PULL-BACK OF QUASI-LOG STRUCTURES OSAMU FUJINO 4 1 0 2 Abstract. Oneofthemainpurposesofthispaperistomakethe n a theory of quasi-log schemes more flexible and more useful. More J precisely, we prove that the pull-back of a quasi-log scheme by a 8 smooth quasi-projective morphism has a natural quasi-log structure after clarifying the definition of quasi-log schemes. We treat ] G some applications to singular Fano varieties. This paper also contains a proof of the simple connectedness of log canonical Fano A varieties. . h t a Contents m [ 1. Introduction 1 2. Preliminaries 3 1 v 3. Pull-back of quasi-log structures 6 4 4. On quasi-log structures 9 5 8 5. Proof of the main theorem 15 1 6. Applications to quasi-log canonical Fano varieties 16 . 1 7. Simple connectedness of log canonical Fano varieties 19 0 8. Appendix: Ambro’s original definition 20 4 1 References 24 : v i X r 1. Introduction a In [A], the notion of quasi-log structures was introduced in order to prove the cone and contraction theorem for (X,∆) where X is a normal variety and ∆ is an effective R-divisor such that K + ∆ is X R-Cartier. Although the theory of quasi-log schemes is very powerful and useful, it may look much harder than the usual X-method for kawamata log terminal pairs. Moreover, the paper [F4] recovers the main theorem of [A] without using the notion of quasi-log structures. So the theory of quasi-log schemes is not yet popular. We note that the Date: 2014/1/8,version 1.02. 2010 Mathematics Subject Classification. Primary 14E30;Secondary 14J45. Keywordsandphrases. quasi-logstructures,Fanovarieties,fundamentalgroups. 1 2 OSAMU FUJINO framework of [F4] is more similar to the theory of algebraic multiplier ideal sheaves than to the traditional X-method. Recently, the author proved that every quasi-projective semi log canonical pair has anatural quasi-log structure in [F7]. The theory of quasi-log schemes seems to be indispensable for the study of semi log canonical pairs. Now the importance of quasi-log structures is increasing. One of the main purposes of this paper is to clarify the definition of quasi-log structures andmakethetheoryofquasi-logschemes moreflexibleandmoreuseful. The following theorem is the main theorem of this paper, which is natural but missing in the literature. For the precise statement, see Theorem 3.11 below. Theorem 1.1 (Pull-backofquasi-logstructures). Let [X,ω]be a quasi- log scheme and let h : X′ → X be a smooth quasi-projective morphism. Then [X′,ω′], where ω′ = h∗ω ⊗ωX′/X with ωX′/X = detΩ1X′/X, has a natural quasi-log structure induced by h. Theorem 1.1 does not directly follow from the original definition of quasi-log schemes. We have to construct a quasi-log resolution of [X′,ω′] suitably. We make an important remark. We do not know whether Theorem 1.1 holds true or not without assuming that h is quasi-projective. As a useful special case of Theorem 1.1, we have: Theorem 1.2 (Finite étale covers). Let [X,ω] be a quasi-log scheme and let h : X′ → X be a finite étale morphism. Then [X′,ω′], where ω′ = h∗ω, has a natural quasi-log structure induced by h. As an easy application of Theorem 1.2 to singular Fano varieties, we obtain: Corollary 1.3. Let [X,ω] be a projective quasi-log canonical pair such that −ω is ample, that is, [X,ω] is a quasi-log canonical Fano variety. Then the algebraic fundamental group of X is trivial, equivalently, X has no non-trivial finite étale covers. By Corollary 1.3, it is natural to conjecture: Conjecture 1.4. Let [X,ω] be a projective quasi-log canonical pair such that −ω is ample. Then X is simply connected. In general, there exists an irreducible projective variety whose algebraic fundamental group is trivial and whose topological fundamental group is non-trivial (see Example 6.4). As a special case of Conjecture 1.4, we have: Conjecture 1.5. Let (X,∆) be a projective semi log canonical pair such that −(K +∆) is ample, that is, (X,∆) is