Adapted differentials as a qfh-sheaf

We consider differential forms associated to Campana’s geometric orbifolds from a new perspective, namely, as a qfh-sheaf on the variety underlying the geometric orbifold. This approach avoids having to choose a covering of the underlying variety, which is one of the drawbacks of a common way to work with these differential forms.

As in the case of smooth projective varieties or logarithmic pairs, one of the fundamental tools to study the geometry of such an object are differential forms on it.Associated to a geometric orbifold we should have differential forms with logarithmic poles of fractional order.These differential forms are only well-defined on a covering space of our geometric orbifold, and this leads to the notion of adapted differentials.This has two disadvantages: • The definition of adapted differentials requires the choice of a suitably ramified cover.This choice is not unique, and the properties of the resulting sheaf, such as the dimension of its space of global sections, depend on the chosen cover.• These differentials do not form an object on a priori, like the sheaf of Kähler differentials on a projective variety or the sheaf of logarithmic differentials on a logarithmic pair do.But there is a natural object on that takes all covers into account at once, namely, the presheaf on Sch/ described in Construction 28.We show in Theorem 1 that this presheaf is a qfh-sheaf on Sch/ in the sense of Suslin and Voevodsky.The qfh-topology is recalled in Section 2.4.1.2.Main result.To make the following discussion a bit more selfcontained, let us just say here that an adapted morphism is a quasi-finite morphism : → of normal varieties of the same dimension such that * Δ has integer coefficients, and that on the domain of such a morphism we can define the sheaf of adapted differential -forms Ω [ ] ( ,Δ, ) , which are the kind of differential forms that we want to consider.See Definition 16.
Theorem 1.Let be a normal variety over C and let Δ be a Weil Q-divisor as above.For every ∈ N, there exists a presheaf Ω ( ,Δ) on Sch/ , unique up to isomorphism, with the following universal property: (a) For every adapted morphism : → there exists a morphism ( ,Δ, ) ( ), and these morphisms are compatible with the pull-back of differential forms.(b) For every presheaf H on Sch/ satisfying (a), there exists a unique morphism H → Ω ( ,Δ) compatible with the morphisms from (a).
Moreover, the morphisms in (a) are isomorphisms.Furthermore, Ω ( ,Δ) is a sheaf with respect to the qfh-topology on Sch/ .
Remark 2. In particular, we can recover Ω [ ] ( ,Δ, ) as the restriction of Ω ( ,Δ) to the small Zariski site on , because every dense open subset of the domain of an adapted morphism induces another adapted morphism.
Remark 3. Adapted differentials are in particular reflexive (logarithmic) differentials on the domain of an adapted morphism.Usual reflexive differentials on normal varieties are known to have qfh-descent, in the sense that Lemma 26 also holds for reflexive differentials [LW09, Théorème 2.2.2].However, reflexive differentials do not form a presheaf on the whole Nor/ , because it is not always possible to define a meaningful pull-back, cf.[Keb13, §1.2] or [KS21, §B.1].Note also that the analogous qfh-descent property is known to fail for usual Kähler differentials [LW09, Exemple 2.2.5].
Remark 4. The universal property in Theorem 1 can be conveniently rephrased as a Kan extension, see Section 2.3 and Remark 27.
Being a presheaf means that we can pull back sections along morphisms, and being a presheaf with transfers means, roughly speaking, that we can more generally pull back sections along finite correspondences.
Corollary 5.In the setting of Theorem 1, the presheaf Ω ( ,Δ) admits a natural structure of presheaf with transfers.
We refer to [CD19,§10] for a more detailed discussion on presheaves with transfers.
1.3.Idea of the proof.Adapted differentials form a Zariski sheaf on the domain of each adapted morphism, see Definition 16.Taking their global sections gives us a presheaf F on the full subcategory Adpt( , Δ) ⊆ Sch/ of adapted morphisms, see Lemma 22.This presheaf can be extended to a presheaf Ω ( ,Δ) on Sch/ as a Kan extension, see Construction 28.The universal property and the description of the sections over adapted morphisms follow directly from the construction.
