<span class="mathjax-formula">$L^2$</span>-theory for the <span class="mathjax-formula">$\protect \overline{\partial }$</span>-operator on complex spaces with isolated singularities

ANNALES

DE LA FACULTÉ DES SCIENCES

Mathématiques

JEANRUPPENTHAL

L²-theory for the∂-operator on complex spaces with isolated singularities

Tome XXVIII, n^o2 (2019), p. 225-258.

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L

-theory for the ∂-operator on complex spaces with isolated singularities

Jean Ruppenthal⁽¹⁾

ABSTRACT. — The present paper is a complement to the L²-theory for the

∂-operator on a Hermitian complex spaceXof pure dimensionnwith isolated sin- gularities, presented in [17] and [13]. The general philosophy is to use a resolution of singularitiesπ:M→X to obtain a “regular” model of theL²-cohomology.

First, we show how the representation of theL²_loc-cohomology ofX on the level of (n, q)-forms in terms of “regular”L²_loc-cohomology onM, given in [17], can be made explicit in terms of differential forms, if a certain reasonable extra condition is satisfied. Second, we prove the analogous statement forL²-cohomology, which is a new result. Finally, we use this in combination with duality observations to give a new proof of the main results from [13], where the resolutionπ:M →X is used to express theL²-cohomology ofX on the level of (0, q)-forms in terms of “regular”

L²-cohomology onM.

RÉSUMÉ. — Le présent article est un complément à la théorie deL²pour l’opéra- teur∂sur un espace complexe hermitienXde dimension purenavec des singularités isolées, présenté dans [17] et [13]. La philosophie générale est d’utiliser une résolution de singularitésπ:M→Xpour obtenir un modèle « régulier » de laL²-cohomologie.

Tout d’abord, nous montrons comment la représentation de laL²_loc-cohomologie deX au niveau de (n, q)-formes en termes de laL²_loc-cohomologie « réguliere » sur M, donnée dans [17], peut être fait explicite en termes de formes différentielles, si une certaine condition supplémentaire raisonnable est remplie. Deuxièmement, nous prouvons la déclaration analogue pour la L²-cohomologie, ce qui est un nouveau résultat. Enfin, nous l’utilisons en combinaison avec des observations de dualité pour donner une nouvelle preuve des principaux résultats de [13], où la résolutionπ:M→ Xest utilisée pour exprimer laL²-cohomologie deXsur le niveau de (0, q)-formes en termes deL²-cohomologie « régulière » surM.

(*)Reçu le 11 septembre 2015, accepté le 28 avril 2017.

Keywords:Cauchy–Riemann equations,L²-theory, singular complex spaces.

2010Mathematics Subject Classification:32J25, 32C35, 32W05.

(1) Department of Mathematics, University of Wuppertal, Gaußstr. 20, 42119 Wuppertal, Germany — ruppenthal@uni-wuppertal.de

Article proposé par Vincent Guedj.

1. Introduction

The present paper is another part of an attempt to create a systematic L²-theory for the∂-operator on singular complex spaces. By a refinement and completion of the techniques introduced in [17], we obtain a better picture for (0, q) and (n, q)-forms on a Hermitian complex spaceX which is of pure dimensionn and has only isolated singularities. The general philosophy is to use a resolution of singularities to obtain a regular model of the L²- cohomology. This complements also the insights of Øvrelid and Vassiliadou in [13].

The key element of our theory is a new kind of canonical sheaf on X introduced in [17] which we denote here byK^s_X. It is the sheaf of germs of holomorphic square-integrable n-forms which satisfy a Dirichlet boundary condition at the singular set SingX. It comes as the kernel of the∂s-operator on square-integrable (n,0)-forms (see (3.5)). The ∂s-operator is a localized version of the L²-closure of the ∂-operator acting on forms with support away from the singular set (see Section 3 for the precise definition). IfX is a Hermitian complex space with only isolated singularities, we showed in [17, Theorem 1.9], that the∂s-complex

0→ K^s_X,→ F^n,0−→ F^{∂s n,1}−→ F^{∂s n,2}−→^∂s . . .−→ F^n,n→0 (1.1) is a fine resolution of K^s_X, where theF^n,q are the sheaves of germs of L²- forms in the domain of the ∂_s-operator. This implies that the cohomology ofK^s_X is represented by theL²-∂_s-cohomology.

The essential idea of ourL²-theory on singular complex spaces is to rep- resent the cohomology of K^s_X also by L²-∂-cohomology on a resolution of singularities. The key to this lies in the following representation ofK_X^s:

Theorem 1.1 ([17, Theorem 1.10]). — Let X be a Hermitian complex space of pure dimension with only isolated singularities. Then there exists a resolution of singularitiesπ:M →X with at most normal crossings and an effective divisorD>Z− |Z| with support on the exceptional set such that:

K^s_X ∼=π∗ KM ⊗ O(−D)

, (1.2)

whereK^s_X is the canonical sheaf for the∂s-operator, KM is the usual canon- ical sheaf onM andZ =π⁻¹(SingX)the unreduced exceptional divisor.

