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DB PAIRS AND VANISHING THEOREMS SÁNDORJKOVÁCS 1 1 0 2 In memoriam Professor Masayoshi Nagata n a J ABSTRACT. The main purposeof thisarticle is to define the notionof 3 Du Bois singularities for pairs and proving a vanishing theorem using 1 thisnewnotion. Themainvanishingtheoremspecializestoanewvan- ishingtheoremforresolutionsoflogcanonialsingularities. ] G A 1. INTRODUCTION . h The class ofrational singularitiesis one ofthe most importantclasses of t singularities. Theiressenceliesinthefactthattheircohomologicalbehavior a m is very similar to that of smooth points. For instance, vanishing theorems [ can be easily extended to varieties with rational singularities. Establishing 2 that a certain class of singularities is rational opens the door to using very v powerfultoolson varietieswiththosesingularities. 6 Du Bois (or DB) singularities are probably somewhat harder to appreci- 3 5 ate at first, but they are equally important. Their main importance comes 3 fromtwofacts: Theyarenottoofarfromrationalsingularities,thatis,they . 9 sharemanyoftheirproperties,buttheclassofDuBoissingularitiesismore 0 inclusivethan that of rational singularities. For instance, log canonical sin- 0 1 gularities are Du Bois, but not necessarily rational. The class of Du Bois : v singularitiesisalsomorestableunderdegeneration. i Recently there has been an effort to extend the notion of rational singu- X larities to pairs. There are at least two approaches; Schwede and Takagi r a [ST08] are dealing with pairs (X,∆) where ⌊∆⌋ = 0 while Kollár and Kovács[KK09]are studyingpairs(X,∆) where ∆isreduced. ThemaingoalofthisarticleistoextendthedefinitionofDuBoissingu- laritiestopairs inthespiritofthelatterapproach. Here isabriefoverviewofthepaper. In section 2 some basic properties of rational and DB singularities are reviewed, a few new ones are introduced, and the DB defect is defined. In Date:January14,2011. 2010MathematicsSubjectClassification. 14J17. Supported in part by NSF Grant DMS-0856185, and the Craig McKibben and Sarah MernerEndowedProfessorshipinMathematicsattheUniversityofWashington. 1 2 SÁNDORJKOVÁCS section 3 I recall the definition and some basic properties of pairs, generalized pairs, and rational pairs. I define the notion of a DB pair and the DB defect of a generalized pair and prove a few basic properties. In section 4 I recall a relevant theorem from Deligne’s Hodge theory and derive a corollary that will be needed later. In section 5 one of the main results is proven. A somewhatweaker versionis the following. See Theorem 5.4 for thestrongerstatement. Theorem 1.1. RationalpairsareDB pairs. Thisgeneralizes[Kov99,TheoremS]and[Sai00,5.4]topairs. Insection 6IprovearathergeneralvanishingtheoremforDBpairsanduseittoderive thefollowingvanishingtheoremforlog canonicalpairs. Theorem 1.2. Let (X,∆) bea Q-factoriallogcanonicalpair,π : X → X e alog resolutionof(X,∆). Let ∆ = (cid:0)π−1⌊∆⌋+Exc (π)(cid:1) . Then e ∗ nklt red Riπ∗OXe(−∆e) = 0 for i > 0. A philosophical consequence one might draw from this theorem is that log canonical pairs are not too far from being rational. One may even view this a vanishing theorem similar to the one in the definition of rational singularities cf. (2.1), (3.4) with a correction term as in vanishing theorems withmultiplierideals. Noticehowever,thatthisisinadualformcompared to Nadel’s vanishing,and hence does not followfrom that, especially since thetarget is notnecessarilyCohen-Macaulay. Theorem 1.2 is also closely related to Steenbrink’s characterization of normalisolatedDuBoissingularities[Ste83,3.6](cf.