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Vanishing results for pairs of Du Bois spaces

Dans le document Differential forms on log canonical spaces (Page 43-47)

PART IV. COHOMOLOGICAL METHODS

13. Vanishing results for pairs of Du Bois spaces

ThenρDα)=m·1DUα.

Proof. — Given an indexα, Claim12.4needs only to be checked on the open set Uα ∩D◦◦⊆Uα∩D. There, it follows from Equation (11.4.1) of Fact11.4.

Claim12.5. — For any indicesα,β, consider the Kähler differential ταβ:=d log(gαβ|D)∈H0

Uα∩Uβ∩D, 1D . Then i1αβ)=σασβ.

Proof. — Given any two indices α, β, Claim 12.5 needs only to be checked on Uα ∩Uβ∩D◦◦. There, we have

the second equality coming from Equation (11.4.2) of Fact11.4, proving Claim12.5.

As an immediate consequence of Claim 12.5, we obtain that δ(m · 1D) ∈ H1(D, 1D)is represented by the ˘Cech-cocycle

13. Vanishing results for pairs of Du Bois spaces

In this section we prove a vanishing theorem for reduced pairs(X,D)where both X and D are Du Bois. A vanishing theorem for ideal sheaves on log canonical pairs (that

are not necessarily reduced) will follow. Du Bois singularities are defined via Deligne’s Hodge theory. We will briefly recall Du Bois’s construction of the generalised de Rham complex, which is called the Deligne-Du Bois complex. Recall, that if X is a smooth complex algebraic variety of dimension n, then the sheaves of differential p-forms with the usual exterior differentiation give a resolution of the constant sheafCX. I.e., one has a complex of sheaves,

OX d

1X d 2X d 3X d · · · d nXωX, which is quasi-isomorphic to the constant sheafCXvia the natural mapCXOXgiven by considering constants as holomorphic functions on X. Recall that this complex is not a complex of quasi-coherent sheaves. The sheaves in the complex are quasi-coherent, but the maps between them are not OX-module morphisms. Notice however that this is actually not a shortcoming; as CX is not a quasi-coherent sheaf, one cannot expect a resolution of it in the category of quasi-coherent sheaves.

The Deligne-Du Bois complex is a generalisation of the de Rham complex to sin-gular varieties. It is a filtered complex of sheaves on X that is quasi-isomorphic to the constant sheaf,CX. The terms of this complex are harder 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 it is indeed a direct generalisation.

The construction of this complex, X, is based on simplicial resolutions. 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 the-ory 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]. The word “hyperresolution” will refer to either a simplicial, polyhedral, or cubic resolution.

Formally, the construction ofXis the same regardless the type of resolution used and no specific aspects of either types will be used. We will actually not use these resolutions here.

They are needed for the construction, but if one is willing to believe the basic properties then one should be able follow the material presented here.

The bare minimum we need is that there exists a filtered complexXunique up to quasi-isomorphism satisfying a number of properties. As a filtered complex, it admits an associated graded complex, which we denote by GrpfiltX. In order to make the formulas work the way they do in the smooth case we need to make a shift. We will actually prefer to use the following notation:

pX:=GrpfiltX[p].

Here “[p]” means that the mth object of the complex pX is defined to be the(m+p)th object of the complex GrpfiltX. In other words, these complexes are almost the same, only

one is a shifted version of the other. They naturally live in the filtered derived category ofOX-modules with differentials of order≤1. For an extensive list of their properties see [DB81] or [KS09, 4.2]. Here we will only recall a few of them.

One of the most important characteristics of the Deligne-Du Bois complex is the existence of a natural morphism in the derived categoryOX0X, cf. [DB81, 4.1]. We will be interested in situations where this map is a quasi-isomorphism. If this is the case and if in addition X is proper overC, the degeneration of the Frölicher spectral sequence at E1, cf. [DB81, 4.5] or [KS09, 4.2.4], implies that the natural map

Hi(Xan,C)→Hi(X,OX)=Hi(X, 0X)

is surjective. HereHistands for hypercohomology of complexes, i.e.,Hi=Ri.

Definition13.1. — A scheme X is said to have Du Bois singularities (or DB singularities for short) if the natural mapOX0Xis a quasi-isomorphism.

Example13.2. — It is easy to see that smooth points are Du Bois. Deligne proved that normal crossing singularities are Du Bois as well cf. [DBJ74, Lemma 2(b)].

We are now ready to state and prove our vanishing results for pairs of Du Bois spaces. While we will only use Corollary13.4in this paper, we believe that these vanishing results are interesting on their own. For instance, based on these observations one may argue that a pair of Du Bois spaces is not too far from a space with rational singularities.

