THE MINIMAL MODEL PROGRAM FOR VARIETIES OF LOG GENERAL TYPE CHRISTOPHER D. HACON Contents 1. Preliminaries 2 1.1. Resolution of singularities 2 1.2. Divisors 2 1.3. Ample divisors 5 1.4. Positivity of divisors 9 2. The Singularities of the Minimal Model Program 11 2.1. Vanishing theorems 15 2.2. Calculus of non Kawamata log terminal centers 17 2.3. Rational Singularities 19 3. Multiplier ideal sheaves 22 3.1. First geometric applications of multiplier ideals 25 3.2. Further properties of multiplier ideal sheaves 26 3.3. The theorem of Anhern and Siu 29 3.4. Asymptotic multiplier ideal sheaves 31 3.5. Adjoint ideal sheaves 35 3.6. Asymptotic multiplier ideal sheaves II 38 4. Extension Theorems and applications 41 4.1. Deformation invariance of plurigenera. 44 5. Pl-flips 45 5.1. pl-flips and the restricted algebra 45 5.2. Zariski decomposition and pl-flips 48 6. The cone theorem 51 6.1. Generalizations 61 7. The minimal model program 61 7.1. Types of models 61 7.2. The minimal model program (traditional) 63 7.3. The minimal model with scaling 65 7.4. Minimal models for varieties of general type 66 7.5. The main induction 67 8. Special termination with scaling 70 Date: March 27, 2013. 1 8.1. lemmas 73 9. Log terminal models 76 9.1. lemmas 79 10. Finiteness of models, the big case 83 10.1. Lemmas 88 11. Non-vanishing 90 11.1. Lemmas 95 12. Finiteness of models, the general case 96 13. The Sarkisov Program 97 13.1. Introduction 97 13.2. Notation and conventions 100 13.3. Preliminaries 100 13.4. The main result 102 14. Further results 109 14.1. Fujita's Approximation Theorem 109 14.2. The Pseudo-effective Cone 111 15. Rationally Connected Fibrations 112 15.1. singular varieties 114 References 115 1. Preliminaries 1.1. Resolution of singularities. We will need the following result of Hironaka on the resolution of singularities. Theorem 1.1. Let X be an irreducible complex projective variety and D be an effective Cartier divisor on X. Then there is a birational mor- phism µ : X0 ! X from a smooth variety X0 given by a finite sequence of blow ups along smooth centers supported over the singularities of D and X such that µ∗D + Exc(µ) is a divisor with simple normal crossings support. The above statement is taken from [11, x4]. For a particularly clear exposition of the proof of this result as well as references to the litera- ture, we refer to [7]. 1.2. Divisors. Let X be a normal complex variety. Definition 1.2. A prime divisor is an irreducible and reduced codi- mension 1 subvariety of X. The group of Weil divisors WDiv(X) P is the set of all finite formal linear combinations D = diDi where 2 di 2 Z and Di are prime divisors with addition defined component by component X X 0 X 0 diDi + diDi = (di + di)Di: A divisor D 2 WDiv(X) is effective (denoted by D ≥ 0) if D = P diDi, with di ≥ 0 and Di prime divisors. Definition 1.3. For any divisor D 2 WDiv(X), one may define the divisorial sheaf OX (D) by setting Γ(U; OX (D)) = ff 2 C(X)j((f) + D)jU ≥ 0g: Remark 1.4. Note that OX (D) is a reflexive sheaf of rank one so that __ ∼ OX (D) = OX (D). Conversly, for any torsion free reflexive sheaf of ∼ rank one F there is a Weil divisor D such that F = OX (D). Notice moreover that if U = X − Xsing and i : U ! X is the inclusion, then i∗OU (DjU ) = OX (D). Definition 1.5. For any rational function 0 6= f 2 C(X), we let (f) 2 Div(X) be the principal divisor corresponding to the zeroes and poles of f. We say that two divisors D; D0 2 Div(X) are linearly equivalent if D − D0 = (f) where f 2 C(X). The complete linear series corresponding to a divisor D 2 Div(X) is given by jDj = fD0 ≥ 0jD0 ∼ Dg: Definition 1.6. For any divisor D 2 Div(X), the base locus of D is given by Bs(D) = \D02jDjSupp(D): (Here Supp(D) is the support of D i.e. the subset of X given by the points of D.) ∼ k ∼ 0 Definition 1.7. If jDj 6= ;, then jDj = P = PH (OX (D)) for some k > 0. We let k be the dimension of jDj and k φjDj : X 99K P be the corresponding rational map. Note that if U = X − Bs(D), then (φjDj)jU is a morphism. More explicitely, if fs0; : : : ; skg is a basis of 0 H (OX (D)), then (φjDj)jU (x) = [s0(x): ::: : sk(x)]: Definition 1.8. A k-cycle on X is a Z-linear combination of irre- ducible subvarieties of dimension k. The set of all k-cycles on X is denoted by Zk(X) and it is an abelian group with respect to addition. Note that Zdim(X)−1(X) = WDiv(X). 