On the F-purity of isolated log canonical singularities
aa r X i v : . [ m a t h . AG ] J u l ON THE F -PURITY OF ISOLATED LOG CANONICALSINGULARITIES OSAMU FUJINO AND SHUNSUKE TAKAGI
Abstract.
A singularity in characteristic zero is said to be of dense F -pure type if its modulo p reduction is locally Frobenius split for infinitely many p . We provethat if x ∈ X is an isolated log canonical singularity with µ ( x ∈ X ) ≤ µ ), then it is of dense F -puretype. As a corollary, we prove the equivalence of log canonicity and being ofdense F -pure type in the case of three-dimensional isolated Q -Gorenstein normalsingularities. Introduction
A singularity in characteristic zero is said to be of dense F -pure type if its modulo p reduction is locally Frobenius split for infinitely many p . The notion of strongly F -regular type is a variant of dense F -pure type and defined similarly using theFrobenius morphism after reduction to characteristic p > Q -Gorenstein singularity in charscteristic zero is log ter-minal if and only if it is of strongly F -regular type. In this paper, as an analogouscharacterization for isolated log canonical singularities, we consider the followingconjecture. Conjecture A n . Let x ∈ X be an n -dimensional normal Q -Gorenstein singularitydefined over an algebraically closed field k of characteristic zero such that x is anisolated non-log-terminal point of X . Then x ∈ X is log canonical if and only if itis of dense F -pure. Hara–Watanabe [12] proved that normal Q -Gorenstein singularities of dense F -pure type are log canonical. Unfortunately, the converse implication is widely openand only a few special cases are known. For example, the two-dimensional casefollows from the results of Mehta–Srinivas [20] and Hara [10], and the case of hy-persurface singularities whose defining polynomials are very general was proved byHern´andez [13]. This problem is now considered as one of the most importantproblems on F -singularities. Making use of recent progress on the minimal modelprogram, we prove Conjecture A .Let x ∈ X be an n -dimensional isolated log canonical singularity defined over analgebraically closed field k of characteristic zero. We suppose that x ∈ X is not logterminal and K X is Cartier at x . Let f : Y → X be a resolution of singularities Mathematics Subject Classification.
Primary 14B05; Secondary 13A35, 14E30.
Key words and phrases. log canonical singularities, F -pure singularities. such that f is an isomorphism outside x and that Supp f − ( x ) is a simple normalcrossing divisor on X . Then we can write K Y = f ∗ K X + F − E, where E and F are effective divisors and have no common irreducible components.In [9], the first author defined the invariant µ ( x ∈ X ) by µ = µ ( x ∈ X ) = min { dim W | W is a stratum of E } and he showed that this invariant plays an important role in the study of x ∈ X .Using his method (which is based on the minimal model program), we can checkthat any minimal stratum W of E is a projective resolution of a µ -dimensionalprojective variety V with only rational singularities such that K V is linearly trivial.Also, running a minimal model program with scaling (see [1] for the minimal modelprogram with scaling), we show that H µ ( V, O V ) can be viewed as the socle of thetop local cohomology module H nx ( O X ) of x ∈ X .Here we introduce the following conjecture. Conjecture B d . Let Z be a d -dimensional projective variety over an algebraicallyclosed field of characteristic zero with only rational singularities such that K Z is lin-early trivial. Then the action induced by the Frobenius morphism on the cohomologygroup H d ( Z p , O Z p ) of its modulo p reduction Z p is bijective for infinitely many p . Conjecture B d is open in general, but it follows from a combination of the resultsof Ogus [22], Bogomolov–Zarhin [2] and Joshi–Rajan [18] that Conjecture B d holdstrue if d ≤ µ is true. Applying Conjecture B µ to V , wesee that the Frobenius action on the cohomology group H µ ( V p , O V p ) of modulo p reduction V p of V is bijective for infinitely many p . On the other hand, by Matlisduality, the F -purity of modulo p reduction x p ∈ X p of x ∈ X is equivalent to theinjectivity of the Frobenius action on H nx p ( O X p ). This injectivity can be checked bythe injectivity of the Frobenius action on its socle H µ ( V p , O V p ). Thus, summing upthe above, we conclude that x ∈ X is of dense F -pure type.A similar argument works in more general settings and our main result is statedas follows. Main Theorem (=Theorem 3.4) . Let x ∈ X be a log canonical singularity definedover an algebraically closed field k of characteristic zero such that x is an isolatednon-log-terminal point of X . If Conjecture B µ holds true where µ = µ ( x ∈ X ) , then x ∈ X is of dense F -pure type. In particular, if µ ( x ∈ X ) ≤ , then x ∈ X is ofdense F -pure type. As a corollary of the above theorem, we show that Conjecture A n +1 is equivalent toConjecture B n (Corollary 3.7). Since Conjecture B is known to be true, ConjectureA holds true. That is, log canonicity is equivalent to being of dense F -pure type inthe case of three-dimensional isolated normal Q -Gorenstein singularities (Corollary3.8). -PURITY OF ISOLATED LOG CANONICAL SINGULARITIES 3 Acknowledgments.
The second author is grateful to Yoshinori Gongyo and MirceaMustat¸˘a for helpful conversations. The first and second authors were partially sup-ported by Grant-in-Aid for Young Scientists (A) 20684001 and (B) 23740024, re-spectively, from JSPS.1.