a semi log canonical X Fano variety. Then X is simply connected. PULL-BACK OF QUASI-LOG STRUCTURES 3 For the details of semi log canonical pairs, see [F7]. Note that every quasi-projective semi log canonical pair has a natural quasi-log structure with only quasi-log canonical singularities. It is the main theorem of [F7]. It is well known that Conjecture 1.5 holds when (X,∆) is kawamata log terminal (see [T]). Kento Fujita pointed out that Conjecture 1.5 holds true when (X,∆) is log canonical (see Theorem 7.1 below). We give Fujita’s proof in Section 7 for the reader’s convenience. We summarize the contents of this paper. Section 2 collects some basic definitions and results. In Section 3, we recall the definition of quasi-log schemes and state the main theorem of this paper precisely (see Theorem 3.11). Section 4 is the main part of this paper. Here we discuss the basic properties of quasi-log schemes. The author believes that Section 4 makes the theory of quasi-log schemes more flexible and more useful than Ambro’s original framework in [A]. Section 5 is de- voted to the proof of the main theorem (see Theorem 1.1 and Theorem 3.11). In Section 6, we treat some applications of the main theorem to singular Fano varieties. We prove that the algebraic fundamental group of a quasi-log canonical Fano variety is always trivial (see Corol- lary 1.3). In Section 7, we prove that a log canonical Fano variety is simply connected (see Theorem 7.1). The proof of Theorem 7.1 is inde- pendent of the theory of quasi-log schemes. Section 8 is an appendix, where we discuss Ambro’s original definition of quasi-log schemes. Acknowledgments. TheauthorwaspartiallysupportedbytheGran- in-Aid for Young Scientists (A) ♯24684002 from JSPS. He would like to thank Takeshi Abe, Kento Fujita, Yuichiro Hoshi, and Tetsushi Ito for answering his questions and giving him useful comments. We will work over C, the complex number field, throughout this paper. For the standard notation of the log minimal model program, see, for example, [F4]. For the basic properties and results of semi log canonical pairs, see [F7]. 2. Preliminaries In this section, we collect some basic results and definitions. Notation 2.1. A pair [X,ω] consists of a scheme X and an R-Cartier R-divisor (or R-line bundle) ω on X. In this paper, a scheme means a separated scheme of finite type over SpecC. Notation 2.2 (Divisors). Let B and B be two R-Cartier R-divisors 1 2 on a scheme X. Then B is linearly (resp. Q-linearly, or R-linearly) 1 4 OSAMU FUJINO equivalent to B , denoted by B ∼ B (resp. B ∼ B , or B ∼ B ) 2 1 2 1 Q 2 1 R 2 if k B1 = B2 +Xri(fi) i=1 such that f ∈ Γ(X,K∗ ) and r ∈ Z (resp. r ∈ Q, or r ∈ R) for every i X i i i i. Here, K is the sheaf of total quotient rings of O and K∗ is the X X X sheaf of invertible elements in the sheaf of rings K . We note that X (f ) is a principal Cartier divisor associated to f , that is, the image of i i f by Γ(X,K∗ ) → Γ(X,K∗ /O∗ ), where O∗ is the sheaf of invertible i X X X X elements in O . X Let D be a Q-divisor (resp. an R-divisor) on an equi-dimensional variety X, that is, D is a finite formal Q-linear (resp. R-linear) combi- nation D = XdiDi i of irreducible reduced subschemes D of codimension one. We define i the round-up ⌈D⌉ = ⌈d ⌉D (resp. round-down ⌊D⌋ = ⌊d ⌋D ), Pi i i Pi i i where 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 put D<1 = XdiDi, D≤1 = XdiDi, and D=1 = XDi. di<1 di≤1 di=1 We can define D≥1, D>1, and so on, analogously. We call D a boundary (resp. subboundary) R-divisor if 0 ≤ d ≤ 1 (resp. d ≤ 1) for every i. i i Notation 2.3 (Singularities of pairs). Let X be a normal variety and let ∆ be an R-divisor on X such that K + ∆ is R-Cartier. Let f : X Y → X be a resolution such that Exc(f)∪f−1∆, where Exc(f) is the ∗ exceptional locus of f and f−1∆ is the strict transform of ∆ on Y, has ∗ a simple normal crossing support. We can write KY = f∗(KX +∆)+XaiEi. i