aa r X i v : . [ m a t h . AG ] N ov ON ISOLATED LOG CANONICAL SINGULARITIESWITH INDEX ONE
OSAMU FUJINO
Dedicated to Professor Shihoko Ishii on the occasion of her sixtieth birthday
Abstract.
We give a method to investigate isolated log canoni-cal singularities with index one which are not log terminal. Ourmethod depends on the minimal model program. One of the mainpurposes is to show that our invariant coincides with Ishii’s Hodgetheoretic invariant.
Contents
1. Introduction 12. Preliminaries 42.1. A criterion of Cohen–Macaulayness 42.2. Basic properties of dlt pairs 52.3. Dlt blow-ups 83. Dlt pairs with torsion log canonical divisor 94. Isolated log canonical singularities with index one 115. Ishii’s Hodge theoretic invariant 16References 201.
Introduction
Let P ∈ X be an n -dimensional isolated log canonical singularitywith index one which is not log terminal. Let f : Y → X be a pro-jective resolution such that f is an isomorphism outside P and thatSupp f − ( P ) is a simple normal crossing divisor on Y . Then we canwrite K Y = f ∗ K X + F − E Date : 2011/11/11, version 1.52.2010
Mathematics Subject Classification.
Primary 14B05; Secondary 14E30.
Key words and phrases. log canonical singularities, Cohen–Macaulay, minimalmodel program, mixed Hodge structures, dual complexes. where E and F are effective Cartier divisors and have no commonirreducible components. The divisor E is sometimes called the essentialdivisor for f (see [I2, Definition 7.4.3] and [I4, Definition 2.5]).In [I1, Propositions 1.4 and 3.7], Shihoko Ishii proves R n − f ∗ O Y ≃ H n − ( E, O E ) ≃ C . For details, see [I2, Propositions 5.3.11, 5.3.12, 7.1.13, 7.4.4, and The-orem 7.1.17]. In this paper, we prove that R i f ∗ O Y ≃ H i ( E, O E )for every i > R n − f ∗ O Y ≃ C ( P )(cf. Remark 4.8). Our proof depends on the minimal model theory andis different from Ishii’s.By Shihoko Ishii, the singularity P ∈ X is said to be of type (0 , i ) ifGr Wk H n − ( E, O E ) = ( C if k = i W is the weight filtration of the mixed Hodge structure on H n − ( E, C ). Note that E is a projective connected simple normalcrossing variety. Therefore, we haveGr Wk H n − ( E, O E ) ≃ Gr Wk Gr F H n − ( E, C ) ≃ Gr F Gr Wk H n − ( E, C )where F is the Hodge filtration. We also note that the type of P ∈ X isindependent of the choice of a resolution f : Y → X by [I1, Proposition4.2] (see also [I2, Proposition 7.4.6]).On the other hand, we define µ ( P ∈ X ) by µ = µ ( P ∈ X ) = min { dim W | W is a stratum of E } (see [F2, Definition 4.12]). We prove that P ∈ X is of type (0 , µ ),that is, Ishii’s Hodge theoretic invariant coincides with our invariant µ (cf. Theorem 5.5). It was first obtained by Shihoko Ishii in [I3].By our method based on the minimal model program, we can provethe following properties of E . Let E = P i E i be the irreducible decom-position. Then P i = i E i | E i has at most two connected components forevery irreducible component E i of E (cf. Remark 4.10). Let W and W be any two minimal strata of E . Then W is birationally equivalentto W (cf. 4.11 and Remark 4.10). These results seem to be out of reachby the Hodge theoretic method. N LOG CANONICAL SINGULARITIES 3
Let Γ be the dual complex of E and let | Γ | be the topological real-ization of Γ. Then the dimension of | Γ | is n − − µ by the definitionof µ .From now on, we assume that µ ( P ∈ X ) = 0. In this case, we canprove that H i ( E, O E ) ≃ H i ( | Γ | , C )for every i . Therefore, P ∈ X is Cohen–Macaulay, equivalently, Goren-stein, if and only if H i ( | Γ | , C ) = ( C if i = 0 , n − , Du Bois singularities ,which is one of the main ingredients of Ishii’s Hodge theoretic approach.We summarize the contents of this paper. Section 2 is a preliminarysection. In Section 2.1, we give a criterion of Cohen–Macaulayness.In Section 2.2, we investigate basic properties of dlt pairs. In Section2.3, we explain the notion of dlt blow-ups , which is very useful in thesubsequent sections. Section 3 is devoted to the study of dlt pairs withtorsion log canonical divisor. In Section 4, we investigate isolated lcsingularities with index one which are not log terminal. In Section5, we prove that our invariant µ coincides with Ishii’s Hodge theoreticinvariant. The main result (cf. Theorem 5.2) in Section 5 can be appliedto special fibers of semi-stable minimal models for varieties with trivialcanonical divisor (cf. [F6]). Notation.
Let X be a normal variety and let B be an effective Q -divisor such that K X + B is Q -Cartier. Then we can define the discrep-ancy a ( E, X, B ) ∈ Q for every prime divisor E over X . If a ( E, X, B ) ≥− > −
1) for every E , then ( X, B ) is called log canonical (resp. kawamata log terminal ). We sometimes abbreviate log canon-ical (resp. kawamata log terminal) to lc (resp. klt ). When ( X,
0) is klt,we simply say that X is log terminal ( lt , for short).Assume that ( X, B ) is log canonical. If E is a prime divisor over X such that a ( E, X, B ) = −
1, then c X ( E ) is called a log canonical center ( lc center , for short) of ( X, B ), where c X ( E ) is the closure of the imageof E on X . OSAMU FUJINO
Let T be a simple normal crossing variety (cf. Definition 2.6) and let T = P i ∈ I T i be the irreducible decomposition. Then a stratum of T isan irreducible component of T i ∩ · · · ∩ T i k for some { i , · · · , i k } ⊂ I .Let r be a rational number. The integral part x r y is the largestinteger ≤ r and the fractional part { r } is defined by r − x r y . We put p r q = − x − r y and call it the round-up of r . Let D = P ri =1 d i D i bea Q -divisor where D i is a prime divisor for every i and D i = D j for i = j . We put x D y = P x d i y D i , p D q = P p d i q D i , { D } = P { d i } D i ,and D =1 = P d i =1 D i . Acknowledgments.