Preliminaries on log canonical singularities
In this section, we work over an algebraically closed field of characteristic zero.We start with the definition of singularities of pairs. Let X be a normal variety and D be an effective Q -divisor on X such that K X + D is Q -Cartier. Definition 1.1.
Let π : e X → X be a birational morphism from a normal variety e X . Then we can write K e X = π ∗ ( K X + D ) + X E a ( E, X, D ) E, where E runs through all the distinct prime divisors on e X and a ( E, X, D ) is arational number. We say that the pair (
X, D ) is canonical (resp. plt , log canonical )if a ( E, X, D ) ≥ a ( E, X, D ) > − a ( E, X, D ) ≥ −
1) for every exceptionaldivisor E over X . If D = 0, we simply say that X has only canonical (resp. logterminal, log canonical) singularities. We say that ( X, D ) is dlt if (
X, D ) is logcanonical and there exists a log resolution π : e X → X such that a ( E, X, D ) > − π -exceptional divisor E on e X . Here, a log resolution π : e X → X of ( X, D )means that π is a proper birational morphism, e X is a smooth variety, Exc( π ) is adivisor and Exc( π ) ∪ Supp π − ∗ D is a simple normal crossing divisor. Definition 1.2.
A subvariety W of X is said to be a log canonical center for thelog canonical pair ( X, D ) if there exist a proper birational morphism π : e X → X from a normal variety e X and a prime divisor E on e X with a ( E, X, D ) = − π ( E ) = W . Then W is denoted by c X ( E ). Remark . Let (
X, D ) be a dlt pair. There then exists a log resolution f : Y → X such that f induces an isomorphism over the generic point of any log canonicalcenter of ( X, D ) and a ( E, X, D ) > − f -exceptional divisor E . This is animmediate consequence of [24, Divisorial Log Terminal Theorem].From now on, let X be a normal Q -Gorenstein algebraic variety and x ∈ X bea germ. The index of X at x is the smallest positive integer r such that rK X isCartier at x . Definition 1.4.
Let x ∈ X be a log canonical singularity such that x is a logcanonical center. First we assume that the index of X at x is one. Take a projectivebirational morphism f : Y → X from a smooth variety Y such that Supp f − ( x )and Exc( f ) are simple normal crossing divisors. Then we can write K Y = f ∗ K X + F − E, where E and F are effective divisors on Y and have no common irreducible com-ponents. By assumption, E is a reduced simple normal crossing divisor on Y . We O. FUJINO and S. TAKAGI define µ ( x ∈ X ) by µ ( x ∈ X ) = min { dim W | W is a stratum of E and f ( W ) = x } . Here we say a subvariety W is a stratum of E = P i ∈ I E i if there exists a subset { i , . . . , i k } ⊆ I such that W is an irreducible component of the intersection E i ∩· · · ∩ E i k . This definition is independent of the choice of the resolution f .In general, we take an index one cover ρ : X ′ → X with x ′ = ρ − ( x ) to define µ ( x ∈ X ) by µ ( x ∈ X ) = µ ( x ′ ∈ X ′ ) . Since the index one cover is unique up to ´etale isomorphisms, the above definitionof µ ( x ∈ X ) is well-defined.We will give in Section 4 a quick overview of the invariant µ and some relatedtopics for the reader’s convenience. Remark . (1) The first author showed in [9, Theorem 5.5] that the invariant µ coincides with Ishii’s Hodge theoretic invariant (see [17] and [9, 5.1] for the defini-tion).(2) By the main result of [4], the index of x ∈ X is bounded if µ ( x ∈ X ) ≤ dlt blow-ups ,which was first introduced by Christopher Hacon. Lemma 1.6 (cf. [9, Lemma 2.9] and [7, Section 4]) . Let X be a log canonical varietyof index one such that X is quasi-projective, x is an isolated non-log-terminal pointof X , and that X is canonical outside x . Then there exists a projective birationalmorphism g : Z → X such that K Z + D = g ∗ K X with D a reduced divisor on Z , thepair ( Z, D ) is a Q -factorial dlt pair and g is a small morphism outside x . Lemma 1.7.
In Lemma 1.6, Z has only canonical singularities.Proof. If a ( E, Z, D ) > −
1, then a ( E, Z, D ) ≥ K Z + D is Cartier. Since K Z is Q -Cartier and D is an effective divisor on Z , one has a ( E, Z, ≥
0. If a ( E, Z, D ) = −
1, then we may assume that Z is a smooth variety and D is a reducedsimple normal crossing divisor on Z by shrinking Z around the log canonical center c Z ( E ). In this case, a ( E, Z, ≥
0. Thus, Z has only canonical singularities. (cid:3) Preliminaries on F -pure singularities In this section, we briefly review the definition of F -pure singularities and itsproperties which we will need later. Definition 2.1 ([16], [14]) . Let x ∈ X be a point of an F -finite integral scheme X of characteristic p > x ∈ X is said to be F -pure if the Frobenius map F : O X,x → F ∗ O X,x a a p splits as an O X,x -module homomorphism. -PURITY OF ISOLATED LOG CANONICAL SINGULARITIES 5 (ii) x ∈ X is said to be strongly F -regular if for every nonzero c ∈ O X,x , thereexists an integer e ≥ cF e : O X,x → F e ∗ O X,x a ca p e splits as an O X,x -module homomorphism.
