SUPPLEMENTS TO NON-LC IDEAL SHEAVES OSAMU FUJINO, KARL SCHWEDE AND SHUNSUKE TAKAGI 1 1 0 Abstract. We consider various definitions of non-lc ideal sheaves – generaliza- 2 tionsofthemultiplieridealsheafwhichdefinethenon-lc(non-logcanonical)locus. Weintroducethemaximalnon-lcidealsheafandintermediatenon-lcidealsheaves n a andconsidertherestrictiontheoremfortheseidealsheaves. Wealsobeginthede- J velopment of the theory of a characteristicp>0 analogof maximalnon-lc ideals, 1 utilizing some recent work of Blickle. ] G A Contents . h 1. Motivation 1 t a 2. Introduction 2 m [ Part 1. Variants of non-lc ideals J 5 NLC 2 3. Lc centers, non-klt centers, and non-lc centers 5 v 4. New cohomological package 7 0 7 5. Non-lc ideal sheaves 7 1 6. Observations towards non-lc ideal sheaves 8 5 7. Maximal non-lc ideal sheaves 10 . 4 8. Intermediate non-lc ideal sheaves 13 0 0 9. Supplementary remarks 16 1 10. Non-klt ideal sheaves 17 : v 11. Differents 18 i X 12. Restriction theorems 20 r 13. The restriction theorem for complete intersections 23 a Part 2. A characteristic p analog of maximal non-lc ideals 31 14. Non-F-pure ideals 31 15. Non-F-pure ideals vs. non-lc ideal sheaves 38 16. The restriction theorem for non-F-pure ideals 41 References 44 1. Motivation In this short section, we explain our motivation for the study of non-lc ideal sheaves. 2000 Mathematics Subject Classification. Primary 14B05,14E15; Secondary 14E30,13A35. 1 2 OSAMU FUJINO,KARLSCHWEDE ANDSHUNSUKETAKAGI 1.1 (Motivation). Let X be a normal variety and ∆ an effective Q-divisor on X such that K +∆ is Q-Cartier. In this situation, we want to define an ideal sheaf X I(X,∆) satisfying the following properties. (A) The pair (X,∆) is log canonical if and only if I(X,∆) = O . X (B) (Kodaira type vanishing theorem). Assume that X is projective. Let D be a Cartier divisor on X such that D −(K +∆) is ample. Then X Hi(X,I(X,∆)⊗O (D)) = 0 X for every i > 0. (C) (Bertini type theorem). Let H be a general member of a free linear system Λ on X. Then I(X,∆) = I(X,∆+H). (D) (Restriction theorem). Assume that ∆ = S + B such that S is a normal prime Weil divisor on X, B is an effective Q-divisor, and that S and B have no common irreducible components. Then I(X,S +B)| = I(S,B ), S S where (K +S +B)| = K +B . X S S S We have already known that the (minimal) non-lc ideal sheaf J (X,∆) intro- NLC duced in[F2]satisfies alltheabove properties. The intermediate non-lc ideal sheaves J′(X,∆) for every negative integer l, which will be defined in Section 8 below, sat- l isfy (A), (B), and (C). However, in general, (D) does not always hold for J′(X,∆) l with l = −1,−2,···. The maximal non-lc ideal sheaf J′(X,∆) also satisfies (A), (B), and (C), and we do not know if (D) holds for J′(X,∆) or not. However, we have some evidence that it is true. In particular, we will give partial answers to this question in Section 12 and in Section 13. In section 13, we also mention a link between J′(X,∆) andideals that appear naturally in the study of the Hodge-theory of singular varieties. We conclude this paper by developing a characteristic p > 0 analog of J′(X,∆), relying heavily on some recent interesting work of Blickle, see [B]. Finally, it should be noted that, in various presentations, S´andor Kova´cs has recently been discussing how the ideal J′(X,∆) is a natural ideal to consider. His work in this direction is independent of the authors although certainly inspired by connections with Du Bois singularities; see Section 13. 