We can also express Ω ( ,Δ) as the Kan extension of G to Sch/ , where G is the Kan extension of F to the category Nor/ of integral normal schemes over , cf.Section 2.3.By [SV96, Lemma 6.4], it suffices to show that G satisfies the following properties: (i) The presheaf G restricts to a Zariski sheaf on any integral normal scheme over .(ii) For any integral normal scheme over and every finite Galois extension /C( ), the restriction along the normalization ′ → of in gives a bijection G( ) G( ′ ) Gal( /C( )) , where the action of the Galois group on G( ′ ) is given by pulling sections back along the corresponding automorphisms of ′ over .Condition (i) is a formal consequence of the construction of the presheaf, cf.Section 3.1.The idea for the surjectivity in condition (ii) is the following.Suppose : ′ → is such a normalization of an integral normal scheme over in a finite Galois field extension with Galois group and suppose we are given some ′ ∈ G( ′ ) .To define ∈ G( ) we need to specify an adapted differential for every : → such that the composition → is an adapted morphism, cf.Construction 28.We consider the commutative diagram ′ ′ ′ in which ′ is the normalization of the fiber product ′ × .Assume for simplicity that ′ is irreducible, so that the composition ′ → is an adapted morphism.We are forced by Construction 28 to take an adapted differential with the property that * ( ) = ′ ′ .The group acts on ′ , and ( , ) is a quotient for this action in the sense of [SGA 1, Exposé V, §1].Therefore, the existence of such an adapted differential is ensured by the property that 1.4.Outline of this paper.In Section 2 we discuss the necessary preliminaries on category theory and on adapted differentials.Most of these preliminaries on adapted differentials are well-known to experts, but we include proofs for convenience.In Section 3 we prove Theorem 1.In Section 4 we compute the cohomology of Ω ( ,Δ) , as well as its sections over some quasi-finite schemes over .1.5.Acknowledgements.I would like to thank Stefan Kebekus for asking the question that motivated this result and for many useful discussions and comments on an earlier version of this paper.I would like to thank Annette Huber-Klawitter for many useful discussions.I would like to thank Shane Kelly for pointing out to me the relevance of presheaves with transfers.I would also like to thank my fellow PhD students in Freiburg, especially Luca Terenzi, Vivien Vogelmann and Giovanni Zaccanelli, for the parallel study sessions which I spent thinking about this question.Finally, I would like to thank an anonymous referee for many useful comments and suggestions.

P
2.1.Notation and conventions.We follow the notation and terminology in [Har77] and [KM98].In particular, varieties are assumed to be irreducible, and if is a Weil Q-divisor on a normal variety , then we denote by ⌈ ⌉ (resp.⌊ ⌋) the divisor obtained by rounding up (resp.down) its coefficients.We work over the complex numbers C and fix a pair ( , Δ) as above once and for all.Some other notation and conventions: • All schemes are assumed to be of finite type over C.
• We denote by Sch/ the category of schemes over .
• We denote by Nor/ the full subcategory of Sch/ consisting of integral normal schemes over .In Section 2.6 we will say that the pull-back of Kähler differentials induces a morphism of reflexive logarithmic differentials to mean that there is a morphism between sheaves of reflexive logarithmic differentials which agrees with the pull-back of Kähler differentials wherever this makes sense, e.g., over the complement of the support of the boundary divisor inside the smooth locus.
2.3.Kan extensions.Kan extensions provide a convenient language to discuss a construction that will be used repeatedly in the proof of Theorem 1.The geometric idea behind this construction is the following.Suppose we are given a presheaf defined only on a basis of open subsets of a topological space.Then, there is a natural way to extend it to a presheaf on the whole topological space.Namely, by setting the sections over any given open subset to be the limit of the sections over basic open subsets contained in that open subset, cf.[EGA I, Chap.0, (3.2.1)].