The first version of this paper was written inNagoya in 2007. The author was partially supported by the Grant-in-Aid for Young Scientists (A) ♯ ♯ C , the complex number field. Wewill freely make use of the standard notation and definition in [KM].2. Preliminaries
In this section, we prove some preliminary results.2.1.
A criterion of Cohen–Macaulayness.
The main purpose ofthis subsection is to prove Corollary 2.3, which seems to be well knownto experts. Here, we give a global proof based on the Kawamata–Viehweg vanishing theorem for the reader’s convenience. See also thearguments in [F5, 4.3.1].
Lemma 2.1.
Let X be a normal variety with an isolated singularity P ∈ X . Let f : Y → X be any resolution. If X is Cohen–Macaulay,then R i f ∗ O Y = 0 for < i < n − , where n = dim X .Proof. Without loss of generality, we may assume that X is projective.We consider the following spectral sequence E p,q = H p ( X, R q f ∗ O Y ⊗ L − ) ⇒ H p + q ( Y, f ∗ L − )for a sufficiently ample line bundle L on X . By the Kawamata–Viehwegvanishing theorem, H p + q ( Y, f ∗ L − ) = 0 for p + q < n . On the other N LOG CANONICAL SINGULARITIES 5 hand, E p, = H p ( X, L − ) = 0 for p < n since X is Cohen–Macaulay.By using the exact sequence0 → E , → E → E , → E , → E → · · · , we obtain E , ≃ E , = 0 when n ≥
3. This implies R f ∗ O Y = 0.We note that Supp R i f ∗ O Y ⊂ { P } for every i >
0. Inductively, weobtain R i f ∗ O Y ≃ H ( X, R i f ∗ O Y ⊗ L − ) = E ,i ≃ E ,i ∞ = 0 for 0 < i Let X be a normal projective n -fold and let f : Y → X be a resolution. Assume that R i f ∗ O Y = 0 for < i < n − . Then X is Cohen–Macaulay.Proof. It is sufficient to prove H i ( X, L − ) = 0 for any ample line bundle L on X for all i < n (see [KM, Corollary 5.72]). We consider thespectral sequence E p,q = H p ( X, R q f ∗ O Y ⊗ L − ) ⇒ H p + q ( Y, f ∗ L − ) . As before, H p + q ( Y, f ∗ L − ) = 0 for p + q < n by the Kawamata–Viehwegvanishing theorem. By the exact sequence0 → E , → E → E , → E , → E → · · · , we obtain H ( X, L − ) = 0 and H ( X, L − ) = 0 if n ≥ 3. Inductively,we can check that H i ( X, L − ) = E i, ≃ E i, ∞ = 0 for i < n . We finishthe proof. (cid:3) Combining the above two lemmas, we obtain the next corollary. Corollary 2.3. Let P ∈ X be a normal isolated singularity and let f : Y → X be a resolution. Then X is Cohen–Macaulay if and only if R i f ∗ O Y = 0 for < i < n − , where n = dim X .Proof. We shrink X and assume that X is affine. Then we compactify X and may assume that X is projective. Therefore, we can applyLemmas 2.1 and 2.2. (cid:3) Basic properties of dlt pairs. In this subsection, we prove sup-plementary results on dlt pairs. For the definition of dlt pairs, see [KM,Definition 2.37, Theorem 2.44]. See also [F4] for details of singularitiesof pairs.The following proposition generalizes [FA, 17.5 Corollary], where itwas only proved that S is semi-normal and S . In the subsequentsections, we will use the arguments in the proof of Proposition 2.4. OSAMU FUJINO Proposition 2.4 (cf. [F5, Theorem 4.4]) . Let ( X, ∆) be a dlt pair andlet x ∆ y =: S = S + · · · + S k be the irreducible decomposition. We put T = S + · · · + S l for ≤ l ≤ k . Then T is semi-normal, Cohen–Macaulay, and has only Du Bois singularities.Proof. We put B = { ∆ } . Let f : Y → X be a resolution such that K Y + S ′ + B ′ = f ∗ ( K X + S + B ) + E with the following properties: (i) S ′ (resp. B ′ ) is the strict transform of S (resp. B ), (ii) Supp( S ′ + B ′ ) ∪ Exc( f ) and Exc( f ) are simple normal crossing divisors on Y , (iii) f isan isomorphism over the generic point of every lc center of ( X, S + B ),and (iv) p E q ≥ 0. We write S = T + U . Let T ′ (resp. U ′ ) be the stricttransform of T (resp. U ) on Y . We consider the following short exactsequence0 → O Y ( − T ′ + p E q ) → O Y ( p E q ) → O T ′ ( p E | T ′ q ) → . Since − T ′ + E ∼ Q ,f K Y + U ′ + B ′ and E ∼ Q ,f K Y + S ′ + B ′ , we have − T ′ + p E q ∼ Q ,f K Y + U ′ + B ′ + {− E } and p E q ∼ Q ,f K Y + S ′ + B ′ + {− E } .By the vanishing theorem of Reid–Fukuda type (see, for example, [F5,Lemma 4.10]), R i f ∗ O Y ( − T ′ + p E q ) = R i f ∗ O Y ( p E q ) = 0for every i > 0. Note that we used the assumption that f is an isomor-phism over the generic point of every lc center of ( X, S + B ). Therefore,we have 0 → f ∗ O Y ( − T ′ + p E q ) → O X → f ∗ O T ′ ( p E | T ′ q ) → R i f ∗ O T ′ ( p E | T ′ q ) = 0 for all i > 0. Note that p E q is effectiveand f -exceptional. Thus, O T ≃ f ∗ O T ′ ≃ f ∗ O T ′ ( p E ′ | T ′ q ). Since T ′ is asimple normal crossing divisor, T is semi-normal. By the above van-ishing result, we obtain Rf ∗ O T ′ ( p E | T ′ q ) ≃ O T in the derived category.Therefore, the composition O T → Rf ∗ O T ′ → Rf ∗ O T ′ ( p E | T ′ q ) ≃ O T isa quasi-isomorphism. Apply R H om T ( , ω • T ) to the quasi-isomorphism O T → Rf ∗ O T ′ → O T . Then the composition ω • T → Rf ∗ ω • T ′ → ω • T isa quasi-isomorphism by the Grothendieck duality. By the vanishingtheorem (see, for example, [F5, Lemma 2.33]), R i f ∗ ω T ′ = 0 for i > h i ( ω • T ) ⊆ R i f ∗ ω • T ′ ≃ R i + d f ∗ ω T ′ , where d = dim T = dim T ′ .Therefore, h i ( ω • T ) = 0 for i > − d . Thus, T is Cohen–Macaulay. Thisargument is the same as the proof of Theorem 1 in [K2]. Since T ′ is asimple normal crossing divisor, T ′ has only Du Bois singularities. Thequasi-isomorphism O T → Rf ∗ O T ′ → O T implies that T has only DuBois singularities (cf. [K1, Corollary 2.4]). Since T ′ is a simple normalcrossing divisor on Y and ω T ′ is an invertible sheaf on T ′ , every asso-ciated prime of ω T ′ is the generic point of some irreducible component N LOG CANONICAL SINGULARITIES 7 of T ′ . By f , every irreducible component of T ′ is mapped birationallyonto an irreducible component of T . Therefore, f ∗ ω T ′ is torsion-freeon T . Since the composition ω T → f ∗ ω T ′ → ω T is an isomorphism,we obtain f ∗ ω T ′ ≃ ω T . It is because f ∗ ω T ′ is torsion-free and f ∗ ω T ′ isgenerically isomorphic to ω T . By the Grothendieck duality, Rf ∗ O T ′ ≃ R H om T ( Rf ∗ ω • T ′ , ω • T ) ≃ R H om T ( ω • T , ω • T ) ≃ O T . So, R i f ∗ O T ′ = 0 for all i > (cid:3) We obtain the following vanishing theorem in the proof of Proposi-tion 2.4. Corollary 2.5. Under the notation in the proof of Proposition 2.4 , R i f ∗ O T ′ = 0 for every i > and f ∗ O T ′ ≃ O T . We close this subsection with a useful lemma for simple normal cross-ing varieties. Definition 2.6 (Normal crossing and simple normal crossing varieties) . A variety X has normal crossing singularities if, for every closed point x ∈ X , b O X,x ≃ C [[ x , · · · , x N ]]( x · · · x k )for some 0 ≤ k ≤ N , where N = dim X . Furthermore, if each irre-ducible component of X is smooth, X is called a simple normal crossing variety. Lemma 2.7. Let f : V → V be a birational morphism between pro-jective simple normal crossing varieties. Assume that there is a Zariskiopen subset U ( resp. U ) of V ( resp. V ) such that U ( resp. U ) con-tains the generic point of any stratum of V ( resp. V ) and that f in-duces an isomorphism between U and U . Then R i f ∗ O V = 0 for every i > and f ∗ O V ≃ O V .Proof. We can write K V = f ∗ K V + E such that E is f -exceptional. We consider the following commutativediagram V ν f ν −−−→ V ν ν y y ν V −−−→ f V where ν : V ν → V and ν : V ν → V are the normalizations. We canwrite K V ν + Θ = ν ∗ K V and K V ν + Θ = ν ∗ K V , where Θ and Θ are OSAMU FUJINO the conductor divisors. By pulling back K V = f ∗ K V + E to V ν by ν ,we have K V ν + Θ = ( f ν ) ∗ ( K V ν + Θ ) + ν ∗ E. Note that V ν is smooth and Θ is a reduced simple normal crossingdivisor on V ν . By the assumption, f ν is an isomorphism over thegeneric point of any lc center of the pair ( V ν , Θ ). Therefore, ν ∗ E iseffective since K V ν + Θ is Cartier. Thus, we obtain that E is effec-tive. We can easily check that f has connected fibers by the assump-tions. Since V is semi-normal and satisfies Serre’s S condition, wehave O V ≃ f ∗ O V and f ∗ O V ( K V ) ≃ O V ( K V ). On the other hand,we obtain R i f ∗ O V ( K V ) = 0 for every i > Rf ∗ O V ( K V ) ≃ O V ( K V ) in the derived category. Since V and V are Gorenstein, we have Rf ∗ O V ≃ O V in the derived categoryby the Grothendieck duality (cf. the proof of Proposition 2.4). (cid:3) Dlt blow-ups. Let us recall the notion of dlt blow-ups . Theorem2.8 was first obtained by Christopher Hacon (cf. [F7, Section 10]). Fora simplified proof, see [F6, Section 4]. Theorem 2.8 (Dlt blow-up) . Let ( X, ∆) be a quasi-projective lc pair.Then we can construct a projective birational morphism f : Y → X such that K Y + ∆ Y = f ∗ ( K X + ∆) with the following properties. (a) ( Y, ∆ Y ) is a Q -factorial dlt pair. (b) a ( E, X, ∆) = − for every f -exceptional divisor E .When ( X, ∆) is dlt, we can make f small and an isomorphism overthe generic point of every lc center of ( X, ∆) . Note that Theorem 2.8 was proved by the minimal model programwith scaling (cf. [BCHM]).As a corollary of Theorem 2.8, we obtain the following useful lemma. Lemma 2.9. Let P ∈ X be an isolated lc singularity with index one,where X is quasi-projective. Then there exists a projective birationalmorphism g : Z → X such that K Z + D = g ∗ K X , ( Z, D ) is a Q -factorial dlt pair, g is an isomorphism outside P , and D is a reduceddivisor on Z . Remark 2.10. If P ∈ X is Q -factorial, then f − ( P ) is a divisor. So,we have Supp D = f − ( P ). In general, we have only Supp D ⊂ f − ( P ).For non-degenerate isolated hypersurface log canonical singularities,we can use the toric geometry to construct dlt blow-ups as in Lemma2.9 (see [FS, Section 6]). N LOG CANONICAL SINGULARITIES 9 Dlt pairs with torsion log canonical divisor This section is a supplement to [F1, Section 2] and [F2, Section 2].We introduce a new invariant for dlt pairs with torsion log canonicaldivisor. Definition 3.1. Let ( X, D ) be a projective dlt pair such that K X + D ∼ Q 0. We put e µ = e µ ( X, D ) = min { dim W | W is an lc center of ( X, D ) } . It is related to the invariant µ , which is defined in [F2] and will playimportant roles in the subsequent sections. See 4.11 below. Remark 3.2. By [CKP, Theorem 1] or [G, Theorem 1.2], K X + D ≡ K X + D ∼ Q Proposition 3.3. Let ( X, D ) be a projective dlt pair such that K X + D ∼ Q . Let W be any minimal lc center of ( X, D ) . Then dim W = e µ ( X, D ) . Moreover, all the minimal lc centers of ( X, D ) are birationaleach other and x D y has at most two connected components.Sketch of the proof. By Theorem 2.8, we may assume that X is Q -factorial. The induction on dimension and [F1, Proposition 2.1] impliesthe desired properties. More precisely, all the minimal lc centers are B -birational each other (cf. [F1, Definition 1.5]). Note that Proof ofClaims in the proof of [F1, Lemma 4.9] may help us understand thisproposition. (cid:3) The next lemma is new. We will use it in Section 4. Lemma 3.4. Let ( X, D ) be an n -dimensional projective dlt pair suchthat K X + D ∼ Q . Assume that x D y = 0 . Then there exists anirreducible component D of x D y such that h i ( X, O X ) ≤ h i ( D , O D ) for every i .Proof. By using the dlt blow-up (cf. Theorem 2.8), we can constructa small projective Q -factorialization of X . So, by replacing X withits Q -factorialization, we may assume that X is Q -factorial. By theassumption, K X + D − ε x D y is not pseudo-effective for 0 < ε ≪ 1. Let H be an effective ample Q -divisor on X such that K X + D − ε x D y + H is nef and klt. Apply the minimal model program on K X + D − ε x D y with scaling of H . Then we obtain a sequence of divisorial contractionsand flips: X = X X · · · X k , and an extremal Fano contraction ϕ : X k → Z (cf. [F6, Section 2]). Bythe construction, there is an irreducible component D of x D y such thatthe strict transform D ′ of D on X k dominates Z . Since X and X k haveonly rational singularities, we have h i ( X, O X ) = h i ( X k , O X k ) for every i . Since R i ϕ ∗ O X k = 0 for every i > 0, we have h i ( X k , O X k ) = h i ( Z, O Z )for every i . Since D and Z have only rational singularities (cf. [F3,Corollary 1.5]), h i ( Z, O Z ) ≤ h i ( D , O D ) for every i (see, for exam-ple, [PS, Theorem 2.29]). Therefore, we have the desired inequality h i ( X, O X ) ≤ h i ( D , O D ) for every i . (cid:3) Example 3.5. Let X = P and let D be an elliptic curve on X = P .Then ( X, D ) is a projective dlt pair such that K X + D ∼ 0. In thiscase, h ( X, O X ) = 0 < h ( D, O D ) = 1.By combining the above results, we obtain the next proposition. Proposition 3.6. Let ( X, D ) be a projective dlt pair such that K X + D ∼ Q . We assume that e µ ( X, D ) = 0 . Then h i ( X, O X ) = 0 for every i > . Moreover, X is rationally connected.Proof. If dim X = 1, then the statement is trivial since X ≃ P . Fromnow on, we assume that dim X ≥ 2. Since e µ ( X, D ) = 0, we obtain that( X, D ) is not klt. Thus we know x D y = 0. Let D be any irreduciblecomponent of x D y . By adjunction, we obtain ( K X + D ) | D = K D + B such that ( D , B ) is dlt, K D + B ∼ Q 0, and e µ ( D , B ) = 0 by Proposi-tion 3.3. By the induction on dimension, we know that every irreduciblecomponent D of x D y is rationally connected and h i ( D , O D ) = 0 forevery i > 0. Thus, by Lemma 3.4, we have that h i ( X, O X ) = 0 for every i > 0. In the proof of Lemma 3.4, Z has only log terminal singularitiesby [F3, Corollary 4.5]. Since D is rationally connected, so is Z by [HM,Corollary 1.5]. On the other hand, the general fiber of ϕ : X k → Z isrationally connected (cf. [Z, Theorem 1] and [HM, Corollaries 1.3 and1.5]). By [GHS, Corollary 1.3], X k is rationally connected. Thus, X isrationally connected by [HM, Corollary 1.5]. (cid:3) By Proposition 3.6, we obtain a corollary: Corollary 3.7. Corollary 3.7. Let ( X, D ) be a projective dlt pair such that K X + D ∼ Q . Let f : Y → X be any resolution such that K Y + D Y = f ∗ ( K X + D ) and that Supp D Y is a simple normal crossing divisor on Y . Assumethat e µ ( X, D ) = 0 . Then every stratum of D =1 Y is rationally connected. N LOG CANONICAL SINGULARITIES 11 Moreover, h i ( W, O W ) = 0 for every