Theory of non-lc ideal sheaves: Basic properties

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1 Theory of non-lc ideal sheaves: Basic properties Osamu Fujino Abstract We introduce the notion of non-lc ideal sheaves. It is an analogue of the notion of multiplier ideal sheaves. We establish the restriction theorem, which seems to be the most important property of non-lc ideal sheaves. 1. Introduction Let X be a smooth complex algebraic variety, and let B be an effective R-divisor on X. Then we can define the multiplier ideal sheaf J X,B. By the definition, X,B is klt if and only if J X,B is trivial. There exist plenty of applications of multiplier ideal sheaves see, e.g., the excellent book [L]. Here we introduce the notion of non-lc ideal sheaves. We denote it by J NLC X,B. By the construction, the ideal sheaf J NLC X,B is trivial if and only if X,B is lc; that is, J NLC X,B defines the non-lc locus of the pair X,B. So, we call J NLC X,B the non-lc ideal sheaf associated to X,B. By the definition of J NLC X,B cf. Definition 2.1, we have the inclusions J X,B J NLC X,B J X,1 εb for every ε>0. Although the ideal sheaf J X,1 εb defines the non-lc locus of the pair X,B for0<ε 1, J X,1 εb does not always coincide with J NLC X,B. This is a very important remark. In [FST] we will discuss various other ideal sheaves that define the non-lc locus of X,B. Let S be a smooth irreducible divisor on X such that S is not contained in the support of B. We put B S = B S. The restriction theorem for multiplier ideal sheaves, which was obtained by Esnault and Viehweg, is one of the key results in the theory of multiplier ideal sheaves. From the analytic point of view, it is a direct consequence of the Ohsawa-Takegoshi L 2 extension theorem see [OT]. For the details, see [Ko] and [L]. Let us recall the restriction theorem here for the reader s convenience. Kyoto Journal of Mathematics, Vol. 50, No , DOI / X , 2010 by Kyoto University Received July 21, Revised October 26, Accepted October 29, Mathematics Subject Classification: Primary 14E15. Secondary 14E30. Author s work partially supported by Japan Society for the Promotion of Science Grant-in-Aid for oung Scientists A no and by the Inamori Foundation.

2 226 Osamu Fujino THEOREM 1.1 RESTRICTION THEOREM FOR MULTIPLIER IDEAL SHEAVES We have an inclusion J S, B S JX,B S. The main result of this article is the following restriction theorem for non-lc ideal sheaves for the precise statement, see Theorem THEOREM 1.2 There is an equality J NLC S, B S =J NLC X,S + B S. In particular, S, B S is lc if and only if X,S + B is lc around S. Once we obtain this powerful restriction theorem for non-lc ideal sheaves, we can translate some results for multiplier ideal sheaves into new results for non-lc ideal sheaves. We prove, for example, a subadditivity theorem for non-lc ideal sheaves. We think that the ideal sheaf J NLC X,B has already appeared implicitly in some articles. However, J NLC X,B was thought to be useless because the Kawamata- Viehweg-Nadel vanishing theorem does not hold for lc pairs. We note that the theory of multiplier ideal sheaves heavily depends on the Kawamata-Viehweg- Nadel vanishing theorem. Fortunately, we have a new cohomological package according to Ambro s formulation, which works for lc pairs see [F5, Chapter 2]. By this new package, we can walk around freely in the world of lc pairs. We prove vanishing theorem and global generation for non-lc ideal sheaves as applications. We hope that the notion of non-lc ideal sheaves will play important roles in various applications. In [F4], we prove the cone and contraction theorem for a pair X,B, where X is a normal variety and B is an effective R-divisor on X such that K X + B is R-Cartier. In that article, we repeatedly use non-lc ideal sheaves. We note that the restriction theorem cf. Theorem 2.14 is not necessary in [F4]. We use only the basic properties of non-lc ideal sheaves. We summarize the contents of this article. In Section 2, we introduce the notion of non-lc ideal sheaves and give various examples. Then we prove the restriction theorem for non-lc ideal sheaves. It produces the subadditivity theorem for non-lc ideal sheaves, and so on. Our proof of the restriction theorem is quite different from the standard arguments in the theory of multiplier ideal sheaves in [L]. It also differs from the usual X-method, which was initiated by Kawamata and is the most important technique in the traditional log minimal model program. So, we explain the proof of the restriction theorem very carefully. In Section 3, we prove the vanishing theorem and the global generation for asymptotic non-lc ideal sheaves. The last section is an appendix, where we quickly review Kawakita s inversion of adjunction on log canonicity and the new cohomological package cf. [F5, Chapter 2].

