On the abundance theorem for numerically trivial canonical divisors in positive characteristic
aa r X i v : . [ m a t h . AG ] F e b ON THE ABUNDANCE THEOREM FORNUMERICALLY TRIVIAL CANONICAL DIVISORSIN POSITIVE CHARACTERISTIC
SHO EJIRI
ABSTRACT.
In this paper, we prove the abundance theorem for numericallytrivial canonical divisors on strongly F -regular varieties, assuming that the geo-metric generic fibers of the Albanese morphisms are strongly F -regular. Introduction
The abundance conjecture predicts that if a variety is minimal then its canonicaldivisor is semi-ample. This conjecture is one of the most important problems in theminimal model theory. In characteristic zero, the conjecture has been verified whenthe canonical divisor is numerically trivial by Kawamata [9, Theorem 8.2]. Thishas been generalized to klt pairs due to Nakayama [18, V.4.8 Theorem], and to lcpairs by Campana–Koziarz–P˘aun [1, Theorem 0.1], Gongyo [5, Theorem 1.2] andKawamata [10, Theorem 1].In this paper we prove the theorem below, which can be viewed as a positivecharacteristic analog of Nakayama’s theorem mentioned above.
Theorem 1.1.
Let X be a normal projective variety over an algebraically closedfield of characteristic p > . Let ∆ be an effective Q -Weil divisor on X such that ( X, ∆) is strongly F -regular. Assume that m ( K X + ∆) is Cartier for an integer m > not divisible by p , and that K X + ∆ is numerically trivial. Let α : X → A bethe Albanese morphism of X and let X η denote the geometric generic fiber of α overits image. If ( X η , ∆ | X η ) is strongly F -regular, then K X + ∆ is Q -linearly trivial. Here, strong F -regularity is a singularity defined only in positive characteristic,which is closely related to klt singularities in characteristic zero.We give a brief explanation of the proof of Theorem 1.1. For simplicity, we supposethat ∆ = 0 and K X is an algebraically trivial Cartier divisor. Then K X ∼ α ∗ L foran algebraically trivial Cartier divisor L on A . Our goal is to show that L ∼ Q
0. Bythe assumption on the geometric generic fiber of α , we can apply [3, Theorem 1.1]and [21, Theorems 4.1 and 9.1], so we see that α is a flat surjective morphism whoseevery fiber is integral and strongly F -regular. Let B be a sufficiently ample Cartierdivisor on X . Let r be the rank of α ∗ O X ( B ). We consider the r -th fiber product Y := X × A · · · × A X of X over A . Let f : Y → A be the natural projection.Applying the argument due to Viehweg [23, Proof of Theorem 6.22], we get an Date : February 9, 2021.2010
Mathematics Subject Classification.
Primary 14E30, Secondary 14J40.
Key words and phrases.
Positive characteristic, Abundance theorem. effective f -ample Cartier divisor C on Y such that det( f ∗ O Y ( C )) ∼ = O A . Using [2,Theorem 6.10 (2)], we see that f ∗ O Y ( C ) is a numerically flat vector bundle. Thenby Theorem 2.4, we find unipotent vector bundles U , . . . , U l and algebraically trivialline bundles N , . . . , N l such that f ∗ O Y ( C ) ∼ = l M i =1 U i ⊗ N i . Applying an argument similar to [4, Proof of Theorem 3.2], we get a ν ∈ Z > suchthat for some µ ∈ Z > and every M ∈ M ( µ ) , where M ( µ ) := ( l O i =1 N n i i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ n i ∈ Z with l X i =1 n i = µ ) , we have M ⊗ O A ( νL ) ∈ M ( µ ) . This means that L ∼ Q
0, since M ( µ ) is a finite set. Acknowledgments.
The author is greatly indebted to Professor Hiromu Tanaka forpointing out some typos and for giving him a simple proof of Lemma 3.1. He wishesto express his thanks to Professors Osamu Fujino, Yoshinori Gongyo, ShunsukeTakagi and Lei Zhang for valuable comments and helpful advice. He would like tothank Professor Kenta Sato for answering his question. He was supported by JSPSKAKENHI Grant Number 18J00171.2.
Preliminaries
Notation and conventions.