In the categorical setting, let : A → B be a functor.Then we have a restriction functor on presheaves : PSh(B) → PSh(A), and this functor has a right adjoint : PSh(A) → PSh(B) which corresponds to the previous geometric construction, cf.[SP,00XF].If is a presheaf on A, then ( ) = Ran ( ) is the right Kan extension of along , cf. [Mac98, §X.3].We will use the following two observations: Remark 6.Let : A → B be a fully faithful functor and let be a presheaf on A. Then, the restriction of the Kan extension Ran ( ) to A is canonically isomorphic to itself.See [Mac98, Corollary X.3.3] and the proof of Lemma 29.

qfh-topology.
A morphism of schemes : → is called a topological epimorphism if the Zariski topology on is the quotient topology of the Zariski topology on , i.e., if is surjective and a subset ⊆ is open if and only if −1 ( ) is open.It is called a universal topological epimorphism if for any → the projection from the fiber product : × → is a topological epimorphism.We recall the definition of the qfh-topology from [Voe96, Definition 3.1.2]:Definition 8 (qfh-topology).The qfh-topology on the category of schemes is the Grothendieck topology associated with the pretopology whose coverings are of the form { : → }, where { } is a finite family of quasi-finite morphisms of finite type such that the morphism ⊔ : ⊔ → is a universal topological epimorphism.
Remark 9. Let be a normal connected scheme and let { : → } ∈ be a finite family of quasi-finite morphisms of finite type such that all are irreducible.Denote by the set of those for which dominates .Then the above family is a qfh-covering if and only if = ∪ ∈ ( ), cf.[SV96, Lemma 10.1].
To prove Theorem 1, we will use the following: Lemma 10 (cf.[SV96, Lemma 6.4]).Let H be a presheaf on Nor/ .Assume that H satisfies the Zariski condition (i) and the Galois condition (ii) spelled out in Section 1.3.Then the right Kan extension of H to Sch/ is a qfh-sheaf.
Proof.This right Kan extension is given by the right adjoint to the restriction functor PSh(Sch/ ) → PSh(Nor/ ), cf.Remark 27.Therefore, it is given by the functor : PSh(Nor/ ) → PSh(Sch/ ) from [SV96, §6].The arguments in the proof of [SV96, Lemma 6.4] do not use the assumption that we are working over a field, but only that we are working with finite type schemes over a field, cf.[SV96, Lemma 10.3].This is still true in our case, because is itself of finite type over C, hence the claim.Definition 11 (Adapted morphism).An adapted morphism with respect to ( , Δ) is a quasi-finite morphism : → such that is a normal variety with dim( ) = dim( ) and such that * Δ is a Weil divisor with integer coefficients.
We denote by Adpt( , Δ) the full subcategory of Sch/ whose objects are adapted morphisms with respect to ( , Δ).
Example 12. Let ( , Δ) be a geometric orbifold with ⌈Δ⌉ − Δ ≠ 0. Then id is a quasi-finite morphism between normal varieties of the same dimension, but it is not an adapted morphism with respect to ( , Δ).

Example 13. Let
is an adapted morphism with respect to ( , Δ) if and only if ∈ 3Z.
We will only consider a single pair ( , Δ), so from now we will simply refer to adapted morphisms with respect to ( , Δ) as adapted morphisms.
Remark 14.If 1 : 1 → is an adapted morphism and : 2 → 1 is a quasi-finite morphism between normal varieties of the same dimension, then 1 • : 2 → is again an adapted morphism.