3 Theory of non-lc ideal sheaves 227 Notation and conventions We work over the complex number field C throughout this article. But we note that by using the Lefschetz principle, we can extend everything to the case where the base field is an algebraically closed field of characteristic zero. We closely follow the presentation of the excellent book [L] in order to make this article more accessible. We use the following notation freely. NOTATION i For an R-Weil divisor D = r j=1 d jd j such that D i is a prime divisor for every i and D i D j for i j, we define the round-up D = r j=1 d j D j resp., the round-down D = r j=1 d j D j, where for every real number x, x resp., x is the integer defined by x x <x+ 1 resp., x 1 < x x. The fractional part {D} of D denotes D D. We define D =1 = D j, D 1 = d j D j, d j=1 d j 1 D = d j d j D j, and D >1 = d j>1 d j D j. We call D a boundary R-divisor if 0 d j 1 for every j. We note that Q resp., R denotes the Q-linear resp., R-linear equivalence of Q-divisors resp., R-divisors. ii For a proper birational morphism f : X, the exceptional locus Excf X is the locus where f is not an isomorphism. iii Let X be a normal variety, and let B be an effective R-divisor on X such that K X + B is R-Cartier. Let f : X be a resolution such that Excf f 1 B has a simple normal crossing support, where f 1 B is the strict transform of B on.wewrite K = f K X + B+ i a i E i and ae i,x,b=a i. We say that X,B islc resp., klt if and only if a i 1 resp., a i > 1 for every i, where lc resp., klt stands for log canonical resp., kawamata log terminal. Note that the discrepancy ae,x,b R can be defined for every prime divisor E over X. By the definition, there exists the largest Zariski open set U of X such that X,B islconu. We put NlcX,B=X \ U and call it the non-lc locus of the pair X,B. We sometimes simply denote NlcX,B byx NLC. iv Let E be a prime divisor over X. The closure of the image of E on X is denoted by c X E and called the center of E on X. We use the same notation as in iii. If ae,x,b= 1 andc X E isnot contained in NlcX,B, then c X E is called an lc center of X,B. We note that our definition of lc centers is slightly different from the usual one.

4 228 Osamu Fujino 2. Non-lc ideal sheaves 2.1. Definitions of non-lc ideal sheaves Let us introduce the notion of non-lc ideal sheaves. DEFINITION 2.1 NON-LC IDEAL SHEAF Let X be a normal variety, and let Δ be an R-divisor on X such that K X +Δ is R-Cartier. Let f : X be a resolution with K +Δ = f K X +Δ such that Supp Δ is simple normal crossing. Then we put J NLC X,Δ = f O Δ Δ>1 = f O Δ +Δ =1 and call it the non-lc ideal sheaf associated to X,Δ. The name comes from the following obvious lemma see also Proposition 2.6. LEMMA 2.2 Let X be a normal variety, and let Δ be an effective R-divisor such that K X +Δ is R-Cartier. Then X,Δ is lc if and only if J NLC X,Δ = O X. REMARK 2.3 In the same notation as in Definition 2.1, we put J X,Δ = f O Δ =f O K f K X +Δ. It is nothing but the well-known multiplier ideal sheaf. J X,Δ J NLC X,Δ. It is obvious that QUESTION 2.4 Let X be a smooth algebraic variety, and let Δ be an effective R-divisor on X. Are there any analytic interpretations of J NLC X,Δ an? Are there any approaches to J NLC X,Δ from the theory of tight closure? DEFINITION 2.5 NON-LC IDEAL SHEAF ASSOCIATED TO AN IDEAL SHEAF Let X be a normal variety, and let Δ be an R-divisor on X such that K X +Δ is R-Cartier. Let a O X be a nonzero ideal sheaf on X, andletc be a real number. Let f : X be a resolution such that K +Δ = f K X +Δ and f 1 a = O F, where SuppΔ + F has a simple normal crossing support. We put J NLC X,Δ; a c = f O Δ + cf Δ + cf >1. We sometimes write J NLC X,Δ; c a=j NLC X,Δ; a c. PROPOSITION 2.6 The ideal sheaves J NLC X,Δ and J NLC X,Δ; a c are well defined; that is, they are independent of the resolution f : X. If Δ is effective and c>0, then J NLC X,Δ O X and J NLC X,Δ; a c O X.