We work over an algebraically closed field k ofcharacteristic p >
0. A variety is an integral separated k -scheme of finite type.Let ϕ : S → T be a morphism of schemes and let U be a T -scheme. Let S U and ϕ U : S U → U denote the fiber product S × T U of S and U over T and itssecond projection, respectively. For an O S -module G , its pullback to S U is denotedby G U . We use the same notation for a Q -Cartier divisor if its pullback is well-defined. Let X be an F p -scheme. The Frobenius morphism of X is denoted by F X : X → X . We denote the source of F eX by X e . Let f : X → Y be a morphism of F p -schemes. When we regard X and Y as the sources of F eX and F eY , respectively, f is denoted by f ( e ) : X e → Y e . We define e -th relative Frobenius morphism of f tobe F ( e ) X/Y := ( F eX , f ( e ) ) : X e → X × Y Y e = X Y e .2.2. Numerically flat vector bundles on abelian varieties.
We recall proper-ties of numerically flat vector bundles on abelian varieties.
Definition 2.1.
Let E be a vector bundle on a projective variety V . We say that E is numerically flat if E and its dual E ∨ are nef. Definition 2.2.
Let E be a vector bundle on an abelian variety A . We say that E is homogeneous if t ∗ a E ∼ = E for every closed point a ∈ A , where t a is the translationmap of A by a . Definition 2.3.
Let E be a vector bundle on an abelian variety A . We say that E is unipotent if E is an iterated extension of O A . he abundance theorem for numerically trivial canonical divisors in char p 3 Theorem 2.4.
Let A be an abelian variety. Let E be a vector bundle on A . Thenthe following are equivalent: (1) E is numerically flat; (2) E is an iterated extension of algebraically trivial line bundles; (3) E is homogeneous vector bundle; (4) E ∼ = L i U i ⊗ L i , where each L i is an algebraically trivial line bundle and each U i is a unipotent vector bundle.Proof. (1) ⇒ (2): This follows from [14, § ⇒ (4): This is easily proved byinduction on the rank. (4) ⇒ (1): This is obvious. (3) ⇒ (4): This has been provedby [16, Theorem 2.3]. (4) ⇒ (3): This has been shown in [17, Theorem 4.17]. (cid:3) Albanese morphisms of varieties with numerically trivialcanonical divisor
In this section, we study the Albanese morphisms of varieties with numericallytrivial canonical divisor.
Lemma 3.1.
Let V be a projective variety over k . Let L be a numerically trivialline bundle on V . Let R be a finitely generated F p -algebra over which the model V R of V and L R of L can be defined. Then there exists a dense open subset S of Spec R such that L µ is numerically trivial for every closed point µ ∈ S .Proof. Let H be an ample line bundle. By [12, Proposition 3], we have that L isnumerically trivial if and only if ( L·H dim V − ) = ( L ·H dim V − ) = 0 . Take S ⊆ Spec R so that the model H S of H over S is defined and H S is relatively ample over S . Sincethe intersection number is independent of the choice of fiber, the lemma follows. (cid:3) Definition 3.2 ([22, Definition 3.1]) . Let X be an affine normal variety and let∆ be an effective Q -Weil divisor on X . We say that the pair ( X, ∆) is strongly F -regular if for every effective divisor D on X , there exists an e ∈ Z > such thatthe composite O X F eX♯ −−→ F eX ∗ O X ι −→ F eX ∗ O X ( ⌈ ( p e − D ⌉ )splits. Here, ι is the natural inclusion.Let X be a normal variety and let ∆ be an effective Q -Weil divisor on X . We saythat ( X, ∆) is strongly F -regular if there exists an affine open cover { U i } of X suchthat ( U i , ∆ | U i ) is strongly F -regular for each i . When ( X,
0) is strongly F -regular,we simply say that X is strongly F -regular .If X is strongly F -regular, then X is Cohen–Macaulay ([8, (6.27) Proposition]). Proposition 3.3.
Let X be a Cohen–Macaulay normal projective variety and let ∆ be an effective Q -Weil divisor on X . Assume that m ( K X + ∆) is a numericallytrivial Cartier divisor for an integer m > not divisible by p . Let α : X → A be theAlbanese morphism of X , and let X η denote the geometric generic fiber of α overits image. If ( X η , ∆ | X η ) is strongly F -regular, then (1) α is a flat surjective morphism, (2) Supp ∆ does not contain any component of any closed fiber of f , and (3) for every closed fiber X y , the pair ( X y , ∆ | X y ) is strongly F -regular. SHO EJIRI
Proof.