We refer to [Del70, II.§3] for the necessary background on logarithmic differentials and to [CP19, §5] for a discussion motivating the following: Definition 16 (Adapted differentials).Let be a set of indices such that Δ = ∈ −1 with distinct prime Weil divisors.We denote by 0 ⊆ the subset of indices such that ∈ N. Let : → be an adapted morphism and assume first that ( , Δ) and ( , * Δ) are both snc pairs.Then is flat, so * O = O * for all ∈ , where * is regarded here as a subscheme of with the appropriate non-reduced structure.Moreover, since * ≥ * ( 1 ) for all ∈ 0 , we have a quotient morphism which is an epimorphism of O -modules.We define the O -module of adapted differential 1-forms as For each ∈ N, the O -module of adapted differential -forms is defined as , which is then locally free because Ω 1 ( ,Δ, ) was locally free.In general, we consider the largest open subset ⊆ such that ( , Δ) is snc over .This is a big open subset, i.e., the codimension of its complement is at least 2. Since is quasi-finite between varieties of the same dimension, . Indeed, by definition we have * Ω 1 X = Ω 1 , and the pull-back of Kähler differentials * Ω 1 → Ω 1 induces an isomorphism onto the subsheaf generated by −1 , which is precisely Ω 1 ( ,Δ, ) .In more general situations, similar computations can be used to argue analytic locally over the smooth loci of ( , Δ) and ( , * Δ), cf.proof of Lemma 21.
2.6.Pull-back of adapted differentials.From now on we fix a ∈ N.
Proof.Again by torsion freeness it suffices to check this over a dense open subset, over which it follows from the corresponding statement for pullback of Kähler differentials along the dominant morphism .
Lemma 25.Let : → be an adapted morphism and let ⊆ be an open subset.Then the restriction map F( ) → F( ) is the usual restriction map of the sheaf Ω [ ] ( ,Δ, ) .Proof.Follows from torsion freeness and the corresponding statement for Kähler differentials as before.
Lemma 26.Let 1 : 1 → and 2 : 2 → be two adapted morphisms and let : 2 → 1 be a morphism over such that there exists a finite group acting on 2 in such a way that ( 1 , ) is a quotient of 2 by .Then the restriction map * induces a bijection F( 1 ) → F( 2 ) .If denotes the automorphism of 2 corresponding to ∈ , then • = , so the previous isomorphism is also an isomorphism of -sheaves with respect to the -sheaf structure induced on the pull-back.Hence, the assumption that ( 1 , ) is a quotient of 2 by and the projection formula imply that and taking global sections yields the desired bijection.

P T 1
To motivate Construction 28, we start with the following observation: Remark 27.If we denote by : PSh(Sch/ ) → PSh(Adpt( , Δ)) the restriction functor, then the universal property in Theorem 1 can be rephrased as follows: (a) There exists a morphism : (Ω ( ,Δ) ) → F. (b) For every presheaf H on Sch/ such that there exists a morphism : (H) → F, there exists a unique morphism : In other words, Ω ( ,Δ) is the right Kan extension of F along the inclusion : Adpt( , Δ) op → (Sch/ ) op .
Construction 28.We define Ω ( ,Δ) as the right Kan extension of F to Sch/ .Then Ω ( ,Δ) satisfies the universal property in Theorem 1 by construction, see Remark 27.Recall from Section 2.3 that we can write Ω ( ,Δ) = (F), where : PSh(Adpt( , Δ)) → PSh(Sch/ ) is the right adjoint to the restriction functor, as described in [SP,00XF].Therefore, we can explicitly describe the sections of this presheaf as (1) So a section = ( ) ∈ Ω ( ,Δ) ( ) consists of a family of adapted differentials indexed by morphisms : → such that the composition → is an adapted morphism.This family is moreover compatible in the sense that for every morphism : ′ → such that the composition ′ → is an adapted morphism.If : ′ → is a morphism and we are given a section ∈ Ω ( ,Δ) ( ), then Ω ( ,Δ) ( )( ) is the family uniquely determined by the property that for every morphism ′ : → ′ such that the composition → is an adapted morphism.
Proof.This is a formal consequence of the inclusion : Adpt( , Δ) → Sch/ being fully faithful, cf.Remark 6.Let us make this explicit in the case at hand.Since the inclusion : Adpt( , Δ) → Sch/ is fully faithful, the indexing category in Equation ( 1) has an initial object, namely the identity on .Hence Ω ( ,Δ) ( ) F( ).More explicitly, the isomorphism is given by sending a family ( ) to the component id Y at id , and the inverse is given by pulling back along the corresponding morphisms.