We say that (X,∆) is sub log canonical (sub lc, for short) if a ≥ −1 for i every i. We usually write a = a(E ,X,∆) and call it the discrepancy i i coefficient of E with respect to (X,∆). It is well known that there i exists the largest Zariski open set U of X such that (U,∆| ) is sub log U canonical. If there exist a resolution f : Y → X and a divisor E on Y such that a(E,X,∆) = −1 and f(E) ∩ U 6= ∅, then f(E) is called a log canonical center (an lc center, for short) with respect to (X,∆). If PULL-BACK OF QUASI-LOG STRUCTURES 5 (X,∆) is sub log canonical and ∆ is effective, then (X,∆) is called log canonical (lc, for short). We note that we can define a(E ,X,∆) in more general settings (see i [K2, Definition 2.4]). Let us recall the definition of simple normal crossing pairs. Definition 2.4 (Simple normal crossing pairs). We say that the pair (X,D) is simple normal crossing at a point a ∈ X if X has a Zariski open neighborhood U of a that can be embedded in a smooth variety Y, where Y has regular system of parameters (x ,··· ,x ,y ,··· ,y ) 1 p 1 r at a = 0 in which U is defined by a monomial equation x ···x = 0 1 p and r D = Xαi(yi = 0)|U, αi ∈ R. i=1 We say that (X,D) is a simple normal crossing pair if it is simple normal crossing at every point of X. We say that a simple normal crossing pair (X,D) is embedded if there exists a closed embedding ι : X → M, where M is a smooth variety of dimX +1. We call M the ambient spaceof(X,D). If(X,0) isa simple normal crossing pair, then X is called a simple normal crossing variety. If X is a simple normal crossing variety, then X has only Gorenstein singularities. Thus, it has aninvertible dualizing sheafω . Therefore, we candefinethecanonical X divisor K such that ω ≃ O (K ). It is a Cartier divisor on X and X X X X is well-defined up to linear equivalence. Let X be a simple normal crossing variety and let X = X be Si∈I i the irreducible decomposition of X. A stratum of X is an irreducible component of X ∩···∩X for some {i ,··· ,i } ⊂ I. i1 ik 1 k Let X be a simple normal crossing variety and let D be a Cartier divisor on X. If (X,D) is a simple normal crossing pair and D is reduced, then D is called a simple normal crossing divisor on X. Let (X,D) be a simple normal crossing pair. Let ν : Xν → X be the normalization. We define Θ by the formula KXν +Θ = ν∗(KX +D). Then a stratum of (X,D) is an irreducible component of X or the ν- image of a log canonical center of (Xν,Θ) (see Notation 2.3). When D = 0, this definition is compatible with the above definition of the strataofX. WhenD isaboundaryR-divisor, W isastratumof(X,D) if and only if W is an slc stratum of (X,D) (see [F7, Definition 2.5]). Note that (X,D) is semi log canonical if D is a boundary R-divisor. 6 OSAMU FUJINO Notation 2.5. π (X) denotes the topological fundamental group of 1 X. 3. Pull-back of quasi-log structures In this section, we give a precise statement of Theorem 1.1 (see Theorem 3.11). First, let us recall the definition of globally embedded simple normal crossing pairs in order to define quasi-log schemes. Definition 3.1 (Globallyembedded simplenormalcrossing pairs). Let Y be a simple normal crossing divisor on a smooth variety M and let D be an R-divisor on M such that Supp(D + Y) is a simple normal crossing divisor on M and that D and Y have no common irreducible components. We put B = D| and consider the pair (Y,B ). We Y Y Y call (Y,B ) a globally embedded simple normal crossing pair and M Y the ambient space of (Y,B ). Y It is obvious that a globally embedded simple normal crossing pair is an embedded simple normal crossing pair in Definition 2.4. Let us define quasi-log schemes (see also Definition 8.2 below). Definition 3.2 (Quasi-log schemes). A quasi-log scheme is a scheme X endowed with an R-Cartier R-divisor (or R-line bundle) ω on X, a proper closed subscheme X ⊂ X, and a finite collection {C} of −∞ reduced and irreducible subschemes of X such