If the exceptional set of the resolution π:M →X has only double self- intersections, which is particularly the case ifdimX = 2or ifX is homoge- neous, then one can takeD=Z− |Z| in (1.2).

By Grauert’s direct image theorem [4], this yields particularly thatK_X^s is a coherent analytic sheaf. Note that hereπ:M →X is not only a resolution ofX, but – in some sense – also a resolution ofK_X^s, making it locally free.

It is moreover not hard to show that π_∗ KM ⊗ O(−Z)

⊂ K^s_X.

This follows directly from the proof of Lemma 6.1 in [17] and makes it reasonable to conjecture that one can always takeD=Z− |Z|in (1.2) (as we also know that this is possible e.g. if dimX = 2 or if the singularities are homogeneous so that they can be resolved by a single blow-up). In the present paper, we show how the L²-theory from [17] can be refined and complemented if we assume that actuallyD=Z− |Z|.⁽¹⁾ So, assume from now on that this is the case.

We need a few notations to explain our results. IfN is any Hermitian complex manifold, let

∂cpt:A^p,q_cpt(N)→A^p,q+1_cpt (N)

be the ∂-operator on smooth forms with compact support in N. Then we denote by

∂max:L^p,q(N)→L^p,q+1(N) the maximal and by

∂_min:L^p,q(N)→L^p,q+1(N)

the minimal closed Hilbert space extension of the operator∂cptas densely de- fined operator fromL²-(p, q)-formsL^p,q(N) toL²-(p, q+1)-formsL^p,q+1(N).

Let H_max^p,q (N) be the L²-Dolbeault cohomology on N with respect to the maximal closed extension ∂max, i.e. the ∂-operator in the sense of distri- butions on N, andH_min^p,q (N) the L²-Dolbeault cohomology with respect to the minimal closed extension∂_min. Our first new main result in the present paper is:

Theorem 1.2. — Let (X, h) be a Hermitian complex space of pure di- mensionn>2 with only isolated singularities and π:M →X a resolution of singularities with only normal crossings. Assume thatn= dimX = 2or that the singularities ofX are all homogeneous.⁽²⁾

(1)We will show e.g. how certain isomorphisms on cohomology from [17] can be realized explicitly by pull-backs or push-forwards of differential forms.

(2)Under this assumption, Theorem 1.1 implies thatK^s_X=π∗(K_M⊗ O(|Z| −Z)), i.e., that one can useD=Z− |Z|in (1.2). Actually, we prove Theorem 1.2 under the mere assumption thatD=Z− |Z|is a possible choice.

Let 06p < n,Ω⊂⊂X a relatively compact domain, Ω :=e π⁻¹(Ω) and Ω^∗ = Ω−SingX. Provide Ωe with a (regular) Hermitian metric which is equivalent toπ^∗hclose to the boundary beΩ.

Let Z := π⁻¹(Ω∩SingX) denote the unreduced exceptional divisor overΩ and let L_|Z|−Z → M be a Hermitian holomorphic line bundle such that holomorphic sections inL_|Z|−Z correspond to holomorphic sections in O(|Z| −Z).

Then the pull-back of formsπ^∗induces a natural injective homomorphism hp:H_min^n,p(Ω^∗)−→H_min^n,p(eΩ, L_|Z|−Z), (1.3) withcokerh_p= Γ(Ω, R^pπ_∗(K_M⊗ O(|Z| −Z))ifp>1, andh₀is an isomor- phism.

Note that singularities in the boundarybΩ of Ω are permitted. Any regu- lar metric onM will do the job if there are no singularities in the boundary of Ω. If Ω =X is compact, then the case p=ncan be included (see Theo- rem 6.11).

The proof of Theorem 1.2 requires a slight variation of Theorem 1.11 in [17], which we give in Section 5 (see Theorem 5.1). We use this oppor- tunity to give a new and more explicit proof of [17, Theorem 1.11], under the hypothesis that D = Z − |Z|. Theorem 1.2 is then deduced by a de- tailed comparison of∂_max- withL²_loc-cohomology, and of∂_min-cohomology with cohomology with compact support (see Section 6). We exploit the fact that the L²- and the L²_loc-Dolbeault cohomology are naturally isomorphic on strongly pseudoconvex domains in complex manifolds. Theorem 1.2 is a completely new result that appears neither in [17] nor in [13].

Let us explain briefly how the hypothesis that one can useD=Z− |Z| in (1.2) enters the proof of Theorem 1.2. We require that the homomor- phism (1.3) and, analogously, the injections (5.1), (5.2) in Theorem 5.1 be- low, are induced by pull-back ofL²-forms under the resolutionπ:M →X. The crucial point is now that we can show that the pull-back of (n, q)-forms in the domain of the∂_s-operator vanish just to the order ofZ− |Z|on the exceptional set of the resolution, so that we obtain homomorphisms into C^n,q_σ (L_Z−|Z|), but not intoC_σ^n,q(L_D) ifD is of higher order thanZ− |Z|(see Lemma 5.2). The fact that the homomorphisms in (1.3), (5.1), (5.2) are in- duced by pull-back of (n, q)-forms is then essentially used in the commutative diagrams in the proofs of Lemma 6.7 and Theorem 6.10.