It remains to show that Ω ( ,Δ) is a qfh-sheaf on Sch/ .As outlined in the introduction, we will apply Lemma 10 to the right Kan extension of F to Nor/ .
Notation 30.We denote by G the right Kan extension of F to Nor/ , where F is the presheaf from Notation 23.By Remark 7 and Remark 6, we may also regard G as the restriction of Ω ( ,Δ) to Nor/ , so its sections are described by the same equations in Construction 28 that describe the sections of Ω ( ,Δ) .

Zariski condition (i).
We need to show that for any integral normal -scheme , the presheaf G defines a sheaf in the small Zariski site on [SV96, Definition 6.1].Arbitrary fiber products do not exist in Adpt( , Δ) nor in Nor/ , but fiber products along open immersions do exist.Moreover, if → is in Adpt( , Δ) (resp. in Nor/ ) and → is an open immersion, then the composition → is again in Adpt( , Δ) (resp. in Nor/ ).Therefore, taking Zariski coverings as coverings, both Adpt( , Δ) and Nor/ are sites in the sense of [SP,00VH], and it suffices to show that G is a sheaf on Nor/ with respect to the corresponding Zariski topology.Since the sheaves of adapted differentials are Zariski sheaves and the restriction maps of F are compatible with the restriction maps of these sheaves by Lemma 25, F is a sheaf with respect to the Zariski topology on Adpt( , Δ).The inclusion functor Adpt( , Δ) → Nor/ is cocontinuous [SP, 00XJ], so the sheaf property is preserved by the right Kan extension [SP, 00XK] and G is a Zariski sheaf on Nor/ .

Galois condition (ii).
In this subsection we fix a normal integralscheme → and a finite Galois extension /C( ) of its function field C( ).We denote by the Galois group of this extension and by : ′ → the normalization of in [GW10, Definition 12.42].
We will use the following observation.This is not a new result, but we include a proof for convenience: Lemma 31.Let and ′ be normal integral schemes and let : ′ → be a morphism.Then is a quotient of ′ by the action of a finite group if and only if induces a finite Galois extension of function fields with Galois group and ′ is the normalization of in this extension of its function field.
Proof.Suppose ( , ) is a quotient of ′ by a finite group in the sense of [SGA 1, Exposé V, §1].Then is finite and the statement that we want to show is local on , so we may assume that both ′ and are affine, say ′ = Spec( ) and = Spec( ).The group acts on and the quotient morphism allows us to identify = , cf. [SGA 1, Exposé V, Proposition 1.1].By [AM69, Exercise 5.12] we have an induced -action on (0) such that (0) = ( (0) ) , so C( ) ⊆ C( ′ ) is a finite Galois extension with Galois group .It remains to show that ′ is the normalization of inside C( ′ ), i.e., we want to show that = { ∈ (0) | integral over }.Every element in is integral over , because ⊆ is integral.And conversely, if ∈ (0) is integral over , then it is also integral over .Since ′ is normal, we deduce that ∈ .
Conversely, suppose that ′ is the normalization of in C( ′ ) and that C( ) ⊆ C( ′ ) is a finite Galois extension with Galois group .We show that acts on ′ with quotient ( , ).The statement is local on and normalization is a finite morphism in our setting, so we may assume again that both ′ = Spec( ) and = Spec( ) are affine.By assumption there is an -action on (0) such that (0) = ( (0) ) , and we want to show that it induces an -action on such that = .Given ∈ non-zero and ℎ ∈ , since ⊆ is an integral extension, there exists a monic polynomial ∈ [ ] such that ( ) = 0. Since the coefficients of are -invariant, this implies that (ℎ • ) = 0, so ℎ • is also integral over , hence over .Since ′ is normal, ℎ • ∈ .So the -action on (0) induces an -action on .Since (0) = ( (0) ) and is normal, we have = .