i > where W is a stratum of D =1 Y .Proof. Let W be a stratum of D =1 Y . Let π : Y ′ → Y be a blow-upat W and let E W be the exceptional divisor of π . Then it is suffi-cient to prove that E W is rationally connected and h i ( E W , O E W ) = 0for every i > 0. Therefore, by replacing Y with Y ′ , we may assumethat W is an irreducible component of D =1 Y . We can construct a dltblow-up f ′ : Y ′ → X such that K Y ′ + D Y ′ = f ′∗ ( K X + D ) and that f ′− ◦ f : Y Y ′ is an isomorphism at the generic point of W (cf. [F6, Section 6]). Since K Y ′ + D Y ′ ∼ Q e µ ( Y ′ , D Y ′ ) = 0 (cf. [F1, Claim ( A n )]), we see that W ′ , the stricttransform of W , is rationally connected and h i ( W ′ , O W ′ ) = 0 for every i > W is rationally connected (cf. [HM,Corollary 1.5]) and h i ( W, O W ) = 0 for every i > (cid:3) Isolated log canonical singularities with index one In this section, we consider when an isolated log canonical singularitywith index one is Cohen–Macaulay or not. Let P ∈ X be an n -dimensional isolated lc singularity with indexone. By the algebraization theorem (cf. [HR], [A1, Corollary 1.6], and[A2, Theorem 3.8]), we always assume that X is an algebraic varietyin this paper (see also [I2, Theorems 3.2.3 and 3.2.4]). Assume that P ∈ X is not lt. We consider a resolution f : Y → X such that (i) f is an isomorphism outside P ∈ X , and (ii) f − ( P ) is a simple normalcrossing divisor on Y . In this setting, we can write K Y = f ∗ K X + F − E, where F and E are both effective Cartier divisors without commonirreducible components. In particular, E is a reduced simple normalcrossing divisor on Y . Lemma 4.2. The cohomology group H i ( E, O E ) is independent of f for every i .Proof. Let f ′ : Y ′ → X be another resolution with K Y ′ = f ′∗ K X + F ′ − E ′ as in 4.1. By the weak factorization theorem (see [M, Theorem 5-4-1] or [AKMW, Theorem 0.3.1(6)]), we may assume that ϕ : Y ′ → Y is ablow-up whose center C ⊂ Supp f − ( P ) is smooth, irreducible, and hassimple normal crossing with Supp f − ( P ). It means that at each point p ∈ Supp f − ( P ) there exists a regular coordinate system { x , · · · , x n } in a neighborhood p ∈ U p such thatSupp f − ( P ) ∩ U p = (Y j ∈ J x j = 0 ) and C ∩ U p = { x i = 0 for i ∈ I } for some subsets I, J ⊂ { , · · · , n } .Thus, we can directly check that H i ( E, O E ) ≃ H i ( E ′ , O E ′ ) for every i . (cid:3) Let Γ be the dual complex of E and let | Γ | be the topological real-ization of Γ. Note that the vertices of Γ correspond to the components E i , the edges correspond to E i ∩ E j , and so on, where E = P i E i isthe irreducible decomposition of E . More precisely, E defines a coni-cal polyhedral complex ∆ (see [KKMS, Chapter II, Definition 5]). By[KKMS, p.70 Remark], we get a compact polyhedral complex ∆ from∆. The dual complex Γ of E is essentially the same as this compactpolyhedral complex ∆ and | Γ | = | ∆ | as topological spaces. See theconstruction of the dual complex in [S] and [P, Section 2] for details.Therefore, we obtain the following lemma. Lemma 4.4. The dual complex Γ is well defined and | Γ | is independentof f .Proof. As we explained above, the well-definedness of Γ is in [KKMS,Chapter II]. By the weak factorization theorem (see [M, Theorem 5-4-1]or [AKMW, Theorem 0.3.1(6)]), we can easily check that the topolog-ical realization | Γ | does not depend on f . (cid:3) Remark 4.5. The paper [S] discusses the dual complex of Supp f − ( P )by the same method. Case 1) in the proof of [S, Lemma] is sufficient forour purposes. Note that we treat the dual complex Γ of E . In general,Supp E ( Supp f − ( P ). Let g : Z → X be a projective birational morphism as in Lemma2.9. Then we have 0 → O Z ( − D ) → O Z → O D → 0. By the vanishingtheorem, we obtain R i g ∗ O Z ( K Z ) = 0 for every i > 0. Therefore, wehave R i g ∗ O Z ≃ R i g ∗ O D ≃ H i ( D, O D )for every i > 0. We note that D is connected since O X ≃ g ∗ O Z → g ∗ O D is surjective. By applying Corollary 2.5, we can construct a resolution h : Y → Z such that K Y + E − F = h ∗ ( K Z + D ) = f ∗ K X , where F and E are both effective Cartier divisors without commonirreducible components, Supp E is a simple normal crossing divisor, N LOG CANONICAL SINGULARITIES 13 f = g ◦ h , h is an isomorphism outside g − ( P ), h is an isomorphismover the generic point of any lc center of ( Z, D ), R i h ∗ O E = 0 forevery i > 0, and h ∗ O E ≃ O D . Therefore, H i ( D, O D ) ≃ H i ( E, O E )for every i . Apply the principalization to the defining ideal sheaf I of f − ( P ). Then we obtain a sequence of blow-ups whose centers havesimple normal crossing with E (cf. [K1, Theorem 3.35]). In this process, H i ( E, O E ) does not change for every