By [3, Theorem 1.1], the morphism α is surjective and X η is integral. By[21, Theorem 4.1], α is equi-dimensional, so α is flat, since X is Cohen–Macaulay[7, Excercise III.10.9]. Thus (1) holds. Furthermore, (2) follows from [21, Proposi-tion 4.2 (2)]. We prove (3). Fix a closed point y ∈ A . We show that X y is integral.Let z ∈ A be a general closed point. Applying [6, Th´eor`eme 12.2.4], we may assumethat X z is integral. Let R be a finitely generated k -algebra over which the models α R : X R → A R , ∆ R , y R and z R of α : X → A , ∆, y and z can be defined, respectively.Then, by Lemma 3.1, there is a dense open subset S ⊆ Spec R such that K X µ +∆ µ isa numerically trivial Q -Cartier divisor for every µ ∈ S . Since the residue field κ ( µ )of µ is finite, we have κ ( µ ) ⊂ F p . Therefore, thanks to [21, Theorem 9.1], we seethat (cid:0) ( X µ ) y µ , (∆ µ ) y µ (cid:1) ∼ = (cid:0) ( X µ ) z µ , (∆ µ ) z µ (cid:1) . Since (cid:0) ( X µ ) z µ , (∆ µ ) z µ (cid:1) is the reductionof ( X z , ∆ | X z ) over µ , we may assume that ( X µ ) z µ is geometrically integral, using [6,Th´eor`eme 12.2.4]. Thus ( X µ ) y µ is geometrically integral, and so X y is integral by[6, Th´eor`eme 12.2.4] again. Applying [20, Coorollary 4.21] and an argument similarto the above, we can prove that ( X y , ∆ | X y ) is strongly F -regular. (cid:3) Proof of the main theorem
In this section, we prove the main theorem in this paper. We first show thefollowing three lemmas that are used in the proof of the main theorem.
Lemma 4.1.
Let V and W be normal varieties, and let Γ and ∆ be effective Q -Weildivisors on V and W , respectively. Assume that ( V, Γ) and ( W, ∆) are strongly F -regular. Then the pair ( V × k W, pr ∗ Γ + pr ∗ ∆) is also strongly F -regular, where pr i is the i -th projection. Note that since pr i is flat, we can define the pullback pr ∗ i D of any Q -divisor D . Proof.
We assume that V and W are affine. Let D (resp. E ) be an effective Cartierdivisor on V (resp. W ) such that ( V D , Γ D ) (resp. ( W E , ∆ E )) is log smooth, where V D := V \ Supp D and Γ D := Γ | V D (resp. W E := W \ Supp E and ∆ E := ∆ | W E ).Here, by log smooth we mean that V D is smooth and Γ D has simple normal crossingsupport. Set C := pr ∗ D + pr ∗ E and Θ := pr ∗ Γ + pr ∗ ∆. Then (cid:0) V × k W \ Supp C, Θ | V × k W \ Supp C (cid:1) = ( V D × k W E , (pr | V D ) ∗ Γ D + (pr | W E ) ∗ ∆ E )is log smooth. Since O V → F eV ∗ O V ( ⌈ ( p e − D ⌉ ) and O W → F eW ∗ O W ( ⌈ ( p e − E ⌉ )split for some e >
0, the morphism O V × W → F eV × W ∗ O V × W ( ⌈ ( p e − C ⌉ )splits. Hence, the assertion follows from [22, Theorem 3.9]. (cid:3) Lemma 4.2.
Let f : V → W be a surjective morphism between projective vari-eties. Let A be an f -ample Cartier divisor on V such that the natural morphism f ∗ f ∗ O V ( A ) → O V ( A ) is surjective. Let F be a coherent sheaf on V . Then thereexists an m ∈ Z > such that the multiplication morphism f ∗ ( F ⊗ O V ( mA + N )) ⊗ f ∗ O V ( nA ) → f ∗ ( F ⊗ O V (( m + n ) A + N )) he abundance theorem for numerically trivial canonical divisors in char p 5 is surjective for each m, n ∈ Z > with m ≥ m and every f -nef Cartier divisor N on V .Proof. By the relative version of Castelnuovo–Mumford regularity [15, Example 1.8.24],it is enough to show that
F ⊗ O V ( mA + N ) is 0-regular with respect to A and f .This follows from the relative Fujita vanishing [11, Theorem 1.5]. (cid:3) Lemma 4.3.