Remark . Strong F -regularity implies F -purity.The following criterion for F -purity is well-known to experts, but we include ithere for the reader’s convenience. Lemma 2.3 (cf. [16]) . Let x ∈ X be a closed point with index one of an n -dimensional F -finite integral scheme X . Then x ∈ X is F -pure if and only if F ( z ) = 0 , where F is the natural Frobenius action on H nx ( O X ) and z is a generatorof the socle (0 : m x ) H nx ( O X ) .Proof. First note that H nx ( O X ) is isomorphic to the injective hull of the residue field O X,x / m x , because O X,x is quasi-Gorenstein. By definition, x ∈ X is F -pure if andonly if F ∨ : Hom O X,x ( F ∗ O X,x , O X,x ) → Hom O X,x ( O X,x , O X,x ) = O X,x is surjective. F ∨ is the Matlis dual of the natural Frobenius action F on H nx ( O X ),so the surjectivity of F ∨ is equivalent to the injectivity of F . Since H nx ( O X ) isan essential extension of the socle (0 : m x ) H nx ( O X ) , F is injective if and only if F | (0: m x ) Hnx ( O X ) is injective. Finally, the latter condition is equivalent to saying that F ( z ) = 0, because the socle (0 : m x ) H nx ( O X ) is a one-dimensional O X,x / m x -vectorspace. (cid:3) We define the notion of F -purity and strong F -regularity in characteristic zero,using reduction from characteristic zero to positive characteristic. Definition 2.4.
Let x ∈ X be a point of a scheme of finite type over a field k ofcharacteristic zero. Choosing a suitable finitely generated Z -subalgebra A ⊆ k , wecan construct a (non-closed) point x A of a scheme X A of finite type over A such that( X A , x A ) × Spec A k ∼ = ( X, x ). By the generic freeness, we may assume that X A and x A are flat over Spec A . We refer to x A ∈ X A as a model of x ∈ X over A . Given aclosed point s ∈ Spec A , we denote by x s ∈ X s the fiber of x ∈ X over s . Then X s is a scheme defined over the residue field κ ( s ) of s , which is a finite field. The readeris referred to [15, Chapter 2] and [21, Section 3.2] for more detail on reduction fromcharacteristic zero to characteristic p .(i) x ∈ X is said to be of strongly F -regular type if there exists a model of x ∈ X over a finitely generated Z -subalgebra A of k and a dense open subset S ⊆ Spec A such that x s ∈ X s is strongly F -regular for all closed points s ∈ S .(ii) x ∈ X is said to be of dense F -pure type if there exists a model of x ∈ X over a finitely generated Z -subalgebra A of k and a dense subset of closedpoints S ⊆ Spec A such that x s ∈ X s is F -pure for all s ∈ S . Remark . The definitions of strongly F -regular type and dense F -pure type areindependent of the choice of a model. O. FUJINO and S. TAKAGI
Theorem 2.6 ([11, Theorem 5.2]) . Let x ∈ X be a normal Q -Gorenstein singularitydefined over a field of characteristic zero. Then x ∈ X is log terminal if and only ifit is of strongly F -regular type. In this paper, we will discuss an analogous statement for log canonical singulari-ties. Especially, we will consider the following conjecture.
Conjecture A n . Let x ∈ X be an n -dimensional normal Q -Gorenstein singularitydefined over an algebraically closed field k of characteristic zero such that x is anisolated non-log-terminal point of X . Then x ∈ X is log canonical if and only if itis of dense F -pure.Remark . Conjecture A n is known to be true when n = 2 (see [10], [20] and[27]) or when x ∈ X is a hypersurface singularity whose defining polynomial is verygeneral (see [13]). The reader is referred to [25, Remark 2.6] for more detail. Definition 2.8.
Let X be an F -finite scheme of characteristic p >
0. If X = Spec R is affine, we denote by R [ F ] the ring R [ F ] = R { F }h r p F − F r | r ∈ R i which is obtained from R by adjoining a non-commutative variable F subject to therelation r p F = F r for all r ∈ R . For a general scheme X , we denote by O X [ F ] thesheaf of rings obtained by gluing the respective rings O X ( U i )[ F ] over an affine opencover X = S i U i . Example 2.9. (1) Let f : Y → X be a morphism of schemes over an F -finite affinescheme Z . Then for all i ≥ H i ( X, O X ) and H i ( Y, O Y ) each has a natural O Z [ F ]-module structure and f induces an O Z [ F ]-module homomorphism f ∗ : H i ( X, O X ) → H i ( Y, O Y ).(2) Let Y be a closed subscheme of a scheme X over an F -finite affine scheme Z .Then for all i ≥
0, we have the following natural exact sequence of O Z [ F ]-modules · · · → H iY ( X, O X ) → H i ( X, O X ) → H i ( X \ Y, O X ) → H i +1 Y ( X, O X ) → · · · . (3) Let X be a scheme over an F -finite affine scheme Z and Y , Y ⊆ X be closedsubschemes. Let Y denote the scheme-theoretic union of Y and Y . Then for all i ≥
0, the Mayer–Vietoris exact sequence · · · → H i ( Y, O Y ) → H i ( Y , O Y ) ⊕ H i ( Y , O Y ) → H i ( Y ∩ Y , O Y ∩ Y ) → H i +1 ( Y, O Y ) → · · · becomes an exact sequence of O Z [ F ]-modules. Proof.