2. Introduction The main purpose of this paper is to consider variants of the non-lc ideal sheaf J introduced in [F2]. We also consider various non-klt ideal sheaves. We will NLC explain our motivation, observations, and some attempts in the study of non-lc ideal sheaves. Let D be an effective R-divisor on a smooth complex variety X. We put J′(X,D) := J(X,(1−ε)D) for 0 < ε ≪ 1, where the right hand side is the multiplier ideal sheaf associated to (1−ε)D and it is independent of ε for sufficiently small 0 < ε ≪ 1. By the definition of J′(X,D), the pair (X,D) is log canonical if and only if J′(X,D) = O . We call X SUPPLEMENTS TO NON-LC IDEAL SHEAVES 3 J′(X,D) the maximal non-lc ideal sheaf of (X,D). We will discuss the definition and the basic properties of J′ in Section 7. In general, J(X,D) ( J(X,(1−ε)D) for 0 < ε ≪ 1 and the relationship between J(X,D) and J′(X,D) is not clear. So, we need new ideas and techniques to handle J′(X,D). We believe that the Kawamata–Viehweg–Nadel vanishing theorem is not powerful enough for the study of J′(X,D). However, the new cohomological package of the first author, explained in Section 4, seems well suited for this task. Let X be a smooth variety and S a smooth irreducible divisor on X. Let B be an effective R-divisor on X such that S 6⊂ Supp B. Then we have the following equality J′(S,B| ) = J′(X,S +B)| . S S See, for example, Theorem 12.7. We call it the restriction theorem. We will par- tially generalize it to the case of singular varieties in Section 13. In [F2], the first author introduced the notion of (minimal) non-lc ideal sheaves J and proved NLC the restriction theorem J (S,B| ) = J (X,S +B)| . NLC S NLC S Both of J′(X,D) and J (X,D) define the non-lc locus of the pair (X,D). NLC However, in general, J′(X,D) does not always coincide with J (X,D). We note NLC that J(X,D) ⊂ J (X,D) ⊂ J′(X,D) NLC holds by the definitions of J, J , and J′. Although J (X,D) seems to be the NLC NLC most natural ideal that defines the non-lc locus of (X,D) from the point of view of the minimal model program (cf. [F4]), J′(X,D) may be more suitable to the theory of multiplier ideal sheaves than J (X,D). NLC More generally, we consider a family of non-lc ideal sheaves. We define interme- diate non-lc ideal sheaves J′(X,D) for every negative integer l. By the definition l of J′(X,D) (which is a sheaf that varies with each negative integer l), J′(X,D) l l defines the non-lc locus of (X,D) and satisfies many of the same useful properties that the first author’s original non-lc ideal J (X,D) enjoys. NLC Furthermore, there are natural inclusions (where again, the l vary of the negative integers) J (X,D) ⊂ ··· ⊂ J′ (X,D) NLC l−1 ⊂ J′(X,D) ⊂ J′ (X,D) ⊂ ··· ⊂ J′(X,D) ⊂ O . l l+1 X Similarly, we also define a family of non-klt ideal sheaves J (X,D) for every non- l positive integer l. These sheaves satisfy J(X,D) ⊂ ··· ⊂ J (X,D) l−1 ⊂ J (X,D) ⊂ J (X,D) ⊂ ··· ⊂ J (X,D) ⊂ O l l+1 0 X and put possibly different scheme structures on the non-klt locus of (X,D). We have natural inclusions J (X,D) ⊂ J′(X,D) l l for every negative integer l, as well as the inclusions J(X,D) ⊂ J (X,D) and J (X,D) ⊂ J′(X,D). NLC 0 4 OSAMU FUJINO,KARLSCHWEDE ANDSHUNSUKETAKAGI Let W be the union of all the lc centers of (X,D) (see our slightly non-standard definition of lc centers in Section 3) and U = X \W. Then J (X,D)| = J′(X,D)| l U l U for every negative integer l, J(X,D)| = J (X,D)| and J (X,D)| = J′(X,D)| . U NLC U 0 U U Because the multiplier ideal sheaf has emerged as such a fundamental tool in higher dimensional algebraic geometry, it is natural to desire a non-lc ideal sheaf which agrees with the multiplier ideal sheaf in as wide a setting as possible. If we as- sume that this is a desired property, then J (X,D) is the right generalization of NLC J(X,D). On the other hand, with regards to condition (B) from Section 1, one way to in- terpret the term I(X,∆) is as a correction term which mitigates for the singularities of ∆. From this point of view, using the maximal non-lc ideal sheaf J′(X,∆) gives a heuristically stronger statement as it says one has to “adjust” to lesser extent. Themultiplier ideal J(X = Spec R,∆)isalsovery closely relatedtothetest