Let E be a vector bundle on a projective variety V . Let H be a Cartierdivisor on V . If O V ( H ) ⊗ F eV ∗ E is nef for infinitely many e ≥ , then E is nef.Proof. Set P := P ( E ) := Proj( L m ≥ S m ( E )). Let π : P → V be the naturalprojection. Then we have the following commutative diagram: P eπ ( e ) , , F ( e ) P/V (cid:15) (cid:15) F eP ❇❇❇❇❇❇❇❇❇ P ( F eV ∗ E ) P V e w ( e ) / / π V e (cid:15) (cid:15) P π (cid:15) (cid:15) V e F eV / / V Let T be a Cartier divisor on P such that O P ( T ) ∼ = O P (1). By the assumption, O P V e ( π ∗ V e H ) ⊗ O P V e (1) is nef. Taking the pullback by F ( e ) P/V , we see that π ∗ H + p e T = p e ( p − e π ∗ H + T )is nef, so p − e π ∗ H + T is nef. Since this holds for infinitely many e by the assumption,we conclude that T is nef. (cid:3) Proof of Theorem 1.1.
By Proposition 3.3, α : X → A is a flat surjective morphismwith strongly F -regular closed fibers. Let s be the Cartier index of K X + ∆. Notethat p ∤ s . By [13, COROLLARY 6.17], there is a numerically trivial Q -Cartierdivisor L on A such that K X + ∆ ∼ Q α ∗ L . Let t be the smallest positive integersuch that t ( K X + ∆) and tL are Cartier and t ( K X + ∆) ∼ Z tα ∗ L . Then s | t . We put I := { i ∈ Z > | ≤ i < t and s | i } . Step 1.
Let B be an ample Cartier divisor on X . Replacing B by lB for l ≫
0, wemay assume that the following conditions hold: • For each m ∈ Z > and each i ∈ I , the sheaf α ∗ O X ( − i ( K X + ∆) + mB ) is locallyfree (note that f is flat). • For each m, n ∈ Z > and every α -nef Cartier divisor N on X , the multiplicationmorphism α ∗ O X ( mB + N ) ⊗ α ∗ O X ( nB ) → α ∗ O X (( m + n ) B + N )is surjective (by Lemma 4.2). • For each e ∈ Z > and every f -nef Cartier divisor N , the morphism α ( e ) ∗ O X ((1 − p e )( K X + ∆) + p e ( B + N )) φ ( e )( X, ∆) /A ( B + N ) −−−−−−−−−→ α A e ∗ O X Ae ( B A e + N A e ) ∼ = F eA ∗ α ∗ O X ( B + N ) SHO EJIRI is surjective. For the construction of φ ( e )( X, ∆) /A ( B + N ), see [19, §
3] or [2, § Step 2.
Let r > α ∗ O X ( B ). We consider the r -th fiber product Y := X × A · · · × A X of X over A . Since α is flat, so is each projection pr i , and hence we can take thepullback pr ∗ i D of every Q -Weil divisor D on X . Set Γ := P ri =1 pr ∗ i ∆. Then we get t ( K Y + Γ) = t ( K Y/A + Γ) ∼ r X i =1 pr ∗ i t ( K X/A + ∆) ∼ r X i =1 pr ∗ i tα ∗ L = rtf ∗ L. (1)Here, f : Y → A is the natural projection. Since ( X a , ∆ | X a ) is strongly F -regularfor all closed fibers X a , the pair ( Y a , Γ | Y a ) is also strongly F -regular by Lemma 4.1.Put B ′ := P ri =1 pr ∗ i B. Since α : X → A is flat, we have f ∗ O Y ( B ′ ) = f ∗ r O i =1 pr ∗ i O X ( B ) ! ∼ = r O α ∗ O X ( B ) . Since r is the rank of α ∗ O X ( B ), we get the morphism ϕ : det ( α ∗ O X ( B )) → r O i =1 α ∗ O X ( B )that is locally defined as x ∧ · · · ∧ x r X σ ∈ S r sign( σ ) · x σ (1) ⊗ · · · ⊗ x σ ( r ) , where S r is the symmetric group of degree r . Put H := det( α ∗ O X ( B )). Let H bea Cartier divisor on A with O A ( H ) ∼ = H . Then ϕ induces the morphism O A → H − ⊗ r O j =1 α ∗ O X ( B ) ! ∼ = H − ⊗ f ∗ O Y ( B ′ ) ∼ = f ∗ O Y ( B ′ − f ∗ H ) . Therefore, there is an effective f -ample Cartier divisor C ∼ B ′ − f ∗ H on Y . Since ϕ splits locally, we see that Supp C does not contain any fiber of f . Step 3.