5 Theory of non-lc ideal sheaves 229 This proposition follows from the next fundamental lemma. LEMMA 2.7 Let f : Z be a proper birational morphism between smooth varieties, and let B be an R-divisor on such that Supp B is a simple normal crossing divisor. Assume that K Z + B Z = f K + B and that Supp B Z is a simple normal crossing divisor. Then we have f O Z B Z B>1 Z O B B>1. By K Z + B Z = f K + B, we obtain K Z =f K + B =1 + {B } + f B + B>1 B Z + B>1 Z B=1 Z {B Z }. If aν,, B =1 + {B } = 1 for a prime divisor ν over, then we can check that aν,, B = 1 by using [KM, Lemma 2.45]. Since f B + B>1 B + B>1 is Cartier, we can easily see that Z Z f B + B>1 = B + B>1 + E, Z where E is an effective f-exceptional divisor. Thus, we obtain f O Z B Z B>1 Z O B B>1. We finish the proof. Z Although the following lemma is not indispensable for the proof of the main theorem, it may be useful. The proof is quite nontrivial. LEMMA 2.8 We use the same notation and assumption as in Lemma 2.7. Let S be a simple normal crossing divisor on such that S Supp B =1. Let T be the union of the irreducible components of B Z =1 which are mapped into S by f. Assume that Supp f 1 B Excf is simple normal crossing on Z. Then we have f O T B T B>1 T O S B S B>1 S, where K Z + B Z T = K T + B T and K + B S = K S + B S. We use the same notation as in the proof of Lemma 2.7. We consider 0 O Z B Z B>1 Z T O Z B Z B>1 Z O T B T B>1 T 0.

6 230 Osamu Fujino Since T = f S F,whereF is an effective f-exceptional divisor, we can easily see that f O Z B Z B>1 Z T O B B>1 S. We note that B Z B>1 Z T K Z + {B Z } +B Z =1 T = f K + B. Therefore, every local section of R 1 f O Z B Z B>1 Z T contains in its support the f-image of some stratum of Z,{B Z }+B Z =1 T by Theorem A.41. CLAIM No strata of Z,{B Z } + B Z =1 T are mapped into S by f. Assume that there is a stratum C of Z,{B Z } + B Z =1 T such that fc S. Note that Supp f S Supp f 1 B Excf and Supp B Z =1 1 Supp f B Excf. Since C is also a stratum of Z,B Z =1andC Supp f S, there exists an irreducible component G of B Z =1 such that C G Supp f S. Therefore, by the definition of T, G is an irreducible component of T because fg S and G is an irreducible component of B Z =1.So,Cis not a stratum of Z,{B Z} + B Z =1 T. It is a contradiction. On the other hand, ft S. Therefore, f O T B T B>1 T R 1 f O Z B Z B>1 Z T is a zero-map by the above claim. Thus, we obtain We finish the proof. f O T B T B>1 T O S B S B>1 S. REMARK 2.9 Let X be an n-dimensional normal variety, and let Δ be an R-divisor on X such that K X +Δ is R-Cartier. Let f : X be a resolution with K +Δ = f K X + Δ such that Supp Δ is a simple normal crossing divisor. We put A = Δ, N = Δ>1,andW =Δ=1.SinceRi f O A N W =0 for i>0 by the Kawamata-Viehweg vanishing theorem, we have and 0 JX,Δ J NLC X,Δ f O W A W N W 0, R i f O A N R i f O W A W N W for every i>0. In general, R i f O A N 0 for 1 i n 1. From now on, we assume that Δ is effective. We put F = W E, wheree is the union of irreducible components of W which are mapped to NlcX,Δ.

7 Theory of non-lc ideal sheaves 231 Then we have f O A N E=f O A N=J NLC X,Δ. Applying f to the short exact sequence 0 O A N W O A N E O F A F N F E F 0, we obtain f O F A F N F E F =f O W A W N W. In particular, J X,Δ = J NLC X,Δ if and only if X,Δ has no lc centers Examples of non-lc ideal sheaves Here we explain some elementary examples. EXAMPLE 2.10 Let X be an n-dimensional smooth variety. Let P X be a closed point, and let m = m P be the corresponding maximal ideal. Let f : X be the blowup at P.Thenf 1 m = O E, where E is the exceptional divisor of f. Ifc>n, then J NLC X; c m=f O n 1 c E = J X; c m=m c n 1. If c<n,then J NLC X; c m=f O n 1 c E = J X; c m=ox. When c = n, we note that J NLC X; c m=f O O X J X; c m=f O E=m. EXAMPLE 2.11 Let X be a smooth variety, and let D be a smooth divisor on X. ThenJ NLC X, D=O X. However, for every 0 <ε 1. On the other hand, for every 0 <ε 1. J NLC X,1 + εd = OX D J X,D=J X,1 + εd = O X D We note the following lemma on the jumping numbers, whose proof is obvious by the definitions cf. [L, Lemma , Definition ]. LEMMA 2.12 JUMPING NUMBERS Let X beasmoothvariety,andletd be an effective Q-divisor resp., R-divisor on X. Letx X be a fixed point contained in the support of D. Then there is an increasing sequence 0 <ξ 0 D; x <ξ 1 D; x <ξ 2 D; x <