[DB81,4.13],[KS09, 6.1]). A weaker version of this theorem was the corner stone of a recent result on extending differential forms to a log resolution [GKKP10]. For details on how this theorem may be applied, see the original article. It is possible thatthecurrent theoremwilllead toastrengtheningofthatresult. Definitions and Notation 1.3. Unless otherwise stated, all objects are assumed to be defined over C, all schemes are assumed to be of finite type overC and a morphism means a morphismbetween schemes of finite type overC. If φ : Y → Z is a birational morphism, then Exc(φ) will denote the exceptionalset ofφ. ForaclosedsubschemeW ⊆ X,theidealsheafofW isdenotedbyI orifnoconfusionislikely,thensimplybyI . Fora W⊆X W pointx ∈ X,κ(x) denotestheresiduefield ofO . X,x For morphismsφ : X → B and ϑ : T → B, the symbol X will denote T X × T and φ : X → T theinducedmorphism. In particular, forb ∈ B B T T we write X = φ−1(b). Of course, by symmetry, we also have the notation b DBPAIRSANDVANISHINGTHEOREMS 3 ϑ : T ≃ X → X and if F is an O -module, then F will denote the X X T X T O -moduleϑ∗ F. XT X Let X be a complex scheme (i.e., a scheme of finite type over C) of di- mension n. Let D (X) denote the derived category of filtered complexes filt of O -modules with differentials of order ≤ 1 and D (X) the subcat- X q filt,coh egory of D (X) of complexes K , such that for all i, the cohomology filt q sheaves of Gri K are coherent cf. [DB81], [GNPP88]. Let D(X) and filt D (X) denotethederivedcategories with thesamedefinition except that coh the complexes are assumed to have the trivial filtration. The superscripts +,−,bcarrytheusualmeaning(boundedbelow,boundedabove,bounded). Isomorphisminthesecategoriesisdenotedby ≃ . AsheafF isalsocon- qis q q sidered as a complex F with F0 = F and Fi = 0 for i 6= 0. If K is a q complexinanyoftheabovecategories,thenhi(K )denotesthei-th coho- q mologysheafofK . The rightderived functorofan additivefunctor F, ifit exists,is denoted byRF andRiF isshortforhi◦RF. Furthermore,Hi,Hi,Hi ,andH iwill c Z Z denote RiΓ, RiΓ , RiΓ , and RiH respectively,where Γ is the functor of c Z Z globalsections,Γ isthefunctorofglobalsectionswithpropersupport,Γ c Z isthefunctorofglobalsectionswithsupportintheclosedsubsetZ,andH Z isthefunctorofthesheafoflocalsectionswithsupportintheclosedsubset Z. Notethataccordingtothisterminology,ifφ: Y → X isamorphismand F isacoherentsheafonY,thenRφ F isthecomplexwhosecohomology ∗ sheavesgiveriseto theusualhigherdirect imagesofF. Wewilloftenusethenotionthatamorphismf : A → Binaderivedcate- goryhasaleftinverse. Thismeansthatthereexistsamorphismfℓ : B → A in the same derived category such that fℓ ◦f : A → A is the identity mor- phismofA. I.e., fℓ isaleft inverseoff. Finally, we will also make the following simplificationin notation. First observethatifι : Σ ֒→ X isaclosedembeddingofschemesthenι isexact ∗ andhence Rι = ι . Thisallowsonetomakethefollowingharmless abuse ∗ ∗ of notation: If A ∈ ObD(Σ), then, as usual for sheaves, we will drop ι ∗ fromthenotationoftheobjectι A. Inotherwords,wewill,withoutfurther ∗ warning,considerA an objectinD(X). ACKNOWLEDGMENT. I would like to thank Donu Arapura for explaining some of the intricacies of the relevant Hodge theory to me, Osamu Fujino forhelpfulremarks, andthereferee foruseful comments. 