The proof is immediate from the fact that every cohomology module in Ex-ample 2.9 can be computed from the ˇCech complex. (cid:3)
The following proposition is a key to prove the main result of this paper.
Proposition 2.10.
Let x ∈ X be an n -dimensional normal singularity with indexone defined over an algebraically closed field k of characteristic zero. Let g : Z → X be a projective birational morphism and D be a reduced Q -Cartier divisor on Z satisfying the following properties: -PURITY OF ISOLATED LOG CANONICAL SINGULARITIES 7 (1) Z has only rational singularities, (2) K Z + D ∼ g , (3) g | Z \ D : Z \ D → X \ { x } is an isomorphism, (4) Supp D = Supp g − ( x ) ,Then x ∈ X is of dense F -pure type if and only if given a model of D over a finitelygenerated Z -subalgebra A of k , there exists a dense subset S ⊆ Spec A such that theaction of Frobenius on H n − ( D s , O D s ) is bijective for every closed point s ∈ S .Proof. Without loss of generality, we may assume that X is affine. Suppose given amodel of ( x ∈ X, Z, D, g ) over a finitely generated Z -subalgebra A of k .First we will show that enlarging A if necessary, we can view H n − ( Z s , O Z s ) asan O X s [ F ]-submodule of H nx s ( O X s ) for all closed points s ∈ Spec A . Since f | Z \ D : Z \ D → X \ { x } is an isomorphism, we have natural isomorphisms H n − ( Z \ D, O Z ) ∼ = H n − ( X \ { x } , O X ) ∼ = H nx ( O X ) . On the other hand, we have the natural exact sequence H n − D ( Z, O Z ) → H n − ( Z, O Z ) → H n − ( Z \ D, O Z )and H n − D ( Z, O Z ) = 0 by the dual form of Grauert–Riemenschneider vanishing the-orem (see, for example, [6, Lemma 4.19 and Remark 4.20]). Hence we can view H n − ( Z, O Z ) as an O X -submodule of H nx ( O X ). By Example 2.9 (1), (2), after pos-sibly enlarging A , we may assume that H n − ( Z s , O Z s ) is an O X s [ F ]-submodule of H nx s ( O X s ) for all closed points s ∈ Spec A .Next we will show that we may assume that H n − ( Z s , O Z s ) ∼ = H n − ( D s , O D s )as an O X s [ F ]-module homomorphism for all closed points s ∈ Spec A . The shortexact sequence 0 → O Z ( − D ) → O Z → O D → H n − ( Z, O Z ( − D )) → H n − ( Z, O Z ) → H n − ( D, O D ) → H n ( Z, O Z ( − D )) = 0of O X -modules. It follows from the Grauert–Riemenschneider vanishing theoremthat H n − ( Z, O Z ( − D )) ∼ = H n − ( Z, O Z ( K Z )) = 0, so we have an O X -module iso-morphism H n − ( Z, O Z ) ∼ = H n − ( D, O D ). By Example 2.9 (1), after possibly enlarg-ing A , we may assume that H n − ( Z s , O Z s ) ∼ = H n − ( D s , O D s ) as an O X s [ F ]-modulehomomorphism for all closed points s ∈ Spec A .Finally, we will check that H d − ( D s , O D s ) is the socle of the O X s ,x s [ F ]-module of H dx s ( O X s ). Since m x · H n − ( Z, O Z ) = H ( Z, O Z ( − D )) · H n − ( Z, O Z ) ⊆ H n − ( Z, O Z ( − D )) = 0 ,H n − ( D, O D ) ∼ = H n − ( Z, O Z ) is contained in the socle of H nx ( O X ). Let ω D be thedualizing sheaf of D . Then we obtain ω D ≃ O Z ( K Z + D ) ⊗ O D since K Z + D isCartier. Therefore, ω D ≃ O D because K Z + D ∼ g g ( D ) = x . By Serre duality(which holds for top cohomology groups even if the variety is not Cohen–Macaulay),one has dim k H n − ( D, O D ) = 1, because H ( D, ω D ) = H ( D, O D ) ∼ = k . The socleof H nx ( O X ) is the one-dimensional k -vector space, so it coincides with H n − ( D, O D ).By the above argument, the bijectivity of the Frobenius action on H n − ( D s , O D s )means that the restriction of the Frobenius action on H nx s ( O X s ) to its socle is injec-tive. This condition is equivalent to saying that X s is F -pure by Lemma 2.3. Thus, O. FUJINO and S. TAKAGI x s ∈ X s is of dense F -pure type if and only if there exists a dense subset of closedpoints S ⊆ Spec A such that the Frobenius action on H n − ( D s , O D s ) is bijective forall s ∈ S . (cid:3) Main result
In order to state our main result, we introduce the following conjecture.