ideal τ (R,∆), a notion that appears in the theory of commutative algebra in positive b characteristic, see for example [T1]. We conclude this paper with several sections which explore a positive characteristic analog of J′(X,∆) which we call the non-F- pure ideal and denote it by σ(X,∆). In order to define this ideal, we rely heavily on some recent work of Blickle, see [B]. In section 15 we then relate the characteristic zero notion J′(X,∆) and the characteristic p > 0 notion σ(R,∆). In the final section, we prove a restriction theorem for σ(X,∆), including a proof that the formation of σ(X,∆) commutes with the restriction to an arbitrary codimension normal F-pure center (which is a characteristic p > 0 analog of a log canonical center). We summarize the contents of this paper. This paper is divided into two parts. Part I, consisting of Section 3–13, is devoted to the study of variants of non-lc ideal sheaves J on complex algebraic varieties. Part II, consisting of Section 14–16, is NLC devoted to the study of a positive characteristic analog of the maximal non-lc ideal sheaves. These two parts are independent to each other, except for the definition of the maximal non-lc ideal sheaves. In Section 3, we define lc centers, non-klt centers, and non-lc centers. It is very important to distinguish these three notions. In Section 4, we recall Ambro’s formu- lation of Koll´ar’s torsion-free and vanishing theorems for the reader’s convenience. In Section 5, we recall the notion of non-lc ideal sheaves introduced in [F2]. In Section 6, we discuss how to define certain non-lc ideal sheaves. We also discuss some properties which should be satisfied by these ideal sheaves. This section is an informal discussion. In Section 7, we will define the maximal non-lc ideal sheaf J′ and investigate basic properties of J′. In Section 8, we introduce the notion of intermediate non-lc ideal sheaves. Section 9 is a supplement to the fundamental theorems for the log minimal model program in [F4]. In Section 10, we discuss various non-klt ideal sheaves. In Section 11, we recall Shokurov’s differents for the restriction theorem discussed in Section 12. Sections 12 and 13 are attempts to prove the restriction theorem for J′. Also in section 13, we explain how J′(X,∆) SUPPLEMENTS TO NON-LC IDEAL SHEAVES 5 appears in the study of the Hodge theory of singular varieties. In Section 14, we introduce a characteristic p analog of the maximal non-lc-ideal J′(X,∆), called a non-F-pure ideal, and investigate its basic properties. In Section 15, we explore the relationship between non-F-pure ideals and maximal non-lc ideals, which is followed by Section 16 where we prove a restriction theorem for non-F-pure ideals. We will work over the complex number field C throughout Part I. 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. Also, we will use the following notation freely. Notation. (i) For an R-Weil divisor D = r d D such that D is a prime divisor j=1 j j j for every j and D 6= D for i 6= j, we define the round-up pDq = r pd qD i j P j=1 j j (resp. the round-down xDy = r xd yD ), where for every real number x, pxq j=1 j j P (resp. xxy) is the integer defined by x ≤ pxq < x+1 (resp. x−1 < xxy ≤ x). The fractional part {D} of D denotePs D −xDy. We define D=k = d D = k D , D≤k = d D , j j j j j dXj=k dXj=k dXj≤k D<k = d D , D≥k = d D and D>k = d D j j j j j j dXj<k dXj≥k dXj>k for every k ∈ R. We put kD = Supp D=k. We note that 0D = Supp D=0 = 0 and 1D = Supp D=1 = D=1. We call D a boundary R-divisor if 0 ≤ d ≤ 1 for every j. We note that ∼ (resp. ∼ ) denotes j Q R the Q-linear (resp. R-linear) equivalence of Q-divisors (resp. R-divisors). (ii) For a proper birational morphism f : X → Y, the exceptional locus Exc(f) ⊂ X is the locus where f is not an isomorphism. Acknowledgments. The first author was partially supported