2. RATIONAL AND DU BOIS SINGULARITIES Definition 2.1. Let X be a normal variety and φ : Y → X a resolution of singularities. X is said to have rational singularities if Riφ O = 0 ∗ Y 4 SÁNDORJKOVÁCS for all i > 0, or equivalently if the natural map O → Rφ O is a quasi- X ∗ Y isomorphism. Du Bois singularities are defined via Deligne’s Hodge theory We will need alittlepreparation beforewecan definethem. The starting point is Du Bois’s construction, following Deligne’s ideas, of the generalized de Rham complex, which we call the Deligne-Du Bois complex. Recall, that if X is a smooth complex algebraic variety of di- mension n, then the sheaves of differential p-forms with the usual exterior differentiation give a resolution of the constant sheaf C . I.e., one has a X filtered complexofsheaves, OX d // Ω1X d // Ω2X d // Ω3X d // ... d// ΩnX ≃ ωX, which is quasi-isomorphic to the constant sheaf C via the natural map X C → O given by considering constants as holomorphic functions on X X X. Recall that this complex is not a complex of quasi-coherent sheaves. Thesheavesinthecomplexarequasi-coherent,butthemapsbetweenthem are not O -module morphisms. Notice however that this is actually not X a shortcoming; as C is not a quasi-coherent sheaf, one cannot expect a X resolutionofitin thecategory ofquasi-coherentsheaves. The Deligne-Du Bois complex is a generalization of the de Rham complex to singular varieties. It is a complex of sheaves on X that is quasi- isomorphicto theconstantsheaf C . The terms ofthis complexare harder X to describe but its properties, especially cohomological properties are very similar to the de Rham complex of smooth varieties. In fact, for a smooth variety the Deligne-Du Bois complex is quasi-isomorphic to the de Rham complex,so itisindeed adirect generalization. q The construction of this complex, Ω , is based on simplicial resolu- X tions. The reader interested in the details is referred to the original article [DB81]. Note also that a simplified construction was later obtained in [Car85] and [GNPP88] via the general theory of polyhedral and cubic resolutions. An easily accessible introduction can be found in [Ste85]. Other useful references are the recent book [PS08] and the survey [KS09]. We will actually not use these resolutions here. They are needed for the construction,butif oneis willingto believethelistedproperties (which follow in a rather straightforward way from the construction) then one should be able follow the material presented here. The interested reader should note that recently Schwede found a simpler alternative construction of (part of) the Deligne-Du Bois complex that does not need a simplicial resolution [Sch07]. For applications of the Deligne-Du Bois complex and Du Bois singularities other than the ones listed here see [Ste83], [Kol95, Chapter 12],[Kov99,Kov00b, KSS10,KK10]. DBPAIRSANDVANISHINGTHEOREMS 5 The word “hyperresolution” will refer to either a simplicial, polyhedral, q orcubicresolution. Formally,theconstructionofΩ isthesameregardless X the type of resolution used and no specific aspects of either types will be used. The next theorem lists the basic properties of the Deligne-Du Bois complex: Theorem2.2[DB81]. LetX beacomplexschemeoffinitetype. Thenthere q exists a functorially defined object Ω ∈ ObD (X) such that using the X filt notation q p p Ω := Gr Ω [p], X filt X itsatisfiesthefollowingproperties (2.2.1) q Ω ≃ C . X qis X q (2.2.2) Ω is functorial, i.e., if φ: Y → X is a morphism of complex (_) schemes of finite type, then there exists a natural map φ∗ of filtered complexes q q φ∗: Ω → Rφ Ω X ∗ Y q Furthermore, Ω ∈ Ob(cid:0)Db (X)(cid:1) and if φ is proper, then φ∗ is X filt,coh amorphisminDb (X). filt,coh (2.2.3) Let U ⊆ X bean opensubschemeof X. Then q q Ω (cid:12) ≃ Ω . X(cid:12)U qis U (2.2.4) IfX isproper,thenthereexistsaspectralsequencedegeneratingat E andabuttingtothesingularcohomologyofX: 1 E pq = Hq(X,Ωp ) ⇒ Hp+q(X,C). 