Conjecture B n . Let V be an n -dimensional projective variety over an algebraicallyclosed field k of characteristic zero with only rational singularities such that K V is linearly trivial. Given a model of V over a finitely generated Z -subalgebra A of k , there exists a dense subset of closed points S ⊆ Spec A such that the naturalFrobenius action on H n ( V s , O V s ) is bijective for every s ∈ S .Remark . (1) An affirmative answer to [21, Conjecture 1.1] implies an affirmativeanswer to Conjecture B n . Indeed, take a resolution of singularities π : e V → V .Since V has only rational singularities, π induces the isomorphism H n ( V, O V ) ∼ = H n ( e V , O e V ). Suppose given a model of π over a finitely generated Z -subalgebra A of k . If [21, Conjecture 1.1] holds true, then there exists a dense subset of closedpoints S ⊆ Spec A such that the Frobenius action on H n ( e V s , O e V s ) is bijective forevery s ∈ S . Since we may assume that H n ( V s , O V s ) ∼ = H n ( e V s , O e V s ) as κ ( s )[ F ]-modules for all s ∈ S by Example 2.9 (1), we obtain the assertion.(2) Let W be an n -dimensional smooth projective variety defined over a perfectfield of characteristic p >
0. If W is ordinary (in the sense of Bloch–Kato), then thenatural Frobenius action on H n ( W, O W ) is bijective (see [21, Remark 5.1]). If W isan abelian variety or a curve, then the converse implication also holds true (see [21,Examples 5.4 and 5.5]). Lemma 3.2.
Conjecture B n +1 implies Conjecture B n .Proof. Let V be an n -dimensional projective variety over an algebraically closedfield k of characteristic zero with only rational singularities such that K V is linearlytrivial. Let C be an elliptic curve over k , and denote W = V × C . We supposegiven a model of ( V, C, W ) over a finitely generated Z -subalgebra A of k . ApplyingConjecture B n +1 to W , we can take a dense subset of closed points S ⊆ Spec A suchthat the Frobenius action on H n +1 ( W s , O W s ) = H n ( V s , O V s ) ⊗ H ( C s , O C s )is bijective for every s ∈ S . This implies that the Frobenius action on H n ( V s , O V s )is bijective for every s ∈ S . (cid:3) Lemma 3.3.
Conjecture B n holds true if n ≤ .Proof. By an argument similar to the proof of [21, Proposition 5.3], we may assumethat k = Q without loss of generality. By Lemma 3.2, it suffices to consider the casewhen n = 2.Let π : e X → X be a minimal resolution. e X is an abelian surface or a K3surface. Suppose given a model of π over a finitely generated Z -subalgebra A of k . Then there exists a dense subset of closed points S ⊆ Spec A such that the -PURITY OF ISOLATED LOG CANONICAL SINGULARITIES 9 Frobenius action on H ( e X s , O e X s ) is bijective for every s ∈ S (the abelian surfacecase follows from a result of Ogus [22] and the K3 surface case follows from a resultof Bogomolov–Zarhin [2] or that of Joshi and Rajan [18]). Since X has only rationalsingularities, by Example 2.9 (1), we may assume that H ( X s , O X s ) ∼ = H ( e X s , O e X s )as κ ( s )[ F ]-modules for all s ∈ S . Thus, we obtain the assertion. (cid:3) Our main result is stated as follows.
Theorem 3.4.
Let x ∈ X be a log canonical singularity defined over an algebraicallyclosed field k of characteristic zero such that x is an isolated non-log-terminal pointof X . If Conjecture B µ holds true where µ = µ ( x ∈ X ) , then x ∈ X is of dense F -pure type. In particular, if µ ( x ∈ X ) ≤ , then x ∈ X is of dense F -pure type. We need the following proposition for the proof of Theorem 3.4.
Proposition 3.5.
Let x ∈ X be an n -dimensional log canonical singularity definedover an algebraically closed field k of characteristic zero. Suppose that the indexof X at x is one and that x is an isolated non-log-terminal point of X . Let g :( Z, D ) → X be a dlt blow-up as in Lemma 1.6. Then there exists a birational model e g : ( e Z, e D ) → X of g which satisfies the following properties: (1) e Z has only canonical singularities, (2) K e Z + e D is linearly trivial over X , (3) e g | e Z \ e D : e Z \ e D → X \ { x } is an isomorphism. (4) Supp e D = Supp e g − ( x ) , (5) given models of D and e D over a finitely generated Z -subalgebra A of k ,enlarging A if necessary, we may assume that H n − ( D s , O D s ) ∼ = H n − ( e D s , O e D s ) as κ ( s )[ F ] -modules for all closed points s ∈ Spec A .Proof. We may assume that X is affine and K X is Cartier. We run a K Z -minimalmodel program over X with scaling (see [1] for the minimal model program withscaling). Then we obtain a sequence of divisorial contractions and flips: Z Z Z Z k − Z k Z ′ D D D D k − D k D ′ φ / / ❴❴❴❴ φ / / ❴❴❴❴❴ φ k − / / ❴❴❴❴ φ k − / / ❴❴❴❴ / / ❴❴❴❴ / / ❴❴❴❴❴ / / ❴❴❴❴ / / ❴❴❴❴ ?(cid:31) O O ?(cid:31) O O ?(cid:31) O O ?(cid:31) O O such that K Z ′ is nef over X . Suppose given a model of the above sequence over afinitely generated Z -subalgebra A of k . Claim . Assume that φ i : Z i Z i +1 is a flip. Enlarging A if necessary, we mayassume that H j ( D i s , O D is ) ∼ = H j ( D i +1 ,s , O D i +1 ,s )as O X s [ F ]-modules for all j and all closed points s ∈ Spec A . Proof of Claim 1.