8 232 Osamu Fujino of rational resp., real numbers ξ i = ξ i D; x characterized by the properties that J X,c D x = J X,ξ i D x for c [ξ i,ξ i+1, while J X,ξ i+1 D x J X,ξ i D x for every i. The rational resp., real numbers ξ i D; x are called the jumping numbers of D at x. We can check the properties that J NLC X,c D x = J NLC X,d D x for c, d ξ i,ξ i+1, while J NLC X,ξ i+1 D x J NLC X,ξ i D x for every i. Moreover, J NLC X,c D x = J X,c D x for c ξ i,ξ i+1 by Remark 2.9. EXAMPLE 2.13 Let X = C 2 =SpecC[z 1,z 2 ]andd =z 1 =0+z 2 =0+z 1 = z 2. Then we can directly check that and J NLC X,D=m 2 J NLC X,1 εd = J X,1 εd = m for 0 <ε 1, where m is the maximal ideal corresponding to 0 C 2. On the other hand, J NLC X,1 + εd = J X,1 + εd JNLC X,D for 0 <ε 1 because D NlcX,1 + εd. Note that for 0 <ε 1. J X,D=J X,1 + εd J NLC X,D 2.3. Main theorem: Restriction theorem The following theorem is the main theorem of this article. THEOREM 2.14 RESTRICTION THEOREM Let X be a normal variety, and let S + B be an effective R-divisor on X such that S is reduced and normal and that S and B have no common irreducible components. Assume that K X + S + B is R-Cartier. Let B S be the different on S such that K S + B S =K X + S + B S. Then we obtain J NLC S, B S =J NLC X,S + B S. In particular, S, B S is log canonical if and only if X,S + B is log canonical around S. REMARK 2.15 The notion of different was introduced by Shokurov in [S, 3]. For the definition and the basic properties, see, for example, [A, Codimension one adjunction] and [F4, Section 14].

9 Theory of non-lc ideal sheaves 233 Before we start the proof of Theorem 2.14, let us see an easy example. EXAMPLE 2.16 Let X = C 2 =SpecC[x, y], S =x =0, and B =y 2 = x 3. We put B S = B S. Then we have K S + B S =K X + S + B S. By direct calculations, we obtain J NLC S, B S =m 2, J NLC X,S + B=n 2, where m resp., n is the maximal ideal corresponding to 0 S resp., 0, 0 X. Of course, we have Let us start the proof of Theorem J NLC S, B S =J NLC X,S + B S. of Theorem 2.14 We take a resolution f : X with the following properties: i Excf is a simple normal crossing divisor on ; ii f 1 X NLC is a simple normal crossing divisor on, where X NLC = NlcX,S + B; iii f 1 S is a simple normal crossing divisor on ; iv f 1 X NLC S is a simple normal crossing divisor on ; v Excf f 1 X NLC f 1 B f 1 S is a divisor with a simple normal crossing support. We put K + B = f K X + S + B. Then Supp B is a simple normal crossing divisor by i and v. Let S be the strict transform of S on. Let T be the union of the components of B =1 S which are mapped into S by f. We can decompose T = T 1 + T 2 as follows. a Any irreducible component of T 2 is mapped into X NLC by f. b No irreducible component of T 1 is mapped into X NLC by f. By ii and v, any stratum of T 1 is not mapped into X NLC by f. We put A = B and N = B>1.ThenAis an effective f-exceptional divisor. Moreover, A S is exceptional with respect to f : S S. Thenwehave and Here we used J NLC X,S + B=f O A N J NLC S, B S =f O S A N. K S +B S S = f K S + B S. It follows from K + B = f K X + S + B by adjunction.

10 234 Osamu Fujino STEP 1 We consider the following short exact sequence: 0 O A N S + T O A N O S +T A N 0. Applying R i f, we obtain We note that 0 f O A N S + T f O A N f O S +T A N R 1 f O A N S + T. A N S + T K + {B } +B =1 S T = f K X + S + B and that any stratum of B =1 S T is not mapped into S by f see conditions iii and v. Therefore, the support of every nonzero local section of R 1 f O A N S + T cannot be contained in S by Theorem A.41. Thus, the connecting homomorphism f O S +T A N R 1 f O A N S + T is a zero-map. Thus, we obtain 0 J J NLC X,S + B I 0, where I := f O S +T A N andj := f O A N S + T. We note that the ideal sheaf J = f O A N S + T O X defines a scheme structure on S = S X NLC. We will check that I O S and I = J NLC X,S + B S by fs + T =S and the following commutative diagrams: 0 J J NLC X,S + B I 0 = 0 J O X O S 0 and 0 J O X = O S α 0 0 O X S O X O S 0 It is sufficient to prove Ker α I = {0}, whereα : O S O S. We note that I = J NLC X,S + B/J and Ker α = O X S/J. It is easy to see that J NLC X,S + B O X S J since fs + T =S. Thus, Ker α I = {0}. ThismeansthatI O S and I = J NLC X,S + B S. Therefore, it is enough to prove I = J NLC S, B S.