by the Grant-in-Aid for Young Scientists (A) ♯20684001 from JSPS and by the Inamori Foundation. The second author was partially supported by an NSF postdoctoral fellowship and by the NSF grant DMS-1064485. The third author was partially supported by the Grant-in-Aid for Young Scientists (B) ♯20740019 from JSPS and by the Program for Improvement of Research Environment for Young Researchers from SCF commis- sioned by MEXT of Japan. They would like the referee for many useful suggestions. A part of this work was done during visits of all the authors to MSRI. They are grateful to MSRI for its hospitality and support. Part 1. Variants of non-lc ideals J NLC This part is devoted to the study of variants of non-lc ideal sheaves J on NLC complex algebraic varieties. 3. Lc centers, non-klt centers, and non-lc centers Inthissection, wequickly recall thenotionoflcandklt parisanddefine lc centers, non-klt centers, and non-lc centers. 6 OSAMU FUJINO,KARLSCHWEDE ANDSHUNSUKETAKAGI 3.1 (Discrepancies, lc and klt pairs, etc.). Let X be a normal variety and B an effective R-divisor on X such that K + B is R-Cartier. Let f : Y → X be a X resolution such that Exc(f) ∪ f−1B has a simple normal crossing support, where ∗ f−1B is the strict transform of B on Y. We write ∗ K = f∗(K +B)+ a E Y X i i i X and a(E ,X,B) = a . We say that (X,B) is i i log canonical (lc, for short) if a ≥ −1 for every i, and, i kawamata log terminal (klt, for short) if a > −1 for every i. ( i Note that the discrepancy a(E,X,B) ∈ R can be defined for every prime divisor E over X. By definition, there exists the largest Zariski open set U (resp. U′) of X such that (X,B) is lc (resp. klt) on U (resp. U′). We put Nlc(X,B) = X \ U (resp. Nklt(X,B) = X \U′) and call it the non-lc locus (resp. non-klt locus) of the pair (X,B). We sometimes simply denote Nlc(X,B) by X . We will discuss NLC various scheme structures on Nlc(X,B) (resp. Nklt(X,B)) in Section 9 (resp. in Section 10). Let E be a prime divisor over X. The closure of the image of E on X is denoted by c (E) and called the center of E on X. X 3.2 (Lc centers, non-klt centers, and non-lc centers). Let X be a normal variety and B an effective R-divisor on X such that K + B is R-Cartier. Let E be a prime X divisor over X. In this paper, we use the following terminology. The center c (E) X is an lc center if a(E,X,B) = −1 and c (E) 6⊂ Nlc(X,B), X a non-klt center if a(E,X,B) ≤ −1, and  a non-lc center if a(E,X,B) < −1. The above terminology is slightly different from the usual one. We note that it is   very important to distinguish lc centers, non-klt centers, and non-lc centers in our theory. In the traditional theory of multiplier ideal sheaves, we can not distinguish among lc centers, non-klt centers, and non-lc centers. In our new framework, the notion of lc centers plays very important roles. It is because our arguments heavily depend on the new cohomological package reviewed in Section 4. It is much more powerful than the Kawamata–Viehweg–Nadel vanishing theorem. We note that an lc center is a non-klt center. The next lemma is almost obvious by the definition of lc centers. Lemma 3.3. The number of lc centers of (X,B) is finite even if (X,B) is not log canonical. We note the following elementary example. Example 3.4. Let X = C2 = Spec C[x,y] and C = (y2 = x3). We consider the pair (X,C). Then we can easily check that there is a prime divisor E over X such that a(E,X,C) = −1 and c (E) is the origin (0,0) of C2 and that (X,C) is not X lc at (0,0). Therefore, the center c (E) is a non-klt center but not an lc center of X (X,C). SUPPLEMENTS TO NON-LC IDEAL SHEAVES 7 4. 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 [F3, Chapter 2]. The paper [F1] may help the reader to understand the proof of Theorem 4.2. We think that it is not so easy to grasp the importance of Theorem 4.2. We recommend the reader to learn how to use Theorem 4.2 in [F2], [F3], [F4], and this paper. 