3 Definition 1.9. A Cartier divisor is a Weil divisor D which is locally defined by the zeroes and poles of a rational function f 2 C(X). The group of Cartier divisors Div(X) is a subgroup of WDiv(X) and it ∗ ∗ may be identified with Γ(X; C(X) =OX ) (here C(X) denotes the sheaf of rational functions). Note that a Weil divisor D is Cartier if and only if the sheaf OX (D) is invertible. If K 2 fQ; R;:::g, then we let WDivK (X) = WDiv(X) ⊗Z K and 0 0 DivK (X) = Div(X) ⊗Z K. If D; D 2 WDiv(X), then D ∼K D if and 0 P only if D − D = di(fi) with fi 2 C(X) and di 2 K. Definition 1.10. If D is a Cartier divisor on X and f : Y ! X is a dominant morphism, then we define the pullback f ∗D of D as ∗ follows: Let Ui be an open covering of X and gi 2 C(X) such that ∗ −1 D \ Ui = (gi) \ Ui, then f D is defined by gi ◦ f on f (Ui). Definition 1.11. If D is Cartier divisor on a proper normal variety X and C ⊂ X is a curve contained in X, then the intersection of D and C is given by D · C = deg(i∗D) where i : C0 ! X is the induced map from the normalization of C to X. Two Cartier divisors D and D0 (or more generally two elements D; D0 2 WDiv(X) such 0 that D − D 2 DivR(X)) are numerically equivalent (denoted by D ≡ D0) if (D − D0) · C = 0 for any curve C ⊂ X. Numerical equivalence generates an equivalence relation in Div(X) and in Z1(X). We let 1 N (X) = DivR(X)= ≡ and N1(X) = (Z1(X) ⊗Z R)= ≡ : 1 Note that N (X) and N1(X) are dual vector spaces over R. Their dimension ρ(X) is the Picard number of X. Definition 1.12. The cone of effective 1-cycles is the cone NE(X) ⊂ N1(X) P generated by f niCi s: t: ni ≥ 0g. Given a proper morphism of normal varieties f : X ! Y and an irreducible curve C ⊂ X, we let f∗(C) = df(C) where d = deg(C ! f(C)). If f(C) is a point, then we set f∗C = 0. One sees that ∗ f D · C = D · f∗C 8D 2 Div(Y ): Extending by linearity, we get an injective homomorphism f ∗ : N 1(Y ) ! N 1(X) and surjective homomorphisms f∗ : N1(X) ! N1(Y );NE(X) ! NE(Y ): 4 P Definition 1.13. If D = diDi 2 WDivR(X), where Di are distinct prime divisors, then we define the round down, the round up and the fractional part of D by the formulas X X X pDq = pdiqDi; xDy = xdiyDi; fDg = fdigDi where xdiy is the biggest integer ≤ di, pdiq is the smallest integer ≥ di and fdig = di − xdiy. 0 Remark 1.14. Note that if D ∼Q D it is not the case that pDq = 0 pD q. If D 2 DivQ(X), and Y ⊂ X is a subvariety, it is also not the case that xDjY y = xDyjY . We have p−Dq = −xDy. Definition 1.15. If D 2 WDiv(X) and jDj 6= ;, we let Fix(D) = P fiFi where fi is the minimum of the multiplicities of any divisor 0 D 2 jDj along the prime divisor Fi. We let Mob(D) = D − Fix(D). Note that Bs(Mob(D)) has codimension at least 2. 1.3. Ample divisors. Definition 1.16. A Cartier divisor D 2 Div(X) is very ample if it N is base point free and φjDj : X ! P is an embedding. A Q-Cartier divisor D 2 DivQ(X) is ample if mD is very ample for some m > 0. Recall the following: Definition 1.17. A coherent sheaf F on a variety X is globally gen- erated if the homomorphism 0 H (X; F) ⊗ OX !F is surjective. Theorem 1.18 (Serre). Let D 2 Div(X) be an ample divisor on a projective scheme and F a coherent sheaf on X. Then there is an integer n0 > 0 such that for all n ≥ n0, F ⊗ OX (nD) is globally generated. Proof. [5] II x5. The idea is that we may assume that X = PN .