The advantage of Theorem 1.2 lies in the fact that it allows to carry over our results to (0, q)-forms by the use of theL²-version of Serre duality.

Besides the L²-Serre duality between the ∂min- and the ∂max-Dolbeault

cohomology, there is (forp>1) another duality between the higher direct image sheavesR^pπ_∗KM⊗O(|Z|−Z) on one hand and the flabby cohomology H_E^n−pofO(Z−|Z|) with support on the exceptional setE=|Z|on the other hand. This is explained in Section 7.1. Combination of these two kinds of duality with Theorem 1.2 leads to the following statement:

Theorem 1.3. — Under the assumptions of Theorem 1.2, let06q6n if Ω =X is compact and 0< q 6n otherwise. Then there exists a natural exact sequence

0→H_E^q(eΩ,O(Z− |Z|))→H_max^0,q (eΩ, L_Z−|Z|)→H_max^0,q (Ω^∗)→0, whereH_E^∗ is the flabby cohomology with support on the exceptional setE=

|Z|. In case q=n,H_Eⁿ(eΩ,O(Z− |Z|))has to be replaced by0.

Note again that singularities in the boundarybΩ of Ω are permitted and that any regular metric onM will do the job if there are no singularities in the boundary of Ω. Note also thatH_E⁰(eΩ,O(Z− |Z|)) = 0 by the identity theorem.

The history of Theorem 1.3 is a bit complicated. The idea to identify the kernel of the natural map

H_max^0,q (eΩ, L_Z−|Z|)→H_max^0,q (Ω^∗)

as the flabby cohomology ofO(Z− |Z|) with support onEis due to Øvrelid and Vassiliadou [13] who proved Theorem 1.3 recently in the casesq=n−1 and q = n ([13, Theorem 1.4 and Corollary 1.6]). After the present pa- per appeared as a preprint, Øvrelid and Vassiliadou added another proof of our Theorem 1.3 in the case 0 < q < n−1 also to their paper (see [13, Remark 4.5.1]). Their method is quite different from our approach via Theo- rem 1.2 and duality.⁽³⁾ Concluding, the statement of Theorem 1.3 appeared already in [13], but we give here another, different proof (under the extra condition thatD=Z− |Z|).

For 0< q < n−1, one can show that H_max^0,q (eΩ, L_Z−|Z|) H_E^q(eΩ,O(Z− |Z|))

∼=H_max^0,q Ωe ,

so that we can recover by use of Theorem 1.3 the following result of Øvrelid and Vassiliadou ([13, Theorem 1.3]):

H_max^0,q Ωe∼=H_max^0,q (Ω^∗), 0< q < n−1.

(3)We may remark that Øvrelid and Vassiliadou also use some results from [17].

The organization of the present paper is as follows. In the Sections 2, 3 and 4, we provide the necessary tools for the proof of Theorem 5.1 in Sec- tion 5. More precisely, in Section 2 we show how to deal with the cohomology of non-exact complexes. Section 3 contains a review of the∂_s-complex as it is introduced in [17]. This comprises the definition of the canonical sheaf of holomorphicn-forms with Dirichlet boundary conditionK^s_X and the exact- ness of the∂s-complex (1.1). In Section 4, we recall from [17] howK^s_X can be represented as the direct image of an invertible sheaf under a resolution of singularities (see Theorem 1.1). In Section 6 we study how the concepts that appear in the context of Theorem 5.1 are related toL²-Dolbeault co- homology and prove Theorem 1.2. Section 7 finally contains the proof of Theorem 1.3.

An outline of the historical development of the topic can be found in the introduction of the previous paper [17] and in [13].

Acknowledgements.The author thanks Nils Øvrelid for many inter- esting and very helpful discussions on the topic. He is also grateful to the unknown referee for many valuable suggestions which helped to improve the exposition of this paper considerably. This research was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), grant RU 1474/2 within DFG’s Emmy Noether Programme.

2. Cohomology of non-exact complexes

LetX be a paracompact Hausdorff space. In this section, we consider a complex of sheaves of abelian groups

0→ A,→ A⁰−→ A^{a0 1}−→ A^{a1 2}−→ A^{a2 3}−→. . . (2.1) overX which is exact atAandA⁰such thatA ∼= kera₀, but not necessarily exact at A^p, p > 1. We denote by K^p = kera_p the kernel of a_p and by I^p = Imap the image ofap. Note thatK⁰ ∼=A. We will now represent the q-th (flabby) cohomology group H^q(X,A) of A over X by use of the q-th cohomology group of the complex (2.1). Since (2.1) is not a resolution ofA, this also involves the quotient sheaves

R^p:=K^p/I^p−1, p>1, (2.2) which are well-defined sinceap◦a_p−1 = 0. To compute the (flabby) coho- mology ofA, we first require that the sheavesA^p are acyclic:

Lemma 2.1. — Assume that the sheavesA^p,p>0, in the complex(2.1) are acyclic. Then there are natural isomorphisms

δ⁰: Γ(X,I^p) ap(Γ(X,A^p))

∼=

−→H¹(X,K^p), (2.3)

δ^q :H^q(X,I^p)−→^∼= H^q+1(X,K^p) (2.4) for allp>0,q>1.