11 Theory of non-lc ideal sheaves 235 STEP 2 In this step, we prove that the natural inclusion f O S +T 1 A N T 2 f O S +T A N=I is an isomorphism. We consider the short exact sequence Applying R i f, we obtain 0 O A N S + T O A N T 2 O S +T 1 A N T J f O A N T 2 f O S +T 1 A N T 2 δ R 1 f O A N S + T. The connecting homomorphism δ is zero by exactly the same reason as in Step 1. Therefore, we obtain the following commutative diagram: 0 J f O A N T 2 f O S +T 1 A N T 2 0 = β 0 J f O A N I 0 The homomorphism β is an isomorphism since ft 2 fn=x NLC. Therefore, we obtain f O S +T 1 A N T 2 =I O S. STEP 3 The inclusion f O S A N T 2 f O S A N=J NLC S, B S O S is obvious. By Kawakita s inversion of adjunction on log canonicity cf. Corollary A.2, we obtain the opposite inclusion Therefore, we obtain f O S A N f O S A N T 2. f O S A N T 2 =f O S A N=J NLC S, B S. STEP 4 We consider the short exact sequence We note that 0 O T1 A N S T 2 O S +T 1 A N T 2 O S A N T 2 0. f O S +T 1 A N T 2 =I O S

12 236 Osamu Fujino by Step 2 and by Step 3. By taking R i f, we obtain f O S A N T 2 =J NLC S, B S 0 I J NLC S, B S R 1 f O T1 A N S T 2. Here we used the fact that f O T1 A N S T 2 =0. Note that no irreducible components of S are dominated by T 1. Since J NLC S, B S O S, we obtain Since we have J NLC S, B S /I O S /I. A N S + T 2 K + T 1 + {B } +B =1 S T = f K X + S + B, A N S + T 2 T1 K T1 +{B } + B =1 S T T1 R f K X + S + B ft1. Therefore, the support of every nonzero local section of R 1 f O T1 A N S T 2 cannot be contained in SuppO S /I SuppO S /I = Supp O X /J NLC X,S + B = X NLC by Theorem A.41. We note that no stratum of T1, {B } + B =1 S T T1 is mapped into X NLC by f see conditions iv and v. Thus, we obtain I = J NLC S, B S. We finish the proof of the main theorem. In some applications, the following corollaries may play important roles. COROLLAR 2.17 We use the notation in the proof of Theorem We have the equalities J NLC S, B S =f O S A N = f O S +T A N=f O S +T 1 A N T 2. COROLLAR 2.18 We use the notation in the proof of Theorem We obtain the short exact sequence 0 J J NLC X,S + B J NLC S, B S 0.

13 Theory of non-lc ideal sheaves 237 Let π : X V be a projective morphism onto an algebraic variety V,andletL be a Cartier divisor on X such that L K X + S + B is π-ample. Then R i π J OX L =0 for all i>0. Inparticular, R i π JNLC X,S + B O X L R i π JNLC S, B S O S L is surjective for i =0 and is an isomorphism for every i 1. Asacorollary,we obtain π JNLC S, B S O S L Im π O X L π O S L. Note that we have f L + A N S + T K + B =1 + {B } S + T = f L K X + S + B. Therefore,R i π f O f L+A N S +T =0fori>0 by Theorem A.42. Thus, R i π J O X L = 0 for all i>0 because J = f O A N S + T. REMARK 2.19 In Corollary 2.18, the ideal J is independent of the resolution f : X by Lemma 2.8. REMARK 2.20 In Corollary 2.18, we can weaken the assumption that L K X + S + B is π-ample as follows. The R-Cartier R-divisor D = L K X + S + B isπ-nef and π-big, and D C is π-big for every lc center C that is not contained in S see the proof of Theorem Direct consequences of the restriction theorem Let us collect some direct consequences of the restriction theorem. PROPOSITION 2.21 Let X be a smooth variety, let D be an effective R-divisor on X, andleth X be a smooth irreducible divisor that does not appear in the support of D. Then J NLC H,D H =J NLC X,H + D H J NLC X,D H. It is obvious. COROLLAR 2.22 Let V be a free linear system, and let H V be a general divisor. Then we