i (cf. the proof of Lemma 4.2).Therefore, we may assume that f − ( P ) is a divisor on Y . We furthertake a sequence of blow-ups whose centers have simple normal crossingwith E . Then we can make Supp f − ( P ) a simple normal crossingdivisor on Y (cf. [BEV, Corollary 7.9] or [K2, Proposition 6]). We notethat we may assume that f is an isomorphism outside P ∈ X . We alsonote that R i g ∗ O Z ≃ R i f ∗ O Y for every i because Z has only rationalsingularities. So, we obtain the next proposition. Proposition 4.7. Let f : Y → X be a resolution as in . Then R i f ∗ O Y ≃ H i ( E, O E ) for every i > . Therefore, P ∈ X is Cohen–Macaulay, equivalently, P ∈ X is Gorenstein, if and only if H i ( E, O E ) =0 for < i < n − .Proof. It is a direct consequence of Lemma 4.2 and Corollary 2.3 by4.6. (cid:3) Remark 4.8. In 4.6, ( K Z + D ) | D = K D ∼ 0. Therefore, H n − ( D, O D )is dual to H ( D, O D ), where n = dim X . So, R n − g ∗ O Z ≃ C ( P ). Thus, P ∈ X is not a rational singularity. Remark 4.9. Shihoko Ishii proves R i f ∗ O Y ≃ H i ( f − ( P ) red , O f − ( P ) red )for every i > Remark 4.10. Let f : Y → X with K Y + E = f ∗ K X + F be as in 4.1.Let H be an effective f -ample Q -divisor on Y such that ( Y, E + H ) isdlt and that K Y + E + H is nef over X . We can run the minimal modelprogram on K Y + E over X with scaling of H . Then we obtain a dltblow-up f ′ : Y ′ → X such that ( Y ′ , E ′ ) is a Q -factorial dlt pair andthat K Y ′ + E ′ = f ′∗ K X where E ′ is the pushforward of E on Y ′ (cf. [F6,Section 4]). We note that each step of the minimal model program Y Y Y · · · Y ′ is an isomorphism at the generic point of any lc center of ( Y, E ). By4.6, R i f ∗ O Y ≃ R i f ′∗ O Y ′ ≃ R i f ′∗ O E ′ ≃ H i ( E ′ , O E ′ ) for every i > 0. Bytaking a common resolution W α ~ ~ }}}}}}}} β BBBBBBBB Y / / _______ Y ′ such that α (resp. β ) is an isomorphism over the generic point of any lccenter of ( Y, E ) (resp. ( Y ′ , E ′ )) and that Exc( α ), Exc( β ), and Exc( α ) ∪ Exc( β ) ∪ Supp α − ∗ E are simple normal crossing divisors on W , we caneasily check that H i ( E, O E ) ≃ H i ( E ′ , O E ′ )for every i because Rα ∗ O T ≃ O E and Rβ ∗ O T ≃ O E ′ (cf. Corollary2.5). Note that K W + ∆ = α ∗ ( K Y + E ) and K W + ∆ = β ∗ ( K Y ′ + E ′ )with ∆ =11 = T = ∆ =12 such that T is a reduced simple normal crossingdivisor on W . Therefore, H i ( E, O E ) ≃ H i ( E ′ , O E ′ ) ≃ R i f ∗ O Y for i > E = P i E i be the irreducible decomposition and let E ′ = P i E ′ i be the corresponding irreducible decomposition. Let E i be an irre-ducible component of E and let T i be the strict transform of E i on W . By applying the connectedness lemma (cf. [KM, Theorem 5.48])to α : T i → E i and β : T i → E ′ i , we know that the numberof the connected components of P i = i E i | E i coincides with that of P i = i E ′ i | E ′ i . Therefore, P i = i E i | E i has at most two connected com-ponents by applying Proposition 3.3 to ( E ′ i , P i = i E ′ i | E ′ i ). Note that( E ′ i , P i = i E ′ i | E ′ i ) is dlt and K E ′ i + P i = i E ′ i | E ′ i ∼ (Invariant µ ) . Let P ∈ X be an isolated lc singularity with indexone which is not lt. Let g : Z → X be a projective birational morphismsuch that K Z + D = g ∗ K X and that ( Z, D ) is a Q -factorial dlt pair.We define µ = µ ( P ∈ X ) = min { dim W | W is an lc center of ( Z, D ) } . This invariant µ was first introduced in [F2, Definition 4.12]. Let D = P i D i be the irreducible decomposition. Then K D i + ∆ i := ( K Z + D ) | D i ∼ D i , ∆ i ) is dlt. By applying Proposition 3.3 to each( D i , ∆ i ), every minimal lc center of ( Z, D ) is µ -dimensional and all theminimal lc centers are birational each other. Note that D is connected. N LOG CANONICAL SINGULARITIES 15 Let g ′ : Z ′ → X be another projective birational morphism such that K Z ′ + D ′ = g ′∗ K X and that ( Z ′ , D ′ ) is a Q -factorial dlt pair. Then itis easy to see that ( Z, D ) ( Z ′ , D ′ ) is B -birational. This means thatthere is a common resolution W α ~ ~ }}}}}}}} β BBBBBBBB Z / / _______ Z ′ such that α ∗ ( K Z + D ) = β ∗ ( K Z ′ + D ′ ). Then we can easily check thatmin { dim W | W is an lc center of ( Z, D ) } = min { dim W ′ | W ′ is an lc center of ( Z ′ , D ′ ) } . See, for example, the proof of [F1, Lemma 4.9]. Therefore, µ ( P ∈ X )is well-defined. Let f : Y → X with K Y = f ∗ K X + F − E be as in 4.1.Then it is easy to see that µ = µ ( P ∈ X ) = min { dim W | W is a stratum of E } by Remark 4.10.Now, the following theorem is not difficult to prove. Theorem 4.12. We use the notation in . We assume µ ( P ∈ X ) = 0 . Then H i ( E, O E ) ≃ H i ( | Γ | , C ) . Therefore, P ∈ X is Cohen–Macaulay, equivalently, P ∈ X is Gorenstein, if and