the Minimal Model Program: Mori's bend and break, which relies on methods in positive characteristic [Mor82], the Cone and Contraction theorem [KM98], the theory of asymptotic multiplier ideals, which was necessary to prove the existence of flips in [HM10]. Moreover, contrary to classical Minimal Model Program, we do not need to work with singular varieties.In [CL10], Corti and the second author recently proved that the Cone and Contraction theorem, and the main result of [BCHM10], follow quickly from one of our main results, Theorem A. Therefore, this paper and [CL10] together give a completely new organisation of the Minimal Model Program.We now briefly describe the strategy of the proof. As part of the induction, we prove the following two theorems.Theorem A. Let X be a smooth projective variety of dimension n. Let B 1 , . . . , B k be Q-divisors on X such that ⌊B i ⌋ = 0 for all i, and such that the support of k i=1 B i has simple normal crossings. Let A be an ample Q-divisor on X, and denote D i = K X + A + B i for every i.Then the adjoint ring 3 We denote by C the topological closure of a set C ⊆ R N . Let X be a smooth projective variety and R ∈ {Z, Q, R}. We denote by Div R (X) the group of R-divisors on X, and ∼ R and ≡ denote R-linear and numerical equiv-Given D ∈ Div R (X) and x ∈ X, mult x D is the order of vanishing of D at x. If S is a prime divisor, mult S D is the order of vanishing of D at the generic ...

Effective finite generation for adjoint rings

“…, D k on a smooth projective variety X, but it will not be used in this paper in this generality (e.g. see [CL10a] for more details).…”

“…The main goal of this paper is to combine together some of these results and study an effective version of the finite generation for adjoint rings. More specifically, given a Kawamata log terminal pair (X, B), the main result of [BCHM10] implies that the canonical ring R(X, K X +B) of K X +B is finitely generated (see also [Siu08,CL10a]). Moreover, if A is an ample Q-divisor and B 1 , .…”

The Minimal Model Program revisited

“…Theorem A, in fact, also gives the affirmative answer to Question 1.1 -details can be found in [CL10a]. This theorem was originally proved in [BCHM10] as a consequence of Mori theory, and later in [Laz09] by induction on the dimension and without Mori theory; for an easy introduction to the latter circle of ideas, see [Cor11].…”

“…Here we survey a different approach to the finite generation problem, recently obtained in [CL10a], which avoids all the standard difficult operations of Mori theory. In that paper, we give a new proof of the following result, only using the Kawamata-Viehweg vanishing and induction on the dimension.…”

Finite generation and geography of models

“…Examples of finitely generated rings. The following is a small variation of the main result of [CL12a], where it is proved by a self-contained argument avoiding the techniques of the MMP. It was first proved in the seminal paper [BCHM10] by MMP methods.…”