1 X (2.2.5) If εq: Xq → X isa hyperresolution,then q q ΩX ≃qisRεq∗ΩXq. In particular,hi(Ωp ) = 0 fori < 0. X (2.2.6) Thereexists anaturalmap, O → Ω0 , compatiblewith(2.2.2). X X (2.2.7) If X isa normalcrossingdivisorin asmoothvariety,then q q Ω ≃ Ω . X qis X In particular, Ωp ≃ Ωp . X qis X (2.2.8) If φ: Y → X isa resolutionof singularities,then ΩdimX ≃ Rφ ω . X qis ∗ Y 6 SÁNDORJKOVÁCS (2.2.9) Let π : X → X be a projective morphism and Σ ⊆ X a reduced e closedsubschemesuchthatπisanisomorphismoutsideofΣ. LetE denote the reduced subscheme of X with support equal to π−1(X). e Then for each p one has an exact triangle of objects in the derived category, ΩpX // ΩpΣ ⊕Rπ∗ΩpXe − // Rπ∗ΩpE +1 // . (2.2.10) Suppose X = Y ∪ Z is the union of two closed subschemes and denote their intersection by W:= Y ∩Z. Then for each p one has anexact triangleofobjectsin thederived category, Ωp // Ωp ⊕Ωp − // Ωp +1 // . X Y Z W ItturnsoutthattheDeligne-DuBoiscomplexbehavesverymuchlikethe de Rham complex for smooth varieties. Observe that (2.2.4) says that the Hodge-to-de Rham (a.k.a. Frölicher) spectral sequence works for singular varieties ifone uses the Deligne-Du Bois complexin place of thede Rham complex. This has far reaching consequences and if the associated graded piecesΩp turnouttobecomputable,thenthissinglepropertyleadstomany X applications. Notice that (2.2.6) gives a natural map O → Ω0 , and we will be in- X X terested in situations when this map is a quasi-isomorphism. When X is properoverC, such aquasi-isomorphismimpliesthatthenatural map Hi(Xan,C) → Hi(X,O ) = Hi(X,Ω0 ) X X is surjective because of the degeneration at E of the spectral sequence in 1 (2.2.4). Notice that this is the condition that is crucial for Kodaira-type vanishingtheorems cf. [Kol95,§9]. Following Du Bois, Steenbrink was the first to study this condition and he christened this property after Du Bois. It should be noted that many of the ideas that play important roles in this theory originated from Deligne. Unfortunatelythenowstandardterminologydoes notreflect this. Definition 2.3. A scheme X is said to have Du Bois singularities (or DB singularitiesforshort)ifthenaturalmapO → Ω0 from(2.2.6)isaquasi- X X isomorphism. REMARK 2.4. If ε : Xq → X is a hyperresolution of X then X has DB singularities if and only if the natural map O → Rεq O q is a quasi- X ∗ X isomorphism. EXAMPLE 2.5. It is easy to see that smooth points are DB and Deligne provedthat normal crossing singularitiesare DB as well cf. (2.2.7), [DJ74, Lemme2(b)]. DBPAIRSANDVANISHINGTHEOREMS 7 In applications it is very useful to be able to take general hyperplane sections. Thenextstatementhelpswiththat. Proposition2.6. LetX beaquasi-projectivevarietyandH ⊂ X ageneral q q memberof a veryamplelinearsystem. Then Ω ≃ Ω ⊗ O . H qis X L H Proof. Let εq : Xq → X be a hyperresolution. Since H is general, the fiber product Xq × H → H provides a hyperresolution of H. Then the X statementfollowsfrom(2.2.5)appliedtobothX and H. (cid:3) We saw in (2.2.5) that hi(cid:0)Ω0 (cid:1) = 0 for i < 0. In fact, there is a corre- X spondingupperboundby[GNPP88,III.1.17],namelythathi(cid:0)Ω0 (cid:1) = 0for X i > dimX. It turnsoutthatonecan makeaslightlybetterestimate. Proposition 2.7 cf. [GKKP10, 13.7], [KSS10, 4.9]. Let X be a positive dimensional variety (i.e., reduced). Then the ith cohomology sheaf of Ωp X vanishesforalli ≥ dimX,i.e.,hi(Ωp ) = 0forallpandforalli ≥ dimX. X Proof. For i > dimX or p > 0, the statement follows from [GNPP88, III.1.17]. The case p = 0 and i = n := dimX follows from either [GKKP10,13.7]or[KSS10, 4.9]. (cid:3) Another,muchsimplerfact that willbeused lateristhefollowing: Corollary 2.8. If dimX = 1, then hi(Ωp ) = 0 for i 6= 0. In particular X X isDB ifandonlyifitis semi-normal. Proof. The first statement is a direct consequence