4.1 (Global embedded simple normal crossing pairs). Let Y be a simple normal crossingdivisoronasmoothvarietyM andD anR-divisoronM suchthatSupp(D+ Y) is simple normal crossing and that D and Y have no common irreducible com- ponents. We put B = D| and consider the pair (Y,B). Let ν : Yν → Y be the Y normalization. We put K +Θ = ν∗(K +B). A stratum of (Y,B) is an irreducible Yν Y component of Y or the image of some lc center of (Yν,Θ=1). When Y is smooth and B is an R-divisor on Y such that Supp B is simple normal crossing, we put M = Y ×A1 and D = B ×A1. Then (Y,B) ≃ (Y ×{0},B ×{0}) satisfies the above conditions. Theorem 4.2. Let (Y,B) be as above. Assume that B is a boundary R-divisor. Let f : Y → X be a proper morphism and L a Cartier divisor on Y. (1) Assume that L−(K +B) is f-semi-ample. Let q be an arbitrary non-negative Y integer. Then every non-zero local section of Rqf O (L) contains in its support the ∗ Y f-image of some stratum of (Y,B). (2) Let π : X → V be a proper morphism and assume that L−(K +B) ∼ f∗H Y R for some π-ample R-Cartier R-divisor H on X. Then, Rqf O (L) is π -acyclic, ∗ Y ∗ that is, Rpπ Rqf O (L) = 0 for every p > 0 and q ≥ 0. ∗ ∗ Y Remark 4.3. It is obvious that the statement of Theorem 4.2 (1) is equivalent to the following one. (1′) Assume that L − (K + B) is f-semi-ample. Let q be an arbitrary non- Y negative integer. Then every associated prime of Rqf O (L) is the generic point of ∗ Y the f-image of some stratum of (Y,B). For the proof of Theorem 4.2, see [F3, Theorem 2.39]. Remark 4.4. In Theorem 4.2 (2), it is sufficient to assume that H is π-nef and π-log big. See [F3, Theorem 2.47]. We omit the technical details on nef and log big divisors in order to keep this paper readable. 5. Non-lc ideal sheaves Let us recall the definition of non-lc ideal sheaves (cf. [F2, Section 2] and [F4, Section 7]). Definition 5.1 (Non-lc ideal sheaf). Let X be a normal variety and B an R- divisor on X such that K +B is R-Cartier. Let f : Y → X be a resolution with X K +B = f∗(K +B) such that Supp B is simple normal crossing. Then we put Y Y X Y J (X,B) = f O (p−(B<1)q−xB>1y) NLC ∗ Y Y Y = f O (−xB y+B=1) ∗ Y Y Y 8 OSAMU FUJINO,KARLSCHWEDE ANDSHUNSUKETAKAGI and call it the (minimal) non-lc ideal sheaf associated to (X,B). If B is effective, then J (X,B) ⊂ O . NLC X The ideal sheaf J (X,B) is independent of the choice of resolution, and thus NLC well-defined, by the following easy lemma. Lemma 5.2. Let g : Z → Y be a proper birational morphism between smooth varieties and B an R-divisor on Y such that Supp B is simple normal crossing. Y Y Assume that K +B = g∗(K +B ) and that Supp B is simple normal crossing. Z Z Y Y Z Then we have g O (p−(B<1)q−xB>1y) ≃ O (p−(B<1)q−xB>1y). ∗ Z Z Z Y Y Y Proof. By K +B = g∗(K +B ), we obtain Z Z Y Y K =g∗(K +B=1 +{B }) Z Y Y Y +g∗(xB<1y+xB>1y)−(xB<1y+xB>1y)−B=1 −{B }. Y Y Z Z Z Z If a(ν,Y,B=1 +{B }) = −1 for a prime divisor ν over Y, then we can check that Y Y a(ν,Y,B ) = −1 by using [KM, Lemma 2.45]. Since g∗(xB<1y+xB>1y)−(xB<1y+ Y Y Y Z xB>1y) is Cartier, we can easily see that Z g∗(xB<1y+xB>1y) = xB<1y+xB>1y+E, Y Y Z Z where E is an effective f-exceptional Cartier divisor. Thus, we obtain g O (p−(B<1)q−xB>1y) ≃ O (p−(B<1)q−xB>1y). ∗ Z Z Z Y Y Y This completes the proof. (cid:3) The next lemma is obvious by definition: Definition 5.1. Lemma 5.3. Let X be a normal variety and B an effective R-divisor on X such that K +B is R-Cartier. Then (X,B) is lc if and only if J (X,B) = O . X NLC X In the following sections, we consider variants of non-lc ideal sheaves. 6. Observations towards non-lc ideal sheaves First, we informally define J′ as a limit of multiplier ideal sheaves. We will call J′(X,B) the maximal non-lc ideal sheaf of the pair (X,B). For the details, see Section 7. 