14 238 Osamu Fujino have J NLC H,D H =J NLC X,D H because J NLC X,D=J NLC X,H + D. It is obvious. COROLLAR 2.23 Let D be an effective R-divisor on the smooth variety X, andlet X be a smooth subvariety that is not contained in the support of D. Then where D = D. J NLC,D J NLC X,D, It is obvious see, e.g., the proof of [L, Corollary 9.5.6]. COROLLAR 2.24 Let f : X be a morphism of smooth irreducible varieties, and let D be an effective R-divisor on X. Assume that the support of D does not contain f. Then one has an inclusion of ideal sheaves on. J NLC,f D f 1 J NLC X,D See, for example, [L, Example 9.5.8]. PROPOSITION 2.25 DIVISORS OF SMALL MULTIPLICIT Let D be an effective R-divisor on a smooth variety X. Suppose that x X is a point at which mult x D 1. Then the ideal J NLC X,D is trivial at x. It is obvious see, e.g., [L, Proposition ]. THEOREM 2.26 GENERIC RESTRICTION Let X and T be smooth irreducible varieties, and let p : X T be a smooth surjective morphism. Consider an effective R-divisor D on X whose support does not contain any of the fibers X t = p 1 t, so that for each t T, the restriction D t = D Xt is defined. Then there is a nonempty Zariski open set U T such that J NLC X t,d t =J NLC X,D t

15 Theory of non-lc ideal sheaves 239 for every t U, wherej NLC X,D t = J NLC X,D O Xt denotes the restriction of the indicated non-lc ideal to the fiber X t. More generally, if t U, then for every c>0. J NLC X t,c D t =J NLC X,c D t We use the same notation as in the proof of [L, Theorem ]. Let U be the nonempty Zariski open set of T which was obtained in the proof of [L, Theorem ]. By shrinking T, we can assume that T = U. We take a general hypersurface H of T passing through t U. ThenJ NLC X,c D=J NLC X,X 1 + c D, where X 1 = p H. By Theorem 2.14, J NLC X,c D X1 = J NLC X,X 1 + c D X1 = J NLC X 1,c D X1. By applying this argument dim T times, we obtain J NLC X t,c D t =J NLC X, c D t. The following corollary is a direct consequence of Theorem COROLLAR 2.27 SEMICONTINUIT Let p : X T be a smooth morphism as in Theorem 2.26, and let D be an effective R-divisor on X satisfying the hypotheses of that statement. Moreover, given a section y : T X of p, write y t = yt X. If y t NlcX t,d t for t 0 T,theny 0 NlcX 0,D 0. See the proof of [L, Corollary ]. We close this subsection with the subadditivity theorem for non-lc ideal sheaves cf. [DEL]. THEOREM 2.28 SUBADDITIVIT Let X beasmoothvariety. 1 Suppose that D 1 and D 2 are any two effective R-divisors on X. Then J NLC X,D 1 + D 2 J NLC X,D 1 J NLC X,D 2. 2 If a, b O X are ideal sheaves, then for any c, d > 0. Inparticular, J NLC X; a c b d J NLC X; a c J NLC X; b d J NLC X; a b J NLC X; a J NLC X; b.

16 240 Osamu Fujino The proof of the subadditivity theorem for multiplier ideal sheaves works for non-lc ideal sheaves see, e.g., the proof of [L, Theorem ]. We leave the details as an exercise for the reader. 3. Miscellaneous results In this section, we collect some basic results of non-lc ideal sheaves Vanishing and global generation theorems Here we state vanishing and global generation theorems explicitly. We can easily check them as applications of Theorem A.4. THEOREM 3.1 VANISHING THEOREM Let X be a smooth projective variety, let D be any R-divisor on X, andletl be any integral divisor such that L D is ample. Then H i X,O X K X + L J NLC X,D =0 for i>0. Let f : X be a resolution with K + B = f K X + D such that Supp B is a simple normal crossing divisor. Then B B>1 + f K X + L K + B =1 + {B }=f L D. Therefore, H i X,R j f O B B>1 +f K X +L = 0 for every i>0 and j 0 by Theorem A.42. In particular, H i X,f O B B>1 + f K X + L =0 for i>0. This is the desired vanishing theorem because J NLC X,D = f O B B>1. We can weaken the assumption in Theorem 3.1. However, Theorem 3.1 is sufficient for our purpose in this article. So, the reader can skip the next difficult theorem. THEOREM 3.2 Let X be a normal variety, and let Δ be an effective R-divisor such that K X +Δ is R-Cartier. Let π : X V be a proper morphism onto an algebraic variety V, and let L be a Cartier divisor on X. Assume that L K X +Δ is π-nef and π-log big with respect to X,Δ; that is, L K X +Δ is π-nef and π-big and L K X +Δ C is π-big for every lc center C of the pair X,Δ. Then we have R i π JNLC X,Δ O X L =0 for all i>0.