In this lemma and throughout the whole section, we may as well consider global sections Γ_cpt and cohomologyH_cpt with compact support.

Proof. — We shortly repeat the proof which is standard. For anyp>0, we consider the short exact sequence

0→ K^p,→ A^p−→ I^{ap p}→0. (2.5) UsingH^r(X,A^p) = 0 forr>1, we obtain the exact cohomology sequences

0→Γ(X,K^p),→Γ(X,A^p)−→^ap Γ(X,I^p) ^δ

0

−→H¹(X,K^p)→0, 0→H^q(X,I^p) ^δ

q

−→H^q+1(X,K^p)→0, q>1, which yield the statement of the lemma.

The assertion that the isomorphisms are natural means the following. Let 0→ B,→ B⁰−→ B^{b0 1}−→ B^{b1 2}−→ B^{b2 3}−→. . . (2.6) be another such complex and

0 //A //

f

A^∗

g

0 //B //B^∗

a morphism of complexes (where we abbreviate the complex (2.1) by 0→ A → A^∗and the complex (2.6) by 0→ B → B^∗). If we denote byL^p= kerb_p the kernel ofb_p and byJ^p = Imb_p the image ofb_p, then

Γ(X,I^p) a_p(Γ(X,A^p))

δ0 //

g_p+1⁰

H¹(X,K^p)

g⁰⁰_p

Γ(X,J^p) bp(Γ(X,B^p))

δ0 //H¹(X,L^p)

is commutative, whereg⁰_p+1andg_p⁰⁰are the maps induced by the commutative diagram

0 //K^p //

gp

A^{p ap} //

gp

I^p //

gp+1

0

0 //L^p //B^{p bp} //J^p //0.

The statement that the isomorphismsδ^q are natural follows analogously.

To go on, we need some assumptions onR^p:

Lemma 2.2. — Forp>1, assume that the quotient sheafR^pof the com- plex(2.1)defined in(2.2)is acyclic and that the natural mappingΓ(X,K^p)→ Γ(X,R^p)is surjective. Then there is a natural isomorphism

H^q(X,I^p−1)−→^∼= H^q(X,K^p) for allq>1 (induced by the inclusionI^p−1,→ K^p).

Proof. — Under the assumptions, the proof follows directly from the long exact cohomology sequence that is obtained from the short exact sequence

0→ I^p−1,→ K^p−→ R^p=K^p/I^p−1→0. (2.7) If 0 → B → B^∗ is another such complex, then we obtain commutative diagrams as in the proof of Lemma 2.1 showing that the isomorphism is

natural.

Note that the assumptions of Lemma 2.2 are trivially fulfilled if the com- plex (2.1) is exact such that R^p = 0 for all p > 1. From Lemma 2.1 and Lemma 2.2, we deduce by induction:

Lemma 2.3. — Under the assumptions of Lemma 2.1 and Lemma 2.2, there are natural isomorphisms

γ^p: Γ(X,I^p−1) ap−1(Γ(X,A^p−1))

∼=

−→H^p(X,A)

for all p>1. Here, natural means the following. If 0→ B → B^∗ is another such complex as in(2.6) and

0 //A //

f

A^∗

g

0 //B //B^∗

a morphism of complexes as in the proof of Lemma 2.1, then we obtain a commutative diagram

Γ(X,I^p−1) ap−1(Γ(X,A^p−1))

γ^p //

g_p⁰

H^p(X,A)

fp

Γ(X,J^p−1) b_p−1(Γ(X,B^p−1))

γ^p //H^p(X,B),

whereJ^p−1 = Imbp−1,fp is the map on cohomology induced by f :A → B andg_p⁰ is the map induced by gp:A^p→ B^p. Iff is an isomorphism, theng_p⁰ is an isomorphism as well.

We can now make the connection to the p-th cohomology group of the complex (2.1) by use of:

Lemma 2.4. — Forp>1, assume that the natural mappingΓ(X,K^p)→ Γ(X,R^p)is surjective. Then there is a natural injective homomorphism

i^p: Γ(X,I^p−1)

ap−1(Γ(X,A^p−1)) −→H^p(Γ(X,A^∗)) = Γ(X,K^p) ap−1(Γ(X,A^p−1)) withcokeri^p= Γ(X,R^p). More precisely, the natural sequence

0→ Γ(X,I^p−1) a_p−1(Γ(X,A^p−1))

i^p

−→H^p(Γ(X,A^∗))−→Γ(X,R^p)→0 is exact.