of (2.7). For the last statement recall that the seminormalization of O is exactly h0(Ω0 ), and X X so X is seminormal if and only if O ≃ h0(Ω0 ) [Sai00, 5.2] (cf. [Sch06, X X 5.4.17],[Sch07, 4.8],and [Sch09, 5.6]). (cid:3) Definition 2.9. The DB defect of X is the mapping cone of the morphism O → Ω0 . ItisdenotedbyΩ×. Asasimpleconsequenceofthedefinition, X X X onehas an exact triangle, O // Ω0 // Ω× +1 // . X X X Noticethath0(Ω×) ≃ h0(Ω0 )/O and hi(Ω×) ≃ hi(Ω0 ) fori > 0. X X X X X Proposition 2.10. Let X be a quasi-projectivevariety and H ⊂ X a gen- eralmemberof avery amplelinearsystem. Then Ω× ≃ Ω× ⊗ O . H qis X L H Proof. Thisfollowseasily fromthedefinitionand 2.6. (cid:3) ThenextsimpleobservationexplainsthenameoftheDB defect. Lemma2.11. AvarietyX isDBifandonlyiftheDBdefectofX isacyclic, thatis,Ω× ≃ 0. X qis Proof. Thisfollowsdirectly fromthedefinition. (cid:3) 8 SÁNDORJKOVÁCS Proposition 2.12. Let X = Y ∪ Z be a union of closed subschemes with intersectionW = Y ∩Z. Then onehasan exact triangleoftheDB defects ofX,Y,Z, andW: Ω× // Ω× ⊕Ω× − // Ω× +1 // . X Y Z W Proof. Recall that there is an analogous exact triangle (a.k.a. a short exact sequence) for the structure sheaves of X,Y,Z, and W, which forms a commutativediagramwiththeexacttriangleof(2.2.10), O // O ⊕O − // O +1 // X Y Z W (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) Ω0 // Ω0 ⊕Ω0 − // Ω0 +1 // . X Y Z W Then the statement follows by the (derived category version of the) 9- lemma. (cid:3) 3. PAIRS AND GENERALIZED PAIRS 3.A. Basic definitions For an arbitrary proper birational morphism, φ : Y → X, Exc(φ) stands for the exceptional locus of φ. A Q-divisor is a Q-linear combination of integralWeildivisors;∆ = Pai∆i,ai ∈ Q,∆i (integral)Weildivisor. For aQ-divisor∆,itsround-downisdefinedbytheformula: ⌊∆⌋ = P⌊ai⌋∆i, where⌊a ⌋isthelargestintegernotlargerthan a . i i A logvarietyorpair (X,∆) consistsofan equidimensionalvariety(i.e., areducedschemeoffinitetypeoverafieldk)X andaneffectiveQ-divisor ∆ ⊆ X. A morphism of pairs φ : (Y,B) → (X,∆) is a morphism φ : Y → X suchthat φ(suppB) ⊆ supp∆. Let (X,∆) be a pair with ∆ a reduced integral divisor. Then (X,∆) is said to have simple normal crossings or to be an snc pair at p ∈ X if X is smooth at p, and there are local coordinates x ,...,x on X in a 1 n neighbourhood of p such that supp∆ ⊆ (x ···x = 0) near p. (X,∆) is 1 n sncifitis sncat everyp ∈ X. Amorphismofpairsφ : (Y,∆ ) → (X,∆)isalogresolutionof(X,∆) Y ifφ : Y → X isproperandbirational,∆ = φ−1∆,and(∆ ) +Exc(φ) Y ∗ Y red isan sncdivisoronY. Notethat we allow(X,∆) to be sncand stillcall a morphismwith these properties a log resolution. Also note that the notion of a log resolution is notusedconsistentlyin theliterature. If (X,∆) is a pair, then ∆ is called a boundary if ⌊(1−ε)∆⌋ = 0 for all 0 < ε < 1, i.e., the coefficients of all irreducible components of ∆ are in the interval [0,1]. For the definition of klt, dlt, and lc pairs see [KM98]. DBPAIRSANDVANISHINGTHEOREMS 9 Let (X,∆) be a pair and µ : Xm → X a proper birational morphism.Let E = PaiEi bethediscrepancydivisor,i.e.,alinearcombinationofexcep- tionaldivisorssuch that KXm +µ−∗1∆ ∼Q µ∗(KX +∆)+E and let ∆m:= µ−∗1∆+Pai≤−1Ei. For an irreducible divisorF on a birational model of X we define its discrepancy as its coefficient in E. Notice thatasdivisorscorrespondtovaluations,thisdiscrepancyisindependentof the model chosen, it only depends on the divisor. A non-klt place of a pair (X,∆)isanirreducibledivisorF overX withdiscrepancyatmost−1 and a non-klt center is the image of any non-klt place. Exc (µ) denotes the nklt unionofthelociofall non-kltplaces ofφ. Note that in theliterature, non-kltplaces and centers are often called log canonicalplacesandcenters. Foramoredetailedandprecisedefinitionsee [HK10, p.37]. Now if (Xm,∆m) is as above, then it is a minimal dlt model of (X,∆) if it is a dlt pair and the discrepancy of every µ-exceptional divisor is at most −1 cf. [KK10]. Note that if (X,∆) is lc with a minimal dlt model (Xm,∆m), thenKXm +∆m ∼Q µ∗(KX +∆). 