We consider the following flipping diagram: Z i φ i / / ❴❴❴❴❴❴❴ ψ i ❆❆❆❆❆❆❆❆ Z i +1 ψ i +1 | | ③③③③③③③③ W i Enlarging A if necessary, we may assume that a model of the above diagram over A is given. Note that K Z i + D i ∼ ψ i K Z i +1 + D i +1 ∼ ψ i +1
0. Then we have thefollowing exact sequences:0 → O Z i ( K Z i ) →O Z i → O D i → , → O Z i +1 ( K Z i +1 ) →O Z i +1 → O D i +1 → . We put C i = ψ i ( D i ) = ψ i +1 ( D i ) ⊆ W i . Since Z i , Z i +1 and W i each has only rationalsingularities, by the Grauert–Riemenschneider vanishing theorem, one has R ψ i ∗ O D i ∼ = O C i ∼ = R ψ i ∗ O D i +1 in the derived category of coherent sheaves on C i . Therefore, ψ i and ψ i +1 inducethe isomorphisms H j ( D i , O D i ) ψ i ∗ ∼ = H j ( C i , O C i ) ψ i +1 ∗ ∼ = H j ( D i +1 , O D i +1 )for all j . By Example 2.9 (1), after possibly enlarging A , we may assume that H j ( D i s , O D is ) ∼ = H j ( D i +1 ,s , O D i +1 ,s )as O X s [ F ]-modules for all closed points s ∈ Spec A . (cid:3) Claim . Assume that φ i : Z i Z i +1 is a divisorial contraction. Enlarging A ifnecessary, we may assume that H j ( D i s , O D is ) ∼ = H j ( D i +1 ,s , O D i +1 ,s )as O X s [ F ]-modules for all j and all closed points s ∈ Spec A . Proof of Claim 2.
Let E be the φ i -exceptional prime divisor on Z i . First we willcheck that φ i ( D i ) = D i +1 . It is obvious when E is not an irreducible componentof D i , so we consider the case when E is an irreducible component of D i . Since K Z i + D i and K Z i +1 + D i +1 both are linearly trivial over X , we have K Z i + D i = φ ∗ i ( K Z i +1 + D i +1 ) . Hence φ i ( E ) is a log canonical center of the pair ( Z i +1 , D i +1 ). Each Z i has onlycanonical singularities, because Z has only canonical singularities by Lemma 1.7and we run a K Z -minimal model program. Thus, φ i ( E ) has to be contained in D i +1 , which implies that φ i ( D i ) = D i +1 .By an argument analogous to the proof of Claim 1 (that is, by the Grauert–Riemenschneider vanishing theorem), we have R φ i ∗ O D i ∼ = O D i +1 in the derivedcategory of coherent sheaves on D i +1 . Therefore, φ i induces the isomorphism H j ( D i , O D i ) φ i ∗ ∼ = H j ( D i +1 , O D i +1 ) -PURITY OF ISOLATED LOG CANONICAL SINGULARITIES 11 for all j . By Example 2.9 (1), after possibly enlarging A , we may assume that H j ( D i s , O D is ) ∼ = H j ( D i +1 ,s , O D i +1 ,s )as O X s [ F ]-modules for all s ∈ Spec A . (cid:3) Let g ′ : ( Z ′ , D ′ ) → X be the output of the minimal model program. By the basepoint free theorem, we obtain the following diagram( Z ′ , D ′ ) π / / g ′ ●●●●●●●●● ( e Z, e D ) e g | | ②②②②②②②②② X such that e Z is the canonical model of Z ′ over X and that K Z ′ + D ′ = π ∗ ( K e Z + e D ) . Enlarging A if necessary, we may assume that a model of the above diagram over A is given. By an argument similar to the proof of Claim 2, we can check that π ( D ′ ) = e D and R π ∗ O D ′ ∼ = O e D in the derived category of coherent sheaves on e D .Thus, π induces the isomorphism H n − ( D ′ , O D ′ ) π ∗ ∼ = H n − ( e D, O e D ) . By Example 2.9 (1), after possibly enlarging A , we may assume that H n − ( D ′ s , O D ′ s ) ∼ = H n − ( e D s , O e D s )as O X s [ F ]-modules for all closed points s ∈ Spec A .Summing up the above arguments, we know that e g : ( e Z, e D ) → X has the followingproperties:(i) e Z has only canonical singularities,(ii) K e Z + e D ∼ e g K e Z is e g -ample,(iv) H n − ( D s , O D s ) ∼ = H n − ( e D s , O e D s ) for all closed points s ∈ Spec A .Since − e D is e g -ample by (i) and (ii), one has Supp e D = Supp e g − ( x ). Therefore, itremains to show that e g is an isomorphism outside x . Note that X \ { x } has onlycanonical singularities. Then we can write K e Z \ e D = g ∗ K X \{ x } + F, where F is a e g -exceptional effective Q -divisor on e Z \ e D . Since K e Z \ e D is e g -ample,one has F = 0. Again, by the e g -ampleness of K e Z \ e D , the birational morphism e g : e X \ e D → X \ { x } has to be finite, that is, an isomorphism. (cid:3) Now we start the proof of Theorem 3.4.
Proof of Theorem 3.4.