17 Theory of non-lc ideal sheaves 241 Let f : X be a resolution with K +Δ = f K X + Δ such that Supp Δ is a simple normal crossing divisor. We put F =Δ =1 E, wheree is the union of irreducible components of Δ =1 which are mapped to X NLC =NlcX,Δ. If needed, we take more blowups and can assume that no strata of F are mapped to X NLC. In this case, we have Since we have J NLC X,Δ = f O Δ Δ>1 E. Δ Δ>1 E + f L K + F + {Δ } = f L K X +Δ, R i π R j f O Δ Δ>1 E + f L =0 for every i>0andj 0 see, e.g., [F5, Theorem 2.47]. So, we obtain for i>0. R i π JNLC X,Δ O X L =0 THEOREM 3.3 GLOBAL GENERATION Let X be a smooth projective variety of dimension n. We fix a globally generated ample divisor B on X. LetD be an effective R-divisor, and let L be an integral divisor on X such that L D is ample or, more generally, nef and log big with respect to X,D. Then O X K X + L + mb J NLC X,D is globally generated as soon as m n. It is obvious by Theorem 3.1 or Theorem 3.2 and Mumford s m-regularity Asymptotic non-lc ideal sheaves Let X be a smooth variety. Let a = {a m } be a graded system of ideals on X. In other words, a consists of a collection of ideal sheaves a k O X satisfying a 0 = O X and a m a l a m+l for all m, l 1. DEFINITION 3.4 NON-LC IDEAL ASSOCIATED TO A GRADED SSTEM OF IDEALS The asymptotic non-lc ideal sheaf of a with coefficient or exponent c, written by either J NLC X; c a or J NLC X; a c, is defined to be the unique maximal member among the family of ideals {J NLC X; c/p a p } for p 1. Thus J NLC X; c a =J NLC X;c/p a p for all sufficiently large and divisible integer p 0.

18 242 Osamu Fujino EXAMPLE 3.5 Let X be a smooth projective variety, and let L be an integral divisor on X of nonnegative Iitaka dimension. We consider the base ideal b k = b kl ofthe complete linear system kl for every k 0. Let Δ be an effective R-divisor on X such that K X +Δ is R-Cartier. Then b is a graded system of ideals on X. We put J NLC X,Δ, L := JNLC X,Δ; b. We note that J NLC X,Δ; b is the unique maximal member among the family of ideals {J NLC X,Δ; 1/p b p } for p 1. Almost all the basic properties of asymptotic multiplier ideal sheaves in [L, 11.1, 11.2.A] can be proved for asymptotic non-lc ideal sheaves by the same arguments. Therefore, we do not repeat them here. We leave them as exercises for the reader. We state only one theorem in this subsection. THEOREM 3.6 Let X be a smooth projective variety, let Δ be an effective Cartier divisor on X, and let L be an integral divisor on X of nonnegative Iitaka dimension. If A is an ample divisor on X, then H i X,O X K X +Δ+mL + A J NLC X,Δ, ml =0 for i>0. Furthermore, we assume that B is a globally generated ample divisor on X. Then for every m 1, O X K X +Δ+lB + A + ml J NLC X,Δ, ml is globally generated as soon as l dim X. Let H kml be a general member for a large and divisible k. ThenJ NLC X, Δ, ml =J NLC X,Δ, 1/kH=J NLC X,Δ+1/kH. On the other hand, Δ+mL + A Δ + 1 k H Q A. Thus, this theorem follows from Theorems 3.1 and 3.3. A. Appendix A.1. Inversionofadjunctiononlogcanonicity We give some comments on the inversion of adjunction on log canonicity. The following theorem is due to Kawakita. Roughly speaking, he proved it by iterating the restriction theorem between adjoint ideal sheaves on X and multiplier ideal sheaves on S for the proof, see [Ka]. THEOREM A.1 KAWAKITA Let X be a normal variety, let S be a reduced divisor on X, andletb be an effective R-divisor on X such that K X + S + B is R-Cartier. Assume that S has

19 Theory of non-lc ideal sheaves 243 no common irreducible component with the support of B. Let ν : S ν S be the normalization, and let B S ν be the different on S ν such that K S ν + B S ν = ν K X + S + B S. Then X,S + B is log canonical around S if and only if S ν,b S ν is log canonical. By adjunction, it is obvious that S ν,b S ν is log canonical if X,S + B islog canonical around S. So the above theorem is usually called the inversion of adjunctiononlogcanonicity. We need the following corollary of Theorem A.1 in the proof of Theorem The proof is obvious. COROLLAR A.2 Let X,S + B be as in Theorem A.1. Let P X be a closed point such that X,S + B is not log canonical at P. Let f : X be a resolution such that K + B = f K X + S + B and that Supp B is simple normal crossing. Then f 1 P S Supp N, wheres = f 1 S and N = B >1. We close this subsection with a remark on the theory of quasi-log varieties. REMARK A.3 We use the notation in Theorem A.1. We note that [X,K X + S + B] has a natural quasi-log structure, which was introduced by Ambro see, e.g., [F5, Chapter 3]. By adjunction, S = S X NLC has a natural quasi-log structure induced by [X,K X +S +B]. More explicitly, the defining ideal sheaf of the quasi-log variety S is J in the proof of Theorem In Step 1 in the proof of Theorem 2.14, we did not use the normality of S. Theorem A.1 says that [S, K X + S + B S ] has only qlc singularities around S if and only if S ν,b S ν islc. A.2. New cohomological package We quickly review Ambro s formulation of torsion-free and vanishing theorems in a simplified form for more advanced topics and the proof, see [F5, Chapter 2]. Let be a simple normal crossing divisor on a smooth variety M, and let D be an R-divisor on M such that SuppD + is simple normal crossing and that D and have no common irreducible components. We put B = D and consider the pair,b. Let ν : ν be the normalization. We put K ν +Θ=ν K + B. A stratum of,b is an irreducible component of or the image of some lc center of ν, Θ =1. When is smooth and B is an R-divisor on such that Supp B is simple normal crossing, we put M = A 1 and D = B A 1. Then,B {0},B {0} satisfies the above conditions. THEOREM A.4 Let,B be as above. Assume that B is a boundary R-divisor. Let f : X be a proper morphism, and let L be a Cartier divisor on.