Indeed, if X has rational singularities and D= ∅, then the result of Theorem13.3follows directly from the definition of rational singularities. Of course, Du Bois singularities are not necessarily rational and hence one cannot expect vanishing theorems for the higher direct images of the structure sheaf, but our result says that there are vanishing results for ideal sheaves of Du Bois subspaces.

Theorem13.3 (Vanishing for ideal sheaves on pairs of Du Bois spaces). — Let(X,D)be a reduced pair such that X and D are both Du Bois, and letπ :X→X be a log resolution of (X,D) withπ-exceptional set E. If we setD:=supp(E+π1(D)), then

RiπOX(−D)=0 for all i>max

dimπ(E)\D,0 . In particular, if X is of dimension n2, then Rn−1πOX(−D)=0.

Corollary13.4 (Vanishing for ideal sheaves on log canonical pairs). — Let (X,D) be a log canonical pair of dimension n2. Letπ :X→X be a log resolution of(X,D)withπ-exceptional set E. If we setD:=supp(E+π−1D), then

Rn1πOX(−D)=0.

Proof. — Recall from [KK10b, Theorem 1.4] that X is Du Bois, and that any finite union of log canonical centres is likewise Du Bois. Since the components of D are log canonical centres, Theorem13.3applies to the reduced pair (X,D)to prove the claim. For this, recall from Lemma 2.15that the morphismπ is a log resolution of the pair(X,D)and therefore satisfies all the conditions listed in Theorem13.3.

13.A. Preparation for the proof of Theorem13.3. — Before we give the proof of Theo-rem13.3in Section13.B, we need the following auxiliary result. This generalises parts of [GNPP88, III.1.17].

Lemma13.5. — Let X be a positive dimensional variety. Then the ith cohomology sheaf of0X vanishes for all idim X, i.e., hi(0X)=0 for all idim X.

Proof. — For i>dim X, the statement follows from [GNPP88, III.1.17], so we only need to prove the case when i=n:=dim X. Let S:=Sing X and π :X→X a strong log resolution with exceptional divisor E. Recall from [DB81, 3.2] that there are natural restriction maps,0X0S and0X0Ethat reduce to the usual restriction of regular functions if the spaces are Du Bois. These maps are connected via an exact triangle by [DB81, Proposition 4.11]:

(13.5.1) 0X 0S⊕Rπ0

X α

0E +

1

.

Since X is smooth and E is an snc divisor, they are both Du Bois, cf. Example 13.2.

Hence, there exist quasi-isomorphisms0XOXand0EOE. It follows thatα(0,_)is the map RπOX→RπOEinduced by the short exact sequence

0→OX(−E)→OXOE→0.

Next, consider the long exact sequence of cohomology sheaves induced by the exact triangle (13.5.1),

· · · →hn1(0S)⊕Rn1πOX αn−1

−→Rn1πOEhn(0X)hn(0S)⊕RnπOX. Since dim S<n, [GNPP88, III.1.17] implies that hn(0S)=0. Furthermore, asπ is bi-rational, the dimension of any fibre ofπ is at most n−1 and hence RnπOX=0. This implies that hn(0X)cokerαn1. The bound on the dimension of the fibres ofπ also im-plies that RnπOX(−E)=0, so taking into account the observation above about the map (α,0), we obtain that αn1(0,_) is surjective, and then naturally so isαn1. Therefore,

hn(0X)cokerαn1=0.

13.B. Proof of Theorem13.3. — Since the divisor D is assumed to be reduced, we simplify notation in this proof and use the symbol D to denote both the divisor and its support. To start the proof, set :=π(E)\D and s :=max(dim,0). Let :=

D∪π(E)and consider the exact triangle from [DB81, 4.11], 0X 0⊕Rπ0X0D +1 .

SinceX is smooth andD is a snc divisor, we have quasi-isomorphisms Rπ 0

XOX

and Rπ0DOD, so this exact triangle induces the following long exact sequence of sheaves:

· · · →hi(0X)hi(0)⊕RiπOX→RiπODhi+1(0X)→ · · · By assumption hi(0X) = hi(0D) = 0 for i > 0. Furthermore, hi(0) = 0 and hi1(0D)=0 for is by Lemma13.5. Hence, hi(0)=0 for is by [DB81, 3.8]. As in the proof of Lemma13.5we obtain that the natural restriction map

RiπOX→RiπOD

is surjective for is and is an isomorphism for i > s. This in turn implies that RiπOX(−D) =0 for i>s as desired.

Dans le document Differential forms on log canonical spaces (Page 43-47)