So in our current setting : ′ → is the quotient of ′ by the induced -action on ′ .For each ∈ we denote by the corresponding automorphism of ′ .Then we have a -action on G( ′ ) given by pulling back along the corresponding automorphism.The restriction map G( ) : G( ) → G( ′ ) has image contained in G( ′ ) , because • = for all ∈ .Our goal in this subsection is to show that this restriction map is a bijection onto G( ′ ) .
We want to show that = ′ , so let : → be a morphism such that the composition → is an adapted morphism and let us show that = ′ .Let ′ be a component of the normalization of the fiber product ′ × that surjects onto , considered as a subscheme with its induced reduced structure, and let : ′ → be the induced surjective morphism, which is also finite because so are the normalization, the closed immersion and the pull-back of .Let ′ : ′ → ′ be the induced morphism, so that Proof.Let ′ ∈ G( ′ ) be a -invariant section, i.e., G( )( ′ ) = ′ for all ∈ , where denotes the automorphism of ′ corresponding to .We want to find a section ∈ G( ) such that G( )( ) = ′ , so let : → be a morphism such that the composition → is an adapted morphism.Let ′ 0 be any irreducible component of the fiber product ′ × , considered as a subscheme with its induced reduced structure.Let 0 be the stabilizer of ′ 0 in , so that the induced morphism ′ 0 → is a quotient of ′ 0 by 0 , cf. [Hub00, Lemma 2.1.11]and [SV96, Corollary 5.10].Let ¯ ′ 0 be the normalization of ′ 0 .Then 0 acts on ¯ ′ 0 as well and the induced morphism 0 : ¯ ′ 0 → is still a quotient for this action, cf.Lemma 31.Let 0 : ¯ ′ 0 → ′ be the induced morphism, so that • 0 = • 0 and the composition ¯ ′ 0 → is an adapted morphism: The desired ∈ G( ) must then have the property that ( 0 ) * ( ) Let now ∈ 0 and let denote the corresponding automorphism of ¯ ′ 0 .From Equation (2), Equation (3) and the equality 0 • = • 0 we deduce that * ( ′ 0 ) = ′ 0 , hence ′ 0 is a 0 -invariant adapted differential on ¯ ′ 0 .By Lemma 26 we can find an adapted differential on such that ( 0 ) * ( ) = ′ 0 .Moreover, the resulting adapted differential is independent of the chosen irreducible component ′ 0 .Indeed, let ′ 1 be another irreducible component and let be an adapted differential on such that ( 1 ) * ( ) = ′ 1 , where 1 and 1 are defined as before but for the irreducible component ′ 1 instead.Since acts transitively on the irreducible components of the fiber product [SV96, Corollary 5.10], we can find some ∈ such that the corresponding automorphism of the fiber product induces an isomorphism ′ 0 → ′ 1 over , hence also an isomorphism : ¯ ′ 0 → ¯ ′ 1 over .From -invariance of ′ , the equality • 0 = 1 • , the equality 0 = 1 • and the usual equations we have hence = by Lemma 24.We show next that the family of adapted differentials constructed in this manner satisfies Equation (2), i.e., that ∈ G( ).Let : → be a morphism such that the composition → is an adapted morphism and let : → be a morphism such that the composition → is an adapted morphism as well.Since : → is dominant, we may find an irreducible component of ′ × which dominates an irreducible component of ′ × , say ′ 0 → ′ 0 .From the universal property of the normalization we obtain now a morphism 0 : ¯ ′ 0 → ¯ ′ 0 such that • 0 = 0 • 0 , where 0 : ¯ ′ 0 → and 0 : ¯ ′ 0 → are defined as in the last paragraph: From Equation (2) for ′ , the equality • 0 = 0 • 0 and the construction of we deduce that ( 0 ) * ( * ( )) = ( 0 ) * ( • ), hence * ( ) = • by Lemma 24.