20 244 Osamu Fujino 1 Assume that H R L K + B is f-semiample. Then every nonzero local section of R q f O L contains in its support the f-image of some stratum of,b. 2 Let q be an arbitrary nonnegative integer. Let π : X V be a proper morphism, and assume that H R f H for some π-ample R-Cartier R-divisor H on X. Then, R q f O L is π -acyclic; that is, R p π R q f O L =0 for every p>0. For the proof, see [F5, Theorem 2.39]. We note that [F2] is a gentle introduction to this new cohomological package. The reader can find various applications in [F1], [F3], [F6], [F5], [F7], and [F4]. Acknowledgments. I am grateful to Professor Hiromichi Takagi for his warm encouragement. Almost all the results were obtained during my short trip to Germany with the book [L] in the autumn of I thank the Max-Planck Institute for Mathematics and Mathematisches Forschungsinstitut Oberwolfach for their hospitality. I am grateful to Vladimir Lazić and Masayuki Kawakita for useful comments. I also thank Professors Mircea Mustaţă and Mihnea Popa for pointing out mistakes. I am grateful to the Mathematical Sciences Research Institute for its hospitality. I thank Shunsuke Takagi and Karl Schwede for many questions, suggestions, and useful comments. References [A] [DEL] [F1] [F2] [F3] [F4] [F5] [F6] [F7] F. Ambro, Non-klt techniques in Flips for 3-folds and 4-folds, Oxford Lecture Ser. Math. Appl. 35, Oxford Univ. Press, Oxford, 2007, J.-P. D ly, L. Ein, and R. Lazarsfeld, A subadditivity property of multiplier ideals, Michigan Math. J , O. Fujino, Effective base point free theorem for log canonical pairs Kollár type theorem, Tohoku Math. J , , On injectivity, vanishing and torsion-free theorems for algebraic varieties, Proc. Japan Acad. Ser. A Math. Sci , , Effective base point free theorem for log canonical pairs, II: Angehrn-Siu type theorems, to appear in Michigan Math. J., Fundamental theorems for the log minimal model program, preprint, 2009., Introduction to the log minimal model program for log canonical pairs, preprint, 2009., Introduction to the theory of quasi-log varieties, preprint, to appear in the proceedings of the Classification of Algebraic Varieties conference, Schiermonnikoog, Netherlands, May 10 15, 2009., Non-vanishing theorem for log canonical pairs, toappearin J. Algebraic Geom.

21 Theory of non-lc ideal sheaves 245 [FST] O. Fujino, K. Schwede and S. Takagi, Supplements to non-lc ideal sheaves, in preparation. [Ka] M. Kawakita, Inversion of adjunction on log canonicity, Invent. Math , [Ko] J. Kollár, Singularities of pairs in Algebraic Geometry Santa Cruz, Calif., 1995, Proc. Sympos. Pure Math. 62, Part 1, Amer. Math. Soc., Providence, 1997, [KM] J. Kollár and S. Mori, Birational Geometry of Algebraic Varieties, withthe collaboration of C. H. Clemens and A. Corti, trans. from the 1998 Japanese original, Cambridge Tracts in Math. 134, Cambridge Univ. Press, Cambridge, [L] R. Lazarsfeld, Positivity in Algebraic Geometry, II: Positivity for Vector Bundles, and Multiplier Ideals, Ergeb. Math. Grenzgeb. 3 49, Springer, Berlin, [OT] T. Ohsawa and K. Takegoshi, On the extension of L 2 holomorphic functions, Math. Z , [S] V. V. Shokurov, Three-dimensional log perestroikas in Russian, appendix in English by ujiro Kawamata, Izv. Ross. Akad. Nauk Ser. Mat , ; English translation in Russian Acad. Sci. Izv. Math , Department of Mathematics, Faculty of Science, Kyoto University, Kyoto , Japan; fujino@math.kyoto-u.ac.jp