that there is a proper morphism f : (Y,B ) → X from a globally embedded simple normal Y crossing pair satisfying the following properties: (1) f∗ω ∼ K +B . R Y Y (2) The natural map O → f O (⌈−(B<1)⌉) induces an isomor- X ∗ Y Y phism I −≃→ f O (⌈−(B<1)⌉−⌊B>1⌋), X−∞ ∗ Y Y Y where I is the defining ideal sheaf of X . X−∞ −∞ (3) The collection of subvarieties {C} coincides with the image of (Y,B )-strata that are not included in X . Y −∞ We simply write [X,ω] to denote the above data X,ω,f : (Y,B ) → X (cid:0) Y (cid:1) if there is no risk of confusion. Note that a quasi-log scheme X is the union of {C} and X . We also note that ω is called the quasi-log −∞ canonical class of [X,ω], which is defined up to R-linear equivalence. A relative quasi-log scheme X/S is a quasi-log scheme X endowed with a proper morphism π : X → S. PULL-BACK OF QUASI-LOG STRUCTURES 7 Remark 3.3. Let Div(Y) be the group of Cartier divisors on Y and let Pic(Y) be the Picard group of Y. Let δ : Div(Y)⊗R → Pic(Y)⊗R Y be the homomorphism induced by A 7→ O (A) where A is a Cartier Y divisor on Y. When ω is an R-line bundle in Definition 3.2, f∗ω ∼ K +B R Y Y means f∗ω = δ (K +B ) Y Y Y in Pic(Y)⊗R. Even when ω is an R-line bundle, we use −ω to denote the inverse of ω in Pic(X)⊗R (see Corollary 1.3 and Conjecture 1.4) if there is no risk of confusion. If ω is an R-Cartier R-divisor on X in Theorem 1.1, h∗ω ⊗detΩ1 X′/X means δX′(h∗ω)⊗detΩ1X′/X in Pic(X′)⊗R where δX′ : Div(X′)⊗R → Pic(X′)⊗R. We give an important remark on Definition 3.2. Remark 3.4 (Schemes versus varieties). A quasi-log scheme in Defini- tion 3.2 is called a quasi-log variety in [A] (see also [F2]). However, X is not always reduced when X 6= ∅ (see Example 3.5 below). There- −∞ fore, we will use the word quasi-log schemes in this paper. Note that X is reduced when X = ∅ (see Remark 3.10 below). −∞ Example 3.5 ([A, Examples 4.3.4]). Let X be an effective Cartier divisor on a smooth variety M. Assume that Y, the reduced part of X, is non-empty. We put ω = (K +X)| . Let X be the union of M X −∞ the non-reduced components of X. We put K +B = (K +X)| . Y Y M Y Let f : Y → X be the closed embedding. Then X,ω,f : (Y,B ) → X (cid:0) Y (cid:1) is a quasi-log scheme. Note that X has non-reduced irreducible components if X 6= ∅. We also note that f is not surjective if X 6= ∅. −∞ −∞ Notation 3.6. In Definition 3.2, we sometimes simply say that [X,ω] is a quasi-log pair. The subvarieties C are called the qlc centers of [X,ω], X is called the non-qlc locus of [X,ω], and f : (Y,B ) → X −∞ Y is called a quasi-log resolution of [X,ω]. We sometimes use Nqlc(X,ω) to denote X . −∞ For various applications, the notion of qlc pairs is very useful. 8 OSAMU FUJINO Definition 3.7 (Qlc pairs). Let [X,ω] be a quasi-log pair. We say that [X,ω] has only quasi-log canonical singularities (qlc singularities, for short) if X = ∅. Assume that [X,ω] is a quasi-log pair with −∞ X = ∅. Then we simply say that [X,ω] is a qlc pair. −∞ We give some important remarks on the non-qlc locus X . −∞ Remark 3.8. Weput A = ⌈−(B<1)⌉andN = ⌊B>1⌋. Then we obtain Y Y the following big commutative diagram. 0 // f∗OY(A−N) // f∗OY(A) // f∗ON(A) OO OO OO α1 α2 α3 0 // f∗OY(−N) // f∗OY // f∗ON OO OO OO β1 β2 β3 0 // I // O // O // 0 X−∞ X X−∞ Note that α is a natural injection for every i. By an easy diagram i chasing, ≃ I −→ f O (A−N) X−∞ ∗ Y factors through f O (−N). Then we obtain β and β . Since α is ∗ Y 1 3 1 injective and α ◦β is an isomorphism, α and β are isomorphisms. 