only if H i ( | Γ | , C ) = (cid:26) C for i = 0 , n − , otherwise.Proof. We use the spectral sequence in 4.13 to calculate H i ( E, O E ). ByCorollary 3.7, H q ( E [ p ] , O E [ p ] ) = 0 for every q > 0. Therefore, we obtain E i, ≃ H i ( | Γ | , C ) for every i and the spectral sequence degenerates at E . Thus we have H i ( E, O E ) ≃ H i ( | Γ | , C ) for every i . (cid:3) Let E be a simple normal crossing variety and let E = P i E i bethe irreducible decomposition. We put E [0] = ` i E i , E [1] = ` i,j ( E i ∩ E j ), · · · , E [ p ] = ` i , ··· ,i p ( E i ∩ · · · ∩ E i p ), · · · . Let a p : E [ p ] → E be theobvious map. Then it is well known that( a ) ∗ O E [0] → ( a ) ∗ O E [1] → · · · → ( a p ) ∗ O E [ p ] → · · · is a resolution of O E . By taking the associated hypercohomology, weobtain a spectral sequence E p,q = H q ( E [ p ] , O E [ p ] ) ⇒ H p + q ( E, O E ) . We close this section with the following obvious two propositions. Proposition 4.14. We assume that the dimension of X is ≥ . By theabove spectral sequence, if P ∈ X is Cohen–Macaulay, then H ( | Γ | , C ) =0 .Proof. By the spectral sequence in 4.13, it is easy to see that H ( | Γ | , C ) =0 implies H ( E, O E ) = 0. (cid:3) Proposition 4.15. Let P ∈ X be an n -dimensional isolated lc singu-larity with index one which is not lt. If P ∈ X is Cohen–Macaulay,then χ ( O E ) := X i ( − i h i ( E, O E ) = 1 + ( − n − = X p,q ( − p + q dim H q ( E [ p ] , O E [ p ] ) . Remark 4.16. Tsuchihashi’s cusp singularities (cf. [T1] and [T2]) giveus many examples of three dimensional index one isolated lc singular-ities with µ = 0 which are not Cohen–Macaulay.5. Ishii’s Hodge theoretic invariant In this section, we give a Hodge theoretic characterization of ourinvariant µ . It shows that our invariant µ coincides with Ishii’s Hodgetheoretic invariant.Let us quickly recall Ishii’s definition of singularities of type (0 , i ).For the details, see [I2, Section 7] and [I4, 2.6 and Definition 2.7]. (Type (0 , i ) singularities due to Shihoko Ishii) . Let P ∈ X be an n -dimensional isolated lc singularity with index one which is not lt. Let f : Y → X be a resolution such that K Y = f ∗ K X + F − E as in 4.1. Shihoko Ishii proves that H n − ( E, O E ) = C (cf. Proposition4.7 and Remark 4.8). In [I2, Definition 7.4.5] and [I4, Definition 2.7],she defines that the singularity P ∈ X is of type (0 , i ) ifGr Wi H n − ( E, O E ) = 0 . Note that E is a projective simple normal crossing variety, W is theweight filtration of the natural mixed Hodge structure on H n − ( E, C ),and that H n − ( E, O E ) ≃ Gr F H n − ( E, C ) where F is the natural Hodgefiltration. Therefore, we haveGr Wk H n − ( E, O E ) ≃ Gr Wk Gr F H n − ( E, C ) ≃ Gr F Gr Wk H n − ( E, C ) N LOG CANONICAL SINGULARITIES 17 By Deligne’s theory of mixed Hodge structures, we know that 0 ≤ i ≤ n − µ ( P ∈ X ) = i where P ∈ X is of type (0 , i ).The following theorem corresponds to [I1, Theorem 4.3] in our frame-work. For the definition of sdlt pairs , see [F1, Definition 1.1]. Let( X, ∆) be an sdlt pair. Then X is S , normal crossing in codimensionone, and every irreducible component of X is normal. Let V be sdlt.Then there is the smallest Zariski closed subset Z of V such that V \ Z is a simple normal crossing variety and the codimension of Z in V is ≥ 2. We define a stratum of V as the closure of a stratum of V \ Z . Theorem 5.2. Let V be an m -dimensional connected projective sdltvariety such that K V ∼ . Let f : V ′ → V be a projective birationalmorphism from a simple normal crossing variety V ′ . Assume that thereis a Zariski open subset U ′ ( resp. U ) of V ′ ( resp. V ) such that U ′ ( resp. U ) contains the generic point of any stratum of V ′ ( resp. V ) andthat f induces an isomorphism between U ′ and U . We further assumethat the exceptional locus Exc( f ) is a simple normal crossing divisoron V ′ ( cf. [F5, Definition 2.11]) and that K V ′ = f ∗ K V + E where E is effective. Then H m ( V ′ , O V ′ ) = C . Moreover, we obtainthat Gr F Gr Wk H m ( V ′ , C ) ≃ Gr Wk Gr F H m ( V ′ , C ) ≃ Gr Wk H m ( V ′ , O V ′ )= ( C if k = µ otherwisewhere µ is the dimension of the minimal stratum of V ′ . Note that F isthe Hodge filtration and W is the weight filtration of the natural mixedHodge structure on H m ( V ′ , C ) .Proof. First we prove that H m ( V ′ , O V ′ ) = C . Step 1. Since V is simple normal crossing in codimension one and S , V is semi-normal. We can easily check that f has connected fibers bythe assumptions. Therefore, we obtain f ∗ O V ′ ≃ O V . We note that E is f -exceptional by the assumptions. Since E is effective, f -exceptional,and V satisfies Serre’s S condition, we see that f ∗ O V ′ ( E ) ≃ O V . Onthe