Proof. — From (2.7) we obtain by use of the assumption the exact se- quence

0→Γ(X,I^p−1)−→Γ(X,K^p)−→Γ(X,R^p)→0, and this induces the natural exact sequence

0→ Γ(X,I^p−1) a_p−1(Γ(X,A^p−1))

i^p

−→ Γ(X,K^p)

a_p−1(Γ(X,A^p−1))−→Γ(X,R^p)→0 since

a_p−1(Γ(X,A^p−1))⊂Γ(X,I^p−1)⊂Γ(X,K^p).

Combining Lemma 2.3 and Lemma 2.4, we conclude finally:

Theorem 2.5. — Under the assumptions of Lemma 2.1 and Lemma 2.2, there is for allp>1 a natural injective homomorphism

i^p◦(γ^p)⁻¹:H^p(X,A)−→H^p(Γ(X,A^∗))

with cokeri^p◦(γ^p)⁻¹ = Γ(X,R^p). More precisely, there is a natural exact sequence

0→H^p(X,A)ⁱ

p◦(γ^p)⁻¹

−→ H^p(Γ(X,A^∗))−→Γ(X,R^p)→0.

Here, natural means the following. If0 → B → B^∗ is another such complex as in (2.6)and

0 //A //

f

A^∗

g

0 //B //B^∗

a morphism of complexes, then we obtain the commutative diagram Γ(X,R^p)

g⁰⁰⁰_p

H^p(Γ(X,A^∗))

oo

[g_p]

Γ(X,I^p−1) ap−1(Γ(X,A^p−1)) i^p

oo ^γp //

g⁰_p

H^p(X,A)

fp

Γ(X,S^p)oo H^p(Γ(X,B^∗)) _b ^Γ(X,Jp−1)

p−1(Γ(X,B^p−1)) i^p

oo ^γp //H^p(X,B),

wherefp is the map on cohomology induced by f :A → B and[gp], g⁰_p,g_p⁰⁰⁰ are the maps induced by gp:A^p→ B^p. The quotient sheavesS^p are defined for the complex0→ B → B^∗ analogously to the sheavesR^p for the complex 0→ A → A^∗. The maps γ^p are bijective and the mapsi^p are injective.

In the present paper, we need the following consequence of Theorem 2.5.

Here, we make use of our general assumption thatX is a paracompact Haus- dorff space, because this implies that fine sheaves are acyclic.

Theorem 2.6. — Let X, M be paracompact Hausdorff spaces and π : M →X a continuous map. LetCbe a sheaf (of abelian groups) overM and 0→ C,→ C⁰−→ C^{c0 1}−→ C^{c1 2}−→ C^{c2 3}−→. . . (2.8) a fine resolution. Let A ∼= π_∗C be a sheaf on X, isomorphic to the direct image ofC, and0→ A → A^∗ a fine resolution ofAoverX.

Let B := π_∗C be the direct image of C and B^∗ =π_∗C^∗ the direct image complex (which is again fine but not necessarily exact). Since(2.8)is a fine resolution, the non-exactness of 0 → B → B^∗ is measured as above by the higher direct image sheavesS^p:=R^pπ_∗C,p>1. Let

0 //A //

∼= f

A^∗

g

0 //B //B^∗

be a morphism of complexes, and assume that the complex 0 → B → B^∗ satisfies the assumption of Lemma 2.2, i.e. that the direct image sheavesS^p are acyclic and that the mapsΓ(X,kerbp)→Γ(X,S^p)are surjective for all p>1.

Theng induces for allp>1 a natural injective homomorphism H^p(Γ(X,A^∗))−→^[gp] H^p(Γ(X,B^∗))

withcoker[gp] = Γ(X,S^p).

More precisely, there is a natural exact sequence

0→H^p(Γ(X,A^∗))−→^[gp] H^p(Γ(X,B^∗))−→Γ(X,S^p)→0.

In this sequence, one can replaceH^p(Γ(X,B^∗))byH^p(Γ(M,C^∗))because Γ(X,B^q) = Γ(π⁻¹(X),C^q) = Γ(M,C^q),

b_q(Γ(X,B^q)) =c_q(Γ(M,C^q)) by definition for allq>0.

Proof. — The proof follows directly from Theorem 2.5 which we apply to the morphism of complexes

(f, g) : (A,A^∗)→(B,B^∗).

Note that the direct image sheavesB^q =π_∗C^q, q>0, are still fine sheaves for one can push forward a partition of unity under the continuous mapπ.

Consider the big commutative diagram in Theorem 2.5. Since 0→ A → A^∗ is a fine resolution, the quotient sheavesR^p,p>1, do vanish such that the mapi^pin the upper line is an isomorphism. By assumption, the induced map on cohomologyfpis an isomorphism, and sog_p⁰ must also be isomorphic (for the mapsγ^p are isomorphisms, as well). But then

0→H^p(Γ(X,A^∗))−→^[gp] H^p(Γ(X,B^∗))−→Γ(X,S^p)→0

is an exact sequence.