6.1. Let D be an effective R-divisor on a smooth variety X. Let f : Y → X be a resolution such that Exc(f) ∪ Supp f−1D is simple normal crossing. Then the ∗ multiplier ideal sheaf J(X,D) ⊂ O associated to D was defined to be X J(X,D) = f O (K −xf∗Dy), ∗ Y Y/X where K = K −f∗K . In this situation, we put Y/X Y X J′(X,D) = J(X,(1−ε)D) for 0 < ε ≪ 1. We note that the right hand side is independent of ε for 0 < ε ≪ 1. Therefore, we can write J′(X,D) = J(X,(1−ε)D) 0<ε \ SUPPLEMENTS TO NON-LC IDEAL SHEAVES 9 since J(X,(1−ε)D) ⊂ J(X,(1−ε′)D) for 0 < ε < ε′. We write K +∆ = f∗(K +D). Then Y Y X J(X,D) = f O (−x∆ y), ∗ Y Y and ∞ J(X,(1−ε)D) = f O (−x∆ y+ k∆ ) ∗ Y Y Y k=−∞ X for 0 < ε ≪ 1. Since k∆ is f-exceptional for k < 0, we can write Y ∞ J′(X,D) = f O (−x∆ y+ k∆ ), ∗ Y Y Y k=1 X This expression is very useful for generalizations. By definition, we can easily check that J(X,(1+ε)D) = J(X,D) for 0 < ε ≪ 1 and that J(X,(1 − ε)D) = J(X,D) for 0 < ε ≪ 1 if and only if t = 1 is not a jumping number of J(X,tD). In this paper, we are mainly interested in the case when D is a reduced divisor. In this case, t = 1 is a jumping number of J(X,tD) and then J′(X,D) ) J(X,D). Next, we observe various properties which should be satisfied by non-lc ideal sheaves. 6.2. Let X be a smooth projective variety and B an effective integral Cartier divisor on X such that Supp B is simple normal crossing. We can write B = ∞ kB , k=1 k where B := kB = Supp B=k. We would like to define an ideal sheaf I(X,B) ⊂ O k X P such that Supp O /I(X,B) = Nlc(X,B). Let us put X ∞ I(X,B) = O (− m B ) X k k k=2 X for some m ≥ 1 for every k ≥ 2. Then I(X,B) defines the non-lc locus of the pair k (X,B). Let L be a Cartier divisor on X such that A := L− (K + B) is ample. X For various geometric applications, we think that it is natural to require Hi(X,O (L)⊗I(X,B)) = 0 X for all i > 0. Since ∞ O (L)⊗I(X,B) = O (K +B +A− m B ) X X X k k k=2 X ∞ = O (K +B + (k −m )B +A), X X 1 k k k=2 X In view of the Norimatsu vanishing theorem (cf. [L, Lemma 4.3.5]), if we hope for vanishing, we should make m equal k or k−1 for every k ≥ 2. If m = k for every k k k ≥ 2, then I(X,B) = J (X,B). NLC 10 OSAMU FUJINO,KARLSCHWEDE ANDSHUNSUKETAKAGI If m = k −1 for every k ≥ 2, then k I(X,B) = J′(X,B). Let f : Y → X be a blow-up along a stratum of Supp B, where a stratum of Supp B means an lc center of (X,Supp B). We put K + B = f∗(K + B). Then it is Y Y X natural to require ∞ I(Y,B ) = O (− n kB ) Y Y k Y k=2 X such that n = k or k −1 for every k ≥ 2 and k f I(Y,B ) = I(X,B). ∗ Y We think that the most natural choices for non-lc ideal sheaves are ∞ I(X,B) = J (X,B) = O (− kB ) NLC X k k=2 X or ∞ I(X,B) = J′(X,B) = O (− (k −1)B ). X k k=2 X The ideal sheaf J (X,B) should be called minimal non-lc ideal sheaf of (X,B) NLC and J′(X,B) should be called maximal non-lc ideal sheaf of (X,B). The smaller B + ∞ (k−m )B is, themore easily we canapply our torsion-free 1 k=2 k k theorem (cf. Theorem 4.2 (1)) to I(X,B). It is one of the main reasons why the P first author adopted J (X,B) to define Nlc(X,B). NLC Finally, we put 1−l ∞ I(X,B) = O (− kB − (k −1)B ) =: J′(X,B) X k k l k=2 k=2−l X X for l = 0,−1,··· ,−∞. Then J (X,B) = J′ (X,B) ⊂ J′(X,B) ⊂ J′(X,B) = J′(X,B) NLC −∞ l 0 and J′(X,B) satisfies all the above desired properties for every l. We will discuss l J′(X,B)foreverynegativeintegerl inSection8. WedonotknowwhetherJ′(X,B) l l with l 6= 0,−∞ is useful or not for geometric applications. 7. Maximal non-lc ideal sheaves Let us define maximal non-lc ideal sheaves. Definition 7.1. Let X be a normal variety and ∆ an R-divisor on X such that K +∆ is R-Cartier. Let f : Y → X be a resolution with K +∆ = f∗(K +∆) X Y Y X such that Supp ∆ is simple normal crossing. Then we put Y J′(X,∆) = f O (pK −f∗(K +∆)+εFq) ∗ Y Y X for 0 < ε ≪ 1, where F = Supp ∆≥1. Y