other hand, we obtain R i f ∗ O V ′ ( E ) ≃ R i f ∗ O V ′ ( K V ′ ) = 0 for every i > Rf ∗ O V ′ ( E ) ≃ O V inthe derived category. By the same arguments as in the proof of Proposi-tion 2.4, we obtain that V is Cohen–Macaulay. Moreover, R i f ∗ O V ′ = 0for every i > f ∗ O V ′ ≃ O V .Thus, H m ( V ′ , O V ′ ) ≃ H m ( V, O V ) = C . We note that K V ∼ V is Cohen–Macaulay.We use the induction on dimension for the latter statement. Thestatement is obvious for a 0-dimensional variety. Step 2. When V is irreducible, the statement is obvious. It is because V ′ is a smooth connected projective variety. So, H m ( V ′ , C ) has thenatural pure Hodge structure of weight m . Step 3. From now on, we assume that V is reducible. Let V ′ be anirreducible component of V ′ and let V be the corresponding irreduciblecomponent of V . We write V ′ = V ′ ∪ V ′ and V = V ∪ V . Considerthe Mayer-Vietoris exact sequence: H m − ( V ′ ∩ V ′ , O V ′ ∩ V ′ ) δ → H m ( V ′ , O V ′ )( ♠ ) → H m ( V ′ , O V ′ ) ⊕ H m ( V ′ , O V ′ ) . By the Serre duality, H m ( V ′ i , O V ′ i ) is dual to H ( V ′ i , O V ′ i ( K V ′ i )). Weput f i = f | V ′ i for i = 1 , 2. We can write K V ′ i + V ′ j | V ′ i = f ∗ i ( K V i + V j | V i ) + E | V ′ i ∼ E | V ′ i =: F i for { i, j } = { , } where F i is an effective f i -exceptional divisor. Wenote that K V i + V j | V i = K V | V i ∼ 0. Let H be an ample Cartier divisoron V . Then ( f ∗ i H ) m − · K V ′ i < V ′ j | V ′ i = 0 for i = 1 , 2. Thus H ( V ′ i , O V ′ i ( K V ′ i )) = 0 for i = 1 , 2. This means that H m ( V ′ i , O V ′ i ) = 0for i = 1 , 2. So the last term in ( ♠ ) is zero. Therefore, we obtain thatGr Wk H m − ( V ′ ∩ V ′ , O V ′ ∩ V ′ ) → Gr Wk H m ( V ′ , O V ′ )is surjective for every k . We note that V ′ ∩ V ′ is an ( m − V ′ ∩ V ′ has at mosttwo connected components by Proposition 3.3 and [KM, Theorem 5.48].Note that ( V , V | V ) is dlt and K V + V | V ∼ 0. Moreover, each con-nected component of V ′ ∩ V ′ satisfies the assumptions of this theoremand the dimension of the minimal stratum of each connected compo-nent of V ′ ∩ V ′ is also µ . Therefore, by the induction on dimension, weobtain that Gr Wk H m ( V ′ , O V ′ ) = 0 if and only if k = µ .We obtain all the desired results. (cid:3) N LOG CANONICAL SINGULARITIES 19 Remark 5.3. By Step 3 in the proof of Theorem 5.2, we obtain thefollowing description. Let C ′ be any minimal stratum of V ′ . Then weobtain an isomorphism C = H µ ( C ′ , O C ′ ) ≃ δ µ · · · ≃ δ k · · · ≃ δ m − H m ( V ′ , O V ′ ) = C where each δ k is the connecting homomorphism of a suitable Mayer–Vietoris exact sequence for µ ≤ k ≤ m − 1. Note that C has onlycanonical singularities with K C ∼ 0, where C = f ( C ′ ). Remark 5.4 (Semi-stable minimal models for varieties with trivialcanonical divisor) . Let f : X → Y be a projective surjective morphismfrom a smooth quasi-projective variety X to a smooth quasi-projectivecurve Y . Assume that f is smooth over Y \ P , K f − ( Q ) ∼ Q ∈ Y \ P , and f ∗ P is a reduced simple normal crossing divisor on X . Then we obtain a relative good minimal model f ′ : X ′ → Y of f : X → Y by [F6, Theorem 1.1]. Then the special fiber S = f ′∗ P isan sdlt variety with K S ∼ 0. So, we can apply Theorem 5.2 to S .As an application of Theorem 5.2, we obtain the following theorem. Theorem 5.5. Let P ∈ X be an isolated lc singularity with index onewhich is not lt. Then P ∈ X is of type (0 , i ) if and only if µ ( P ∈ X ) = i .Proof. We use the notations in Remark 4.10. Let f : Y → X be as in4.1. First, we apply Theorem 5.2 to β : T → E ′ . Then we obtainGr Wµ H n − ( T, O T ) = 0where µ = µ ( P ∈ X ). Next, we consider α : T → E . Let C be aminimal stratum of E and let C ′ be the corresponding stratum of T .By Step 3 in the proof of Theorem 5.2, Remark 5.3, and Lemma 2.7,we can construct the following commutative diagram. C = H µ ( C ′ , O C ′ ) δ −−−→ H n − ( T, O T ) = C α | ∗ C ′ x x α | ∗ T C = H µ ( C, O C ) δ −−−→ H n − ( E, O E ) = C Note that δ and δ are isomorphisms, which are the compositions of theconnecting homomorphisms of suitable Mayer–Vietoris exact sequences(cf. Remark 5.3), and that α | ∗ C ′ and α | ∗ T are isomorphisms (cf. Lemma2.7). By taking Gr W , we obtain thatGr Wµ H n − ( E, O E ) = 0 . This means that P ∈ X is of type (0 , µ ). We note thatGr Wµ H µ ( C, O C ) = H µ ( C, O C )since C is smooth and projective. (cid:3) We note that Theorem 5.5 also follows from [I2, Proposition 7.4.8]and [I3] (see [F2, Remark 4.13]).Anyway, by Theorem 5.5, our approach in [F2] and this paper iscompatible with Ishii’s theory developed in [I1], [I2], and [I4]. References [AKMW] D. Abramovich, K. Karu, K. 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