We prove that det( f ∗ O Y ( C )) ∼ = O A . This follows from the following calcu-lation: det( f ∗ O Y ( C )) ∼ = det( f ∗ O Y ( B ′ − f ∗ H )) ∼ = det (cid:0) H − ⊗ f ∗ O Y ( B ′ ) (cid:1) ∼ = det H − ⊗ r O α ∗ O Y ( B ) !! ∼ = H − r r ⊗ det r O α ∗ O Y ( B ) ! = H − r r ⊗ H r r ∼ = O A . he abundance theorem for numerically trivial canonical divisors in char p 7 Step 4.
We show that f ∗ O Y ( − i ( K Y + Γ) + C ) is a nef vector bundle for each i ∈ I .Note that the sheaves are vector bundles because of the choice of B . Take l ∈ Z > so that ( Y a , (Γ + l − C ) | Y a ) is strongly F -regular for each l ≥ l and every closedfiber Y a of f . Note that we can find such an l thanks to [20, Corollary 4.21]. By[2, Theorem 6.10 (2)], for each m ≫
0, the sheaf f ∗ O Y ( lm ( K Y + Γ + l − C )) = f ∗ O Y ( lm ( K Y + Γ) + mC )is weakly positive over A . Let m be divided by t and fix such an m . Then f ∗ O Y ( lm ( K Y + Γ) + mC ) by (1) ∼ = f ∗ O Y ( lmrf ∗ L + mC ) ∼ = O A ( lmrL ) ⊗ f ∗ O Y ( mC ) . Since L is numerically trivial, we see that f ∗ O Y ( mC ) is weakly positive over A , soit is a nef vector bundle. Note that f ∗ O Y ( mC ) is a vector bundle by the choice of B . Take e ∈ Z > so that (1 − p e )( K Y + Γ) is Cartier (i.e., s | ( p e − a, b beintegers such that p e = am + b and 0 ≤ b < m . Fix i ∈ I . We consider the followingsequence of morphisms: f ∗ O Y ((1 − p e − ip e )( K Y + Γ) + bC ) ⊗ a O f ∗ O Y ( mC ) ! ∼ = H − p e ⊗ f ∗ O Y ((1 − p e − ip e )( K Y + Γ) + bB ′ ) ⊗ a O f ∗ O Y ( mB ′ ) ! ∼ = H − p e ⊗ r O α ∗ O X ((1 − p e − ip e )( K X + ∆) + bB ) ⊗ a O α ∗ O X ( mB ) !!! µ −→H − p e ⊗ r O ( α ∗ O X ((1 − p e − ip e )( K X + ∆) + p e B )) ! ∼ = H − p e ⊗ ( f ∗ O Y ((1 − p e − ip e )( K Y + Γ) + p e B ′ )) ∼ = f ∗ O Y ((1 − p e − ip e )( K Y + Γ) + p e C ) . Here, by the choice of B , the multiplication morphism µ is surjective. Note that − ( K X + ∆) is nef. By the choice of B again, we get the surjection f ∗ O Y ((1 − p e − ip e )( K Y + Γ) + p e C ) ։ F eA ∗ f ∗ O A ( − i ( K Y + Γ) + C ) . (2)Combining this with the above sequence of morphisms, we obtain the surjection f ∗ O Y ((1 − p e − ip e )( K Y + Γ) + bC ) ⊗ a O f ∗ O Y ( mC ) ! (3) ։ F eA ∗ f ∗ O A ( − i ( K Y + Γ) + C ) . (4)For each i ∈ I , let c i , d i be integers such that ip e + p e − c i t + d i and 0 ≤ d i < t .We may assume that e ≫ c i > i ∈ I . Then f ∗ O Y ((1 − p e − ip e )( K Y + Γ) + bC ) ∼ = O A ( − c i rtL ) ⊗ f ∗ O Y ( − d i ( K Y + Γ) + bC ) . SHO EJIRI
Note that s | d , since s | ( p e −