3. Review of the ∂s-complex

3.1. Two ∂-complexes on singular spaces

Let us recall some of the essential constructions from [17]. LetXbe always a (singular) Hermitian complex space of pure dimensionn and U ⊂X an open subset. On a singular space, it is most fruitful to consider forms that are square-integrable up to the singular set. Hence, we will use the following concept of locally square-integrable forms:

L^p,q_loc(U) :={f ∈L^p,q_loc(U−SingX) :f|K ∈L^p,q(K−SingX)∀K⊂⊂U}.

It is easy to check that the presheaves given as L^p,q(U) :=L^p,q_loc(U)

are already sheavesL^p,q→X. OnL^p,q_loc(U), we denote by

∂w:L^p,q_loc(U)→L^p,q+1_loc (U)

the∂-operator in the sense of distributions on U −SingX which is closed and densely defined. The subscript refers to ∂w as an operator in a weak sense. We can define the presheaves of germs of forms in the domain of∂w,

C^p,q :=L^p,q∩∂⁻¹_w L^p,q+1,

given byC^p,q(U) =L^p,q(U)∩Dom∂w(U). It is not hard to check that these are actually already sheaves.

Moreover, it is easy to see that the sheavesC^p,q admit partitions of unity, and so we obtain fine sequences

C^p,0−→ C^{∂w p,1}−→ C^{∂w p,2}−→^∂w . . . (3.1) We will see later, when we deal with resolution of singularities, that

KX:= ker∂w⊂ C^n,0

is just the canonical sheaf of Grauert and Riemenschneider since the L²- property of (n,0)-forms remains invariant under modifications of the metric.

TheL^2,loc-Dolbeault cohomology with respect to the∂_w-operator on an open setU ⊂X is the cohomology of the complex (3.1) which is denoted by H^q(Γ(U,C^p,∗)).

Secondly, we need a suitable local realization of a minimal version of the

∂-operator. This is the ∂-operator with a Dirichlet boundary condition at the singular set SingX ofX. Let

∂_s:L^p,q_loc(U)→L^p,q+1_loc (U)

be defined as follows. We say that f ∈ Dom∂_w is in the domain of ∂_s if there exists a sequence of forms{f_j}_j ⊂Dom∂_w ⊂L^p,q_loc(U) with essential support away from the singular set,

suppfj∩SingX =∅, such that

fj →f inL^p,q(K−SingX), (3.2)

∂_wf_j→∂_wf inL^p,q+1(K−SingX) (3.3) for each compact subsetK⊂⊂U. The subscript refers to∂sas an extension in a strong sense. Note that we can assume without loss of generality (by

use of cut-off functions and smoothing with Dirac sequences) that the forms f_j are smooth with compact support in U−SingX.

We define the presheaves of germs of forms in the domain of∂_s, F^p,q :=L^p,q∩∂⁻¹_s L^p,q+1,

given byF^p,q(U) =L^p,q(U)∩Dom∂_s(U). It is not hard to see that theF^p,q are already sheaves (see [17, Section 6.1]).

As forC^p,q, it is clear that the sheavesF^p,q are fine, and we obtain fine sequences

F^p,0−→ F^{∂s p,1}−→ F^{∂s p,2}−→^∂s . . . (3.4) We can now introduce the sheaf

K^s_X:= ker∂s⊂ F^n,0 (3.5)

which we may call the canonical sheaf of holomorphicn-forms with Dirichlet boundary condition. One of the main objectives of the present paper is to compare different representations of the cohomology of K^s_X. One of them will be theL^2,loc-Dolbeault cohomology with respect to the ∂_s-operator on open setsU ⊂X, i.e. the cohomology of the complex (3.4) which is denoted byH^q(Γ(U,F^p,∗)).

3.2. LocalL²-solvability for (n, q)-forms

It is clearly interesting to study wether the sequences (3.1) and (3.4) are exact, which is well-known to be the case in regular points ofX where the

∂w- and the ∂s-operator coincide. In singular points, the situation is quite complicated for forms of arbitrary degree and not completely understood.

However, the∂w-equation is locally solvable in theL²-sense at arbitrary sin- gularities for forms of degree (n, q), q > 0 (see [14, Proposition 2.1]), and for forms of degree (p, q), p+q > n, at isolated singularities (see [3, The- orem 1.2]). Since we are concerned with canonical sheaves, we may restrict our attention to the case of (n, q)-forms and conclude:

Theorem 3.1. — Let X be a Hermitian complex space of pure dimen- sionn. Then

0→ K_X,→ C^n,0−→ C^{∂w n,1}−→ C^{∂w n,2}−→^∂w . . .−→ C^n,n→0 (3.6) is a fine resolution. For an open setU ⊂X, it follows that

H^q(U,KX)∼=H^q(Γ(U,C^n,∗)), H_cpt^q (U,KX)∼=H^q(Γcpt(U,C^n,∗)).

Concerning the∂_s-equation, localL²-solvability for forms of degree (n, q) is known to hold on spaces with isolated singularities (see [17, Theorem 1.9]), but the problem is open at arbitrary singularities. If X has only isolated singularities, then the ∂_s-equation is locally exact on (n, q)-forms for 1 6 q6n by [17, Lemma 5.5]. The statement was deduced from the results of Fornæss, Øvrelid and Vassiliadou [3].