To finish the proof we need to check that G( )( ) = ′ , so let : → ′ be a morphism such that the composition → is an adapted morphism.Let 0 : ¯ ′ 0 → and 0 : ¯ ′ 0 → ′ be constructed as above with respect to the morphism := • .For every ∈ , denote by the equalizer of • 0 and • 0 , which is a closed subscheme of ¯ ′ 0 because is separated [GW10, Definition and Proposition 9.7.(ii)].We claim that ¯ ′ 0 = ∪ ∈ as sets.Indeed, it suffices to show that every closed point of ¯ ′ 0 belongs to for some ∈ , because all are closed subspaces and the set of closed points is dense in ¯ ′ 0 .But we are working over an algebraically closed field, so this follows from [GW10, Exercise 9.7] and from the equality • 0 = • • 0 , in which is a quotient of ′ by .Since ¯ ′ 0 is irreducible and is finite, there exists some ∈ such that ¯ ′ 0 = as topological spaces.But ¯ ′ 0 is reduced, so we must have ¯ ′ 0 = and thus 1 be a normalized component of the fiber product such that induces an isomorphism : ¯ ′ 0 → ¯ ′ 1 with 1 • = • 0 , notation again as above.Then the following diagram remains commutative after adding the dashed arrow: G).In Section 3.1 and Section 3.2 we have seen that G is a qfh-sheaf on Nor/ in the sense of Suslin-Voevodsky, i.e., that it satisfies conditions (i) and (ii) from the introduction.Therefore, by Lemma 10, its Kan extension Ω ( ,Δ) is a qfh-sheaf on Sch/ .

S
The value Ω ( ,Δ) ( ) can be described more explicitly for quasi-finite separated -schemes → .In Proposition 34 we compute the restriction of Ω ( ,Δ) to the small Zariski site on , which allows us to compute the cohomology of Ω ( ,Δ) in Corollary 35.The cases dim( ) < dim( ) and dim( ) = dim( ) are briefly discussed in Remark 36 and Remark 37 respectively.

Ω
[ ] (log ⌊Δ⌋) → Ω ( ,Δ) | of sheaves on the topological space .We show that it is also surjective.Suppose we are given a family ( ) ∈ Ω ( ,Δ) ( ).Let 0 ⊆ be the largest open subset over which and Supp(Δ) are smooth, which is a big open subset contained in the open subset ⊆ from Definition 16.Surjectivity is local on , so after possibly shrinking , we may assume that we are in the setting of [EV92,§3.5].Therefore, we may assume that there exists an adapted morphism 0 : 0 → factoring through 0 ⊆ such that there exists a finite group acting on 0 so that 0 → 0 is a quotient of 0 by .Equation (2) ensures that the logarithmic differential 0 isinvariant, hence Hurwitz's formula [EV92, Lemma 3.16] implies that there exists a logarithmic differential ∈ Ω 0 (log ⌊Δ| 0 ⌋)( 0 ) such that * 0 = 0 .This logarithmic differential defines a reflexive logarithmic differential in Ω [ ] (log ⌊Δ⌋)( ) which is the desired preimage of the given section of Ω ( ,Δ) .The same argument works for sections defined over smaller open subsets ⊆ , so the morphism above is surjective.
Corollary 35.In the setting of Theorem 1 we have q , Ω ( ,Δ) , Ω [ ] (log ⌊Δ⌋) for all ∈ N, where the left-hand side denotes cohomology with respect to the qfh-topology on Sch/ and the right-hand side denotes usual sheaf cohomology on .
Proof.Since normal schemes are geometrically unibranch [SP, 0BQ3] and qfh-sheaves are étale sheaves, we can apply [CD19, Proposition 10.5.12] and [CD19, Theorem 10.5.10] to compute the left-hand side with respect to the étale topology on Sch/ .Étale cohomology groups can be computed in the small étale site instead [SP,03YX].By Proposition 34, Ω ( ,Δ) | is a quasi-coherent sheaf on , so we can in turn compute its étale cohomology groups as usual Zariski sheaf cohomology on [SP,03DW].