1) and s | t . Put G := M ≤ b For every coherent sheaf G on A , we set L ( G ) := (cid:8) N ∈ Pic ( A ) (cid:12)(cid:12) there is a non-zero morphism G → N (cid:9) . By Step 4, the sheaf F := L i ∈ I f ∗ O Y ( − i ( K Y + Γ) + C ) is a nef vector bundle, so wesee that L ( F ) is a finite set, by considering the Harder–Narasimhan filtration of F .Pick N ∈ L ( F ). Then there is a non-zero morphism f ∗ O Y ( − i ( K Y + Γ) + C ) → N for some i ∈ I . We consider the following morphisms that are generically surjective: O A ( − c i rtL ) ⊗ F ⊗ p e − O f ∗ O Y ( C ) ! γ ։ F eY ∗ f ∗ O Y ( − i ( K Y + Γ) + C ) → N p e . (5)Here, γ is constructed by the same construction as that in Step 4. By Steps 3 and 4,we see that f ∗ O Y ( C ) is numerically flat, so Theorem 2.4 tells us that there arealgebraically trivial line bundles N , . . . , N l and unipotent vector bundles U , . . . , U l such that f ∗ O Y ( C ) ∼ = l M j =1 U j ⊗ N j . (6)Then one can easily check that(i) L ( f ∗ O Y ( C )) = {N , . . . , N l } , and(ii) f ∗ O Y ( C ) has a filtration whose each quotient is isomorphic to N j for some j . he abundance theorem for numerically trivial canonical divisors in char p 9 Hence, from (ii) and morphism (5), we obtain the non-zero morphism O A ( − c i rtL ) ⊗ F ⊗ P → N p e , (7)where P is an element of the set M ( p e − that is defined as follows: for each µ ∈ Z > ,we define M ( µ ) by M ( µ ) := ( λ O j =1 M n j j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M j ∈ L ( F ) and 0 ≤ n j ∈ Z with λ X j =1 n j = µ ) . From morphism (7), we obtain that Q := O A ( c i rtL ) ⊗ N p e ⊗ P − ∈ L ( F ) , which means that N p e ⊗ O A ( c i rtL ) ∼ = P ⊗ Q ∈ M ( p e ) . Thus, we see that for each N ∈ L ( F ), there is i ∈ I such that N p e ⊗ O A ( c i rtL ) ∈ M ( p e ) . Set λ := | L ( F ) | and µ := ( p e − λ + 1. Take M ∈ M ( µ ) . Then, we obtain from thepigeonhole principle that there is N ∈ L ( F ) with M ⊗ N − p e ∈ M ( µ − p e ) , and so M ⊗ O A ( c i rtL ) ∼ = M ⊗ N − p e | {z } ∈ M ( µ − pe ) ⊗ N p e ⊗ O A ( c i rtL ) | {z } ∈ M ( pe ) ∈ M ( µ ) for some c i . By the same argument, we see that there is i ∈ I such that M ⊗ O A (( c i + c i ) rtL ) ∈ M ( µ ) . Recall that c i > i ∈ I by the choice of e . Repeating this argument, weget that the set C := (cid:8) c ∈ Z > (cid:12)(cid:12) M ⊗ O A ( crtL ) ∈ M ( µ ) (cid:9) is an infinite set. Since M ( µ ) is a finite set, there is c, c ′ ∈ C with c < c ′ such that M ⊗ O A ( crtL ) ∼ = M ⊗ O A ( c ′ rtL ), which means that O A (( c ′ − c ) rtL ) ∼ = O A . Thus,we conclude that L ∼ Q (cid:3) References [1] F. Campana, V. Koziarz, and M. P˘aun. Numerical character of the effectivity of adjoint linebundles. Annales de l’institut Fourier , 62(1):107–119, 2012.[2] S. Ejiri. 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