3.B. Rational pairs Recall the definition of a log resolution from (3.A): A morphism of pairs φ : (Y,∆ ) → (X,∆)isalogresolutionof(X,∆)ifφ : Y → X isproper Y andbirational,∆ = φ−1∆, and (∆ ) +Exc(φ)is an sncdivisoron Y. Y ∗ Y red Definition3.1. Let(X,∆)beapairand∆anintegraldivisor. Then(X,∆) iscalledanormalpairifthereexistsalogresolutionφ : (Y,∆ ) → (X,∆) Y such that the natural morphism φ# : O (−∆) → φ O (−∆ ) is an iso- X ∗ Y Y morphism. Definition 3.2. A pair(X,∆) with∆ an integraldivisoris called a weakly rationalpairifthereisalogresolutionφ : (Y,∆ ) → (X,∆)suchthatthe Y naturalmorphismO (−∆) → Rφ O (−∆ ) has aleft inverse. X ∗ Y Y Lemma3.3. Let(X,∆)beaweaklyrationalpair. Thenitisanormalpair. Proof. The0thcohomologyoftheleftinverseofO (−∆) → Rφ O (−∆ ) X ∗ Y Y gives a left inverse of φ# : O (−∆) → φ O (−∆ ). As the morphism X ∗ Y Y φ is birational, the kernel of the left inverse of φ# is a torsion sheaf. How- ever, since φ O (−∆ ) is torsion-free, this implies that φ# is an isomor- ∗ Y Y phism. (cid:3) Definition 3.4. [KK09] Let (X,∆) be a pair where ∆ is an integral divisor. Then (X,∆) is called a rational pair if there exists a log resolution φ : (Y,∆ ) → (X,∆) suchthat Y 10 SÁNDORJKOVÁCS (3.4.1) O (−∆) ≃ φ O (−∆ ), i.e.,(X,∆) is normal, X ∗ Y Y (3.4.2) Riφ O (−∆ ) = 0fori > 0,and ∗ Y Y (3.4.3) Riφ ω (∆ ) = 0 fori > 0. ∗ Y Y Lemma3.5. Let(X,∆)beapairwhere∆isanintegraldivisor. Thenitis arationalpairifandonlyifitisaweaklyrationalpairandRiφ ω (∆ ) = 0 ∗ Y Y fori > 0. Proof. Thisfollowsdirectly from[KK09, 105]. (cid:3) REMARK 3.6. Notethatthenotionofarationalpair describesthe“singu- larity” of the relationship between X and ∆. From the definition it is not clearforinstancewhether(X,∆)beingrationalimpliesthatX hasrational singularities. REMARK 3.7. If∆ = ∅,then(3.4.3)followsfromGrauert-Riemenschneider vanishingandX isweakly rationalifandonlyifitisrationalby[Kov00a]. 3.C. Generalized pairs Definition 3.8. A generalized pair (X,Σ) consists of an equidimensional variety (i.e., a reduced scheme of finite type over a field k) X and a sub- schemeΣ ⊆ X. A morphismofgeneralized pairs φ : (Y,Γ) → (X,Σ) is a morphism φ : Y → X such that φ(Γ) ⊆ Σ. A reduced generalized pair is ageneralized pair(X,Σ) suchthat Σis reduced. The log resolution of a generalized pair (X,W) is a proper birational morphism π : X → X such that Exc(π) is a divisor and π−1W +Exc(π) e isan sncdivisor. Let X be a complex scheme and Σ a closed subscheme whose comple- ment in X is dense. Then (Xq,Σq) → (X,Σ) is a good hyperresolution if Xq → X is a hyperresolution, and if Uq = Xq × (X \ Σ) and X Σq = Xq \ Uq, then, for all α, either Σ is a divisor with normal cross- α ingson X or Σ = X . Noticethat it is possiblethat Xq has components α α α that map into Σ. These component are contained in Σq. For more details and the existence of such hyperresolutions see [DB81, 6.2] and [GNPP88, IV.1.21,IV.1.25,IV.2.1]. Foraprimeronhyperresolutionsseetheappendix of[KS09]. Let (X,Σ) be areduced generalized pair. ConsidertheDeligne-Du Bois complexof(X,Σ) defined bySteenbrink [Ste85, §3]: Definition 3.9. The Deligne-Du Bois complex of the reduced generalized q q pair (X,Σ) is the mapping cone of the natural morphism ̺ : Ω → Ω q X Σ twisted by (−1). In other words, it is an object Ω in D (X) such that X,Σ filt