4. Resolution of(X,K^s_X) 4.1. Desingularization and comparison of metrics

Let π : M → X be a resolution of singularities (which exists due to Hironaka [8]), i.e. a proper holomorphic surjection such that

π|_M−E:M −E→X−SingX

is biholomorphic, whereE =|π⁻¹(SingX)|is the exceptional set. We may assume that E is a divisor with only normal crossings, i.e. the irreducible components of E are regular and meet complex transversely. Let Z :=

π⁻¹(SingX) be the unreduced exceptional divisor. For the topic of desingu- larization, we refer to [1], [2] and [6]. Let γ :=π^∗h be the pullback of the Hermitian metrichofX toM. γis positive semidefinite (a pseudo-metric) with degeneracy locusE.

We giveM the structure of a Hermitian manifold with a freely chosen (positive definite) metricσ. Then γ.σ and γ∼σ on compact subsets of M −E. For an open set U ⊂ M, we denote by L^p,q_γ (U) and L^p,q_σ (U) the spaces of square-integrable (p, q)-forms with respect to the (pseudo-)metrics γandσ, respectively.

Sinceσ is positive definite andγ is positive semi-definite, a straightfor- ward comparison of the volume forms (see [17, Section 2.2]) leads to the inclusions

L^n,q_γ (U)⊂L^n,q_σ (U), (4.1) L^0,q_σ (U)⊂L^0,q_γ (U). (4.2) for relatively compact open setsU ⊂⊂M and all 06q6n.

For an open set Ω ⊂ X, Ω^∗ = Ω−SingX, Ω :=e π⁻¹(Ω), pullback of forms underπgives the isometry

π^∗:L^p,q(Ω^∗)−→L^p,q_γ (eΩ−E)∼=L^p,q_γ (eΩ), (4.3) where the last identification is by trivial extension of forms over the thin exceptional setE.

4.2. Representation ofK_X^s under desingularization

By use of (4.3), both complexes, the resolution (C^n,∗, ∂_w) ofK_X and the resolution (F^n,∗, ∂s) ofK^s_X, can be studied as well on the complex manifold M. This is the point of view that was taken in [17] where we considered the sheavesL^p,q_γ onM given by

L^p,q_γ (U) :=L^p,q_γ,loc(U) and

C_γ,E^p,q :=L^p,q_γ ∩∂⁻¹_w,EL^p,q+1_γ , (4.4) where ∂w,E is the ∂-operator in the sense of distributions with respect to compact subsets ofM−E. (4.4) is given by the presheafC_γ,E^p,q(U) =L^p,q_γ (U)∩

Dom∂_w,E(U). It follows from (4.3) that (C^p,∗, ∂_w) can be canonically iden- tified with the direct image complex (π_∗C_γ,E^p,∗, π_∗∂_w,E). SinceL^n,0_γ =L^n,0_σ for the regular metric σ on M by use of (4.1) and (4.2), we can use the fact that the∂-equation in the sense of distributions forL²_σ-forms extends over exceptional sets (see e.g. [16, Lemma 2.1]) to conclude that

KM := ker∂_w,E ⊂ L^n,0_γ =L^n,0_σ

is just the usual canonical sheaf on the complex manifoldM, and thatK_X∼= π_∗K_M is in fact the canonical sheaf of Grauert–Riemenschneider.

Analogously, we consider now the∂_s-complex. Let∂_s,Ebe the∂-operator acting onL^p,q_γ -forms, defined as the∂_s-operator onX above, but with the exceptional setE in place of the singular set SingX. Let

F_γ,E^p,q :=L^p,q_γ ∩∂⁻¹_s,EL^p,q+1_γ .

Then it follows from (4.3) that (F^p,∗, ∂s) can be canonically identified with the direct image complex (π∗F_γ,E^p,∗, π∗∂s,E).

It remains to study ker∂s,E inL^n,0_γ =L^n,0_σ becauseπ∗(F_γ,E^n,0∩ker∂s,E) = K^s_X. This was a central point [17] (see [17, Section 6.3]): There exists a resolution of singularities π : M → X with only normal crossings and an effective divisor D > Z− |Z| with support on the exceptional set, where Z=π⁻¹(SingX) is the unreduced exceptional divisor, such that

K^s_X =π_∗ K_M ⊗ O(−D)

. (4.5)

In many situations, e.g., if dimX = 2 or ifX has only homogeneous singu- larities, then one can takeD=Z− |Z|. In the present paper, we will assume from now on that this is actually the case, and show how the L²-theory from [17] can be improved under this assumption.

So, for the rest of the paper, we assume that (4.5) holds withD=Z−|Z|.

5. Resolution ofL²_loc-cohomology of (n, q)-forms

In this Section, we prove a slight variation of Theorem 1.11 in [17]:

Theorem 5.1. — Let X be a Hermitian complex space of pure dimen- sionn >2 with only isolated singularities and π :M →X a resolution of singularities with only normal crossings such that

K^s_X∼=π_∗ KM⊗ O(|Z| −Z) ,

whereK^s_X is the canonical sheaf for the∂s-operator (i.e. the canonical sheaf of holomorphic (n,0)-forms with Dirichlet boundary condition), KM is the usual canonical sheaf onM andZ =π⁻¹(SingX)the unreduced exceptional divisor.