Since F -purity and log canonicity are preserved under indexone covers (see [26] for F -purity and [19, Proposition 5.20] for log canonicity), wemay assume that the index of X at x is one. We can also assume that X is affineand K X is Cartier. By Lemma 1.6, there exists a birational projective morphism g : Z → X and areduced divisor D on Z such that K Z + D = g ∗ K X , ( Z, D ) is a Q -factorial dlt pair and g is a small morphism outside x . By Remark 1.3, there exists a projective birationalmorphism h : Y → Z from a smooth variety Y with the following properties:(1) Exc( h ) and Exc( h ) ∪ Supp h − ∗ D are simple normal crossing divisors on Y ,(2) h is an isomorphism over the generic point of any log canonical center of thepair ( Z, D ),(3) a ( E, Z, D ) > − h -exceptional divisor E .Then we can write K Y = h ∗ ( K Z + D ) + F − E, where E and F are effective divisors on Y which have no common irreducible com-ponents. By the construction of h , E is a reduced simple normal crossing divisoron Y and E = h − ∗ D . It follows from [6, Corollary 4.15] or [9, Corollary 2.5] that R h ∗ O E ∼ = O D in the derived category of coherent sheaves on D . Therefore, we havethe isomorphism H i ( E, O E ) h ∗ ∼ = H i ( D, O D )for every i . Suppose given models of D and E over a finitely generated Z -subalgebra A of k . By Example 2.9 (1), after possibly enlarging A , we may assume that H i ( E s , O E s ) ∼ = H i ( D s , O D s )as O X s [ F ]-modules for all closed points s ∈ Spec A .Let W be a minimal stratum of a simple normal crossing variety E . By anargument similar to [9, 4.11] (see also Section 4), one has dim W = µ . Since K Z + D is linearly trivial over X and D is a g -exceptional divisor on Z , by the adjunctionformula, one has K D ∼
0. We also note that D is sdlt (see [3, Definition 1.1] for thedefinition of sdlt varieties). Applying [9, Remark 5.3] to h : E → D , we obtain thefollowing claim. Claim.
Suppose that models of W and E over A are given. Then after possiblyenlarging A , we may assume that H n − ( E s , O E s ) ∼ = H µ ( W s , O W s )as O X s [ F ]-modules for all closed points s ∈ Spec A . Proof of Claim.
It follows from [9, Theorem 5.2 and Remark 5.3] that H µ ( W, O W ) ∼ = . . . ∼ = H n − ( E, O E ) , where each isomorphism is the connecting homomorphism of a suitable Mayer–Vietoris exact sequence. Then by Example 2.9 (3), after possibly enlarging A , wemay assume that H n − ( E s , O E s ) ∼ = · · · ∼ = H µ ( W s , O W s )as O X s [ F ]-modules for all closed points s ∈ Spec A . (cid:3) Let V = h ( W ) ⊆ D . Then V is a minimal log canonical center of the pair ( Z, D ).On the other hand, by adjunction formula for dlt pairs, we obtain K V = ( K Z + -PURITY OF ISOLATED LOG CANONICAL SINGULARITIES 13 D ) | V ∼
0. Thus, V has only Gorenstein rational singularities. Since h : W → V isbirational by the construction of h , one has the isomorphism H µ ( W, O W ) h ∗ ∼ = H µ ( V, O V ) . Suppose models of W and V are given over A . By Example 2.9 (1), after possiblyenlarging A , we may assume that H µ ( W s , O W s ) ∼ = H µ ( V s , O V s )as O X s [ F ]-modules for all closed points s ∈ Spec A .Now we sum up the above arguments together with Proposition 3.5 (we use thesame notation as in Proposition 3.5). Suppose given models of e D and V over afinitely generated Z -subalgebra A of k . Then after possibly enlarging A , we mayassume that H µ ( V s , O V s ) ∼ = H n − ( e D s , O e D s )as O X s [ F ]-modules for all closed points s ∈ Spec A . It follows from an applicationof Conjecture B µ to V that there exists a dense subset of closed points S ⊆ Spec A such that the natural Frobenius action on H µ ( V s , O V s ) is bijective for all s ∈ S .Then the Frobenius action on H n − ( e D s , O e D s ) is also bijective for all closed points s ∈ S , which implies by Proposition 2.10 that x ∈ X is of dense F -pure type. (cid:3) Remark . Let f : Y → X be any resolution as in Definition 1.4. By the uniquenessof the relative canonical model, we have e Z ∼ = Proj M m ≥ f ∗ O Y ( mK Y )over X . Unfortunately, by this construction, it is not clear how to relate the coho-mology group H n − ( D s , O D s ) to H n − ( e D s , O e D s ). Moreover, the relationship between e D and a minimal stratum of E in Definition 1.4 is also not clear. Therefore, we takea dlt blow-up and run a minimal model program with scaling to construct e Z . Corollary 3.7.