Remark 37.If → is quasi-finite with dim( ) = dim( ), then the normalization ¯ → induces a canonical isomorphism Ω ( ,Δ) ( ) Ω ( ,Δ) ( ¯ ) as a consequence of its universal property [SP,035Q].Combining this with the Zariski sheaf property one can express Ω ( ,Δ) ( ) as the product of the sections over its irreducible components, and with arguments similar to those in Proposition 34 this allows one to compute the sections of Ω ( ,Δ) over any quasi-finite separated -scheme → .
Motivation.Geometric orbifolds play a central role in Campana's programme for the birational classification of algebraic varieties [Cam04].They are pairs ( , Δ) consisting of a normal variety and a Weil Q-divisor of the form Δ = −1

2. 5 .
Adapted morphisms and differentials.Quasi-finite morphisms between normal varieties of the same dimension are open and dominant.We can pull back Cartier divisors along dominant morphisms [SP, 02OO], and Weil divisors on a normal variety are Cartier over the regular locus, which is a big open subset.The preimage of a big open subset under a quasi-finite morphism of varieties of the same dimension is again a big open subset.Therefore, quasi-finite morphisms between normal varieties of the same dimension induce a pull-back homomorphism on Weil divisors.This allows us to define: −1 ( ) is a big open subset as well.We can then consider the largest open subset ⊆ −1 ( ) such that ( , * Δ) is snc over , which is a big open subset of .We define the O -module of adapted differentials as Ω [ ] ( ,Δ, ) := * Ω ( ,Δ| , | ) , where : → denotes the open immersion.Remark 17.The sheaves Ω [ ] ( ,Δ, ) are coherent and reflexive, because they are the push-forward of locally free sheaves defined over a big open subset, see [SP, 0AY6].Example 18.If Δ = 0, then any quasi-finite morphism : → between normal varieties of the same dimension is an adapted morphism, and in this case we have [ * ] Ω [ ] Ω [ ] ( ,Δ, ) for all ∈ N, cf. also Lemma 21 below.Example 19.In the setting of Example 13, if = 3 for some ∈ N >0 , then Ω 1 ( ,Δ, ) is the subsheaf of Ω 1 A 1 generated by −1 .Remark 20.Root stacks have been used in the context of symmetric differentials on geometric orbifolds, see [Rou12, Proposition 3.4].If the construction of the (iterated) root stack of the boundary divisors is possible, e.g., if is locally factorial, then we can also understand adapted differentials in terms of root stacks.Namely, adapted differentials are the pull-back of differential forms defined on the root stack.To showcase the argument, let us look at a simple example.Let = Spec(C[ ]), Δ = −1 { = 0} for some ∈ N ≥2 , = Spec(C[ ]) and : → given by ↦ → with = for ∈ N >0 .Let = Spec(C[ ]) Spec(C[ , ]/( − )) and consider the cyclic quotient :→ given by ↦ → .Denoting by ⊆ C × the group of -th roots of unity, we can describe the -th root stack of { = 0} ⊆ as the quotient stack X = [ / ], where the action of on is given by • = [Ols16, Theorem 10.3.10].The morphism factors as • , where : X → is the coarse moduli space and : → X is an atlas of the DM stack X.Since * { = 0} = { = 0} is an -th power, the morphism lifts to a morphism ˜ : → X over .This can in turn be lifted to the morphism : → given by ↦ → , and we obtain the following commutative diagram: X .˜Inthis situation, the pull-back of Kähler differentials induces an isomorph- Proof.The -action on 2 induces a -sheaf structure on logarithmic differentials [GKKP11, Fact 10.5], hence on adapted differentials by Lemma 21.The sheaf( * Ω [ ] ( ,Δ, 2 ) ) is then reflexive [GKKP11, Lemma A.4],so we may restrict our attention to big open subsets and assume that ( , Δ), ( 1 , * 1 Δ) and ( 2 , * 2 Δ) are all snc.By Lemma 21, pull-back of Kähler differentials induces an isomorphism * Ω ( ,Δ, 1 ) Ω ( ,Δ, 2 ) .