Then the pull-back of forms under π induces for p > 1 natural exact sequences

0→H^p(X,K^s_X)^[π

∗ p]

−→H^p(M,KM⊗ O(|Z| −Z))−→Γ(X,R^p)→0, (5.1) 0→H_cpt^p (X,K^s_X)^[π

∗ p]

−→H_cpt^p (M,K_M ⊗ O(|Z| −Z))−→Γ(X,R^p)→0, (5.2) whereR^p is the higher direct image sheafR^pπ_∗(KM⊗ O(|Z| −Z)).

In [17], we stated Theorem 5.1 under the additional assumption thatX is compact (see [17, Theorem 1.11]), but the proof in [17] actually gives also the statement of Theorem 5.1 above. New in the present paper is the proof of Theorem 5.1. Whereas in [17], we just gave an abstract proof by use of the Leray spectral sequence, we give here an explicit realization of all the mappings in (5.1) and (5.2) in terms of differential forms. Clearly, statement (5.2) is also new because it is contained in (5.1) ifX is compact.

The proof of Theorem 5.1 is based on the following observation. IfC^n,∗

is a fine resolution ofKM⊗ O(|Z| −Z), then the non-exactness of the direct image complexπ∗C^n,∗ can be expressed by the higher direct image sheaves R^p =R^pπ_∗KM⊗ O(|Z| −Z), p>1. These are skyscraper sheaves because X has only isolated singularities. So, they are acyclic. On the other hand, global sections inR^p can be expressed globally by L²-forms with compact support. These two properties allow to express the cohomology of the canon- ical sheafK_X^s in terms of the cohomology of the direct image complexπ_∗C^n,∗

modulo global sections in R^p. Another difficulty is to show that the exact sequences in Theorem 5.1 are actually induced by the pull-back ofL²-forms (see Lemma 5.2). We will also see that the surjections in (5.1) and (5.2) are simply induced by taking germs of differential forms. So, we obtain an explicit and localized proof of Theorem 1.11 in [17].

5.1. Proof of Theorem 5.1

We can use Theorem 2.6 to represent the cohomology groups H^q(X,K^s_X)∼=H^q(Γ(X,F^n,∗)),

H_cpt^q (X,K^s_X)∼=H^q(Γcpt(X,F^n,∗))

in terms of cohomology groups on the resolution π : M → X, where we letπ : M → X be a resolution of singularities as in Theorem 1.1 so that D = Z − |Z|. Recall also that X is a Hermitian complex space of pure dimensionnwith only isolated singularities.

Clearly, we intend to use Theorem 2.6 withA=K^s_X and (A^∗, a∗) = (F^n,∗, ∂s),

so that 0→ A → A^∗ is a fine resolution ofAoverX by [17, Theorem 1.9].

By assumption,

A=K^s_X∼=π∗ KM⊗ O(|Z| −Z) , so that we can choose

C=KM ⊗ O(|Z| −Z)

for the application of Theorem 2.6. It remains to choose a suitable fine res- olution 0 → C → C^∗. Since (M, σ) is an ordinary Hermitian manifold, we can use the usualL²_σ-complex of forms with values inO(|Z| −Z). To realize that, we can adopt two different points of view. First, letL_|Z|−Z →M be the holomorphic line bundle associated to the divisor|Z| −Zsuch that holo- morphic sections of L_|Z|−Z correspond to sections of O(|Z| −Z), and give L_|Z|−Z the structure of a Hermitian line bundle by choosing an arbitrary positive definite Hermitian metric. Then, denote by

L^p,q_σ (U, L_|Z|−Z), L^p,q_σ,loc(U, L_|Z|−Z)

the spaces of (locally) square-integrable (p, q)-forms with values in L_|Z|−Z (with respect to the metricσon M and the chosen metric on L_|Z|−Z). We can then define the sheaves of germs of square-integrable (p, q)-forms with values inL_|Z|−Z,L^p,q_σ (L_|Z|−Z), by the assignment

L^p,q_σ (L_|Z|−Z)(U) =L^p,q_σ,loc(U, L_|Z|−Z).

The second point of view is to use the sheavesL^p,q_σ ⊗ O(|Z| −Z) which are canonically isomorphic to the sheavesL^p,q_σ (L_|Z|−Z). Let us keep both points of view in mind. As in (4.4), let

C_σ,E^p,q(L_|Z|−Z) :=L^p,q_σ (L_|Z|−Z)∩∂⁻¹_w,EL^p,q+1_σ (L_|Z|−Z), (5.3) where ∂w,E is the ∂-operator in the sense of distributions with respect to compact subsets of M−E (and for forms with values inL_|Z|−Z). Since σ is positive definite, the∂-equation in the sense of distributions forL²_σ-forms