Conjecture A n +1 is equivalent to Conjecture B n .Proof. First we will show that Conjecture B n implies Conjecture A n +1 . Let x ∈ X bean ( n +1)-dimensional normal Q -Gorenstein singularity defined over an algebraicallyclosed field k of characteristic zero such that x is an isolated non-log-terminal pointof X . If x ∈ X is of dense F -pure type, then by [12, Theorem 3.9], it is logcanonical. Conversely, suppose that x ∈ X is a log canonical singularity. Since µ := µ ( x ∈ X ) ≤ dim X − n , by Lemma 3.2, Conjecture B µ holds true. It thenfollows from Theorem 3.4 that x ∈ X is of dense F -pure type.Next we will prove that Conjecture A n +1 implies Conjecture B n . Let V be an n -dimensional projective variety over an algebraically closed field k of characteristiczero with only rational singularities such that K V ∼
0. Take any ample Cartierdivisor D on V and consider its section ring R = R ( V, D ) = L m ≥ H ( V, O V ( mD )).By [23, Proposition 5.4], the affine cone Spec R of V has only quasi-Gorensteinlog canonical singularities and its vertex is an isolated non-log-terminal point ofSpec R . It then follows from Conjecture A n +1 that given a model of ( V, D, R ) overa finitely generated Z -subalgebra A of k , there exists a dense subset of closed points S ⊆ Spec A such that Spec R s is F -pure for all s ∈ S . Note that after replacing S by a smaller dense subset if necessary, we may assume that R s = R ( V s , D s ) for all s ∈ S . Since Spec R s is F -pure, the natural Frobenius action on the local cohomologymodule H n +1 m Rs ( R s ) is injective, where m R s = L m ≥ H ( V s , O V s ( mD s )) is the uniquehomogeneous maximal ideal of R s . Then the Frobenius action on H n ( V s , O V s ) isalso injective, because H n ( V s , O V s ) is the degree zero part of H n +1 m Rs ( R s ). (cid:3) Since Conjecture B is known to be true (see Lemma 3.3), Conjecture A holdstrue. Corollary 3.8.
Let x ∈ X be a three-dimensional normal Q -Gorenstein singularitydefined over an algebraically closed field of characteristic zero such that x is anisolated non-log-terminal point of X . Then x ∈ X is log canonical if and only if itis of dense F -pure type. Appendix: A quick review of [4] and [9]In this appendix, we quickly review the invariant µ and related topics in [4]and [9] for the reader’s convenience. After [4] was written, the minimal modelprogram has developed drastically (cf. [1]). In [9], we only treat isolated log canonicalsingularities. Here, we survey the basic properties of µ and some related results inthe framework of [9]. For the details, see [4] and [9].Let X be a quasi-projective log canonical variety defined over an algebraicallyclosed field k of characteristic zero with index one. Assume that x ∈ X is a logcanonical center. Let f : Y → X be a projective birational morphism from asmooth variety Y such that K Y = f ∗ K X + F − E where E and F are effective Cartier divisors on Y and have no common irreduciblecomponents. We further assume that f − ( x ) and Supp( E + F ) are simple normalcrossing divisors on Y . Let E = P i ∈ I E i be the irreducible decomposition. Notethat E is a reduced simple normal crossing divisor on Y . We put J = { i ∈ I | f ( E i ) = x } ⊂ I and G = X i ∈ J E i . Then, by [8, Proposition 8.2], we obtain f ∗ O G ∼ = κ ( x ) . In particular, G is connected. We apply a ( K Y + E )-minimal model program withscaling over X (cf. [1] and [7, Section 4]). Then we obtain a projective birationalmorphism f ′ : Y ′ → X such that ( Y ′ , E ′ ) is a Q -factorial dlt pair and that K Y ′ + E ′ = f ′∗ K X where E ′ isthe pushforward of E on Y ′ . It is a dlt blow-up of X (cf. Lemma 1.6). Note that -PURITY OF ISOLATED LOG CANONICAL SINGULARITIES 15 each step of the minimal model program is an isomorphism at the generic point ofany log canonical center of ( Y, E ) because K Y + E = f ∗ K X + F. Therefore, we obtain µ ( x ∈ X ) = min { dim W | W is a log canonical center of ( Y ′ , E ′ ) with f ′ ( W ) = x } . By the proof of [8, Theorem 10.5 (iv)], we have f ′∗ O G ′ ∼ = κ ( x )where G ′ is the pushforward of G on Y ′ . In particular, G ′ is connected. By applying[9, Proposition 3.3] to each irreducible component of G ′ , we can check that if W isa minimal log canonical center of ( Y ′ , E ′ ) with f ′ ( W ) = x then dim W = µ ( x ∈ X ).By this observation, every minimal stratum of E which is mapped to x by f is µ ( x ∈ X )-dimensional and µ ( x ∈ X ) is independent of the choice of the resolution f (cf. [9, 4.11]), that is, µ ( x ∈ X ) is well-defined.Let W and W be any minimal log canonical centers of ( Y ′ , E ′ ) such that f ′ ( W ) = f ′ ( W ) = x . Then we can check that W is birationally equivalent to W (cf. [9,Proposition 3.3]). Therefore, all the minimal stratum of E mapped to x by f arebirational each other. More precisely, we can take a common resolution W α } } ④④④④④④④④ α ! ! ❈❈❈❈❈❈❈❈ W o o / / ❴❴❴❴❴❴❴ W such that α ∗ K W = α ∗ K W (cf. [9, Proposition 3.3]).By the adjunction formula for dlt pairs (cf. [5, Proposition 3.9.2]), we can checkthat K W = ( K Y ′ + E ′ ) | W ∼ W has only canonical Gorenstein singularities if W is a minimal log canon-ical center of ( Y ′ , E ′ ) with f ′ ( W ) = x .Let V be any minimal stratum of E . Then we can prove that H µ ( V, O V ) δ ∼ = H n − ( E, O E )when f ( E ) = x , equivalently, x ∈ X is an isolated non-log-terminal point, where µ = µ ( x ∈ X ) and n = dim X . The isomorphism δ is a composition of connectinghomomorphisms of suitable Mayer–Vietoris exact sequences. For the details, see [9,Section 5]. 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