1 1 1 1 Therefore, we obtain that f(Y) ∩ X = f(N). Note that f is not −∞ always surjective when X 6= ∅. It sometimes happens that X −∞ −∞ contains some irreducible components of X. See, for example, Example 3.5. Remark 3.9 (Semi-normality). By restricting the isomorphism ≃ I −→ f O (A−N) X−∞ ∗ Y to the open subset U = X \X , we obtain −∞ ≃ OU −→ f∗Of−1(U)(A). This implies that ≃ OU −→ f∗Of−1(U) because A is effective. Therefore, f : f−1(U) → U is surjective and has connected fibers. Note that f−1(U) is a simple normal crossing variety. Thus, U is semi-normal. In particular, U = X \X is reduced. −∞ Remark 3.10. Ifthepair[X,ω]hasonlyqlcsingularities, equivalently, X = ∅, then X is reduced and semi-normal by Remark 3.9. Note −∞ that f(Y)∩X = ∅ if and only if B = B≤1, equivalently, B>1 = 0, −∞ Y Y Y by the descriptions in Remark 3.8. PULL-BACK OF QUASI-LOG STRUCTURES 9 Let us state the main theorem of this paper precisely. Theorem 3.11 (Main theorem). Let [X,ω] be a quasi-log pair as in Definition 3.2. Let X′ be a scheme and let h : X′ → X be a smooth quasi-projective morphism. Then [X′,ω′], where ω′ = h∗ω ⊗ ωX′/X with ωX′/X = detΩ1X′/X, has a natural quasi-log structure induced by h. More precisely, we have the following properties: (i) (Non-qlclocus). There is a proper closed subscheme X′ ⊂ X′. −∞ (ii) (Quasi-log resolution). There exists a proper morphism f′ : (Y′,BY′) → X′ from a globally embedded simple normal crossing pair (Y′,BY′) such that f′∗ω′ ∼R KY′ +BY′, and the natural map OX′ → f∗′OY′(⌈−(BY<′1)⌉) induces an isomorphism IX−′∞ −≃→ f∗′OY′(⌈−(BY<′1)⌉−⌊BY>′1⌋) where IX−′∞ is the defining ideal sheaf of X−′ ∞ and IX−′∞ = h∗IX−∞. (iii) (Qlc centers). There is a finite collection {C′} of reduced and irreducible subschemes of X′ such that {C′} = {f−1(C)} and that the collection of subvarieties {C′} coincides with the images of (Y′,BY′)-strata that are not included in X−′ ∞. Remark3.12. Forthedefinitionandbasicpropertiesofquasi-projective morphisms, see [G, Chapitre II §5.3. Morphismes quasi-projectifs]. We will prove Theorem 3.11 in Section 5 after we prepare various useful lemmas in Section 4. We recommend the reader to see [F3] for some basic applications of thetheoryofquasi-logschemes. The adjunctionandvanishing theorem (see, for example, [F3, Theorem 3.6]) is a key result for qlc pairs. 4. On quasi-log structures The following proposition makes the theory of quasi-log schemes more flexible. It is a key result in this paper. Proposition 4.1 ([F2, Proposition 3.50]). Let f : V → W be a proper birational morphism between smooth varieties and let B be an R- W divisor on W such that SuppB is a simple normal crossing divisor W 10 OSAMU FUJINO on W. Assume that K +B = f∗(K +B ) V V W W and that SuppB is a simple normal crossing divisor on V. Then we V have f O (⌈−(B<1)⌉−⌊B>1⌋) ≃ O (⌈−(B<1)⌉−⌊B>1⌋). ∗ V V V W W W Furthermore, let S be a simple normal crossing divisor on W such that S ⊂ SuppB=1. Let T be the union of the irreducible components of W B=1 that are mapped into S by f. Assume that Suppf−1B ∪Exc(f) V ∗ W is a simple normal crossing divisor on V. Then we have f O (⌈−(B<1)⌉−⌊B>1⌋) ≃ O (⌈−(B<1)⌉−⌊B>1⌋), ∗ T T T S S S where (K +B )| = K +B and (K +B )| = K +B . V V T T T W W S S S Proof. By K +B = f∗(K +B ), we obtain V V W W K =f∗(K +B=1 +{B }) V W W W +f∗(⌊B<1⌋+⌊B>1⌋)−(⌊B<1⌋+⌊B>1⌋)−B=1 −{B }. W W V V V V If a(ν,W,B=1 + {B }) = −1 for a prime divisor ν over W, then we W W can check that a(ν,W,B ) = −1 by using [KM, Lemma 2.45]. Since W f∗(⌊B<1⌋+⌊B>1⌋)−(⌊B<1⌋+⌊B>1⌋) W W V V is Cartier, we can easily see that f∗(⌊B<1⌋+⌊B>1⌋) = ⌊B<1⌋+⌊B>1⌋+E, W W V V where E is an effective f-exceptional divisor. Thus, we obtain f O (⌈−(B<1)⌉−⌊B>1⌋) ≃ O (⌈−(B<1)⌉−⌊B>1⌋). ∗ V V V W W W Next, we consider the short exact sequence: 0 → O (⌈−(B<1)⌉−⌊B>1⌋−T) V V V → O (⌈−(B<1)⌉−⌊B>1⌋) → O (⌈−(B<1)⌉−⌊B>1⌋) → 0. V V V T T T Since T = f∗S −F, where F is an effective f-exceptional divisor, we can easily see that f O (⌈−(B<1)⌉−⌊B>1⌋−T) ≃ O (⌈−(B<1)⌉−⌊B>1⌋−S). ∗ V V V W W W We note that (⌈−(B<1)⌉−⌊B>1⌋−T)−(K +{B }+B=1 −T) V V V V V = −f∗(K +B ). W W Therefore, every associated prime of R1f O (⌈−(B<1)⌉−⌊B>1⌋−T) is ∗ V V V thegenericpointofthef-imageofsomestratumof(V,{B }+B=1−T) V V (see, for example, [F4, Theorem 6.3 (i)]).

Pull-back of quasi-log structures

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