Philippe Eyssidieux
email: philippe.eyssidieux@ujf-grenoble.fr

Francesco Sala

INSTANTONS AND FRAMED SHEAVES ON K ÄHLER DELIGNE-MUMFORD STACKS

We provide stacky generalizations of classical gauge-theoretic results inspired by Donaldson, the Uhlenbeck-Yau theorem and variants due to Bando and his collaborators.

Introduction

It is a general principle (or a metatheorem), which was explained to us by C. Simpson, that the basic results of Differential Geometry (Real or Complex) obtained through the analysis of Geometric Partial Differential Equations extend to Deligne-Mumford stacks 1 . The main obstacle to carry out this extension is to set up the stacky definitions correctly. The orbifold versions are (in principle) well-known but nontrivial generic isotropy is not a very serious technical obstacle. Doing the proof of such a result for stacks amounts to checking that the extension to stacks of the constructions used in the proof is possible and sufficiently well documented in the literature. In this article, we enforce this principle and state stacky generalizations of classical gauge-theoretic results inspired by Donaldson, the Uhlenbeck-Yau theorem [START_REF] Uhlenbeck | On the existence of Hermitian-Yang-Mills connections in stable vector bundles[END_REF] and variants due to Bando and his collaborators.

Our policy with respect to writing-up these results will be to give no details on their deep gauge-theoretic or analytical aspects and concentrate on purely stack-theoretic issues. Perhaps, one may regret that no comprehensive reference on global analytic methods on stacks in the style of [START_REF] Laumon | Ergebnisse der Mathematik und ihrer Grenzgebiete[END_REF] is available. If more applications of this method in the realm of ordinary complex/differential geometry are developed, it may become inevitable to lay down these foundations.

Date: April 14, 2014. 1 Complex differential geometry on Artin stacks is much more delicate but may have more applicationshowever we will not pursue this direction here. Our motivation was to generalize the following correspondence due to Donaldson to instantons on ALE spaces. In [START_REF] Donaldson | Instantons and geometric invariant theory[END_REF] Donaldson proved that there is a one-to-one correspondence between U (r)-instantons of charge n on R 4 (modulo gauge equivalence) and framed vector bundles of rank r and second Chern class n on the complex projective plane P 2 = C 2 ∪ l ∞ . A framed vector bundle is a pair (E, φ E ), where E is a vector bundle on P 2 of rank r and c 2 (E) = n and φ E : E |l∞ ∼ -→ O ⊕r l∞ a trivialization along the line l ∞ . Donaldson's correspondence was generalized to instantons on the blowup C 2 of C 2 at n points (for n = 1 by King [START_REF] King | Instantons and holomorphic bundles on the blow-up plane[END_REF] and for n ≥ 2 by Buchdahl [START_REF] Buchdahl | Instantons on nCP2[END_REF]). In all these examples, in order to state a correspondence between instantons and framed vector bundles one needs first to endow the noncompact manifold of a Kähler structure, compactify it by adding a projective line, and consider framed vector bundles on the corresponding smooth projective surface.

A natural question to ask is if this algebro-geometric approach can be applied to other noncompact 4-dimensional Riemannian manifolds. In particular, we are interested in ALE spaces of type A k-1 , where k ≥ 2 is an integer. A first result in the ALE case is due to Bando: in [START_REF] Bando | Einstein-Hermitian metrics on noncompact Kähler manifolds[END_REF], as a consequence of a generalization of Donaldson's characterization of instantons on R 4 to general Kähler manifolds, he proved that if one can compactify an ALE space X • of type A k-1 by adding a smooth divisor D which has positive normal line bundle so that one obtains a compact Kähler manifold X := X • ∪ D, the following correspondence holds: holomorphic vector bundles on X unitary flat along D are in a one-to-one correspondence with holomorphic vector bundles on X • endowed with a Hermite-Einstein metric with trivial holonomy at infinity. If such compactification X exists, the smooth curve D would be connected (since X • has only one end), would have genus zero (because the fundamental group of this end is finite) hence the holonomy at infinity would be the trivial representation of the fundamental group of the end. But this fundamental group is non trivial. The way we propose to circumvent this obstacle is to consider orbifold compactifications of X • .

Let us fix X • = X k , where X k is the minimal resolution of the A k-1 toric singularity of C 2 /Z k . In [START_REF] Bruzzo | Framed sheaves and SUSY gauge theories on stacky ALE spaces[END_REF] a compactification X k of X k is constructed, which turns out to be a projective toric orbifold. In particular X k \ X k is a smooth effective Cartier divisor D ∞ with ample and positive normal line bundle.

In this paper we generalize Bando's result by proving the following:

Theorem. There is a one-to-one correspondence between holomorphic vector bundles on X k , which are isomorphic along D ∞ to a fixed vector bundle F ∞ endowed with a flat unitary connection ∇, and vector bundles on X k endowed with an Hermite-Einstein metric such that the curvature is square integrable and the holonomy at infinity is given by the holonomy of ∇.

Let us describe the organisation of this article. In Sections 2 and 3, we set up the definitions of the basic ingredients of the differential geometry of Deligne-Mumford stacks with an emphasis on the Kähler case. In Sections 4 and 5, we extend to Deligne-Mumford stacks the Uhlenbeck-Yau and Bando theorems. In Section 6, we recall the construction of the orbifold compactification of X k and conclude with the proof of the above theorem (Theorem 6.9). 1.1. Acknowledgements. We thank C. Simpson for useful suggestions and for his interest in our work and U. Bruzzo for reading and commenting on a draft of this paper. We are also grateful to M. Pedrini, R. J. Szabo and R. Thomas for helpful discussions and correspondence. The last draft of the paper was written while the second author was visiting IH ÉS and Université Joseph Fourier. He thanks both institutions for hospitality and support.

Topological, differentiable and smooth analytic stacks

In this section we briefly describe topological, differentiable and smooth analytic stacks. Our main references are [START_REF] Behrend | Cohomology of stacks[END_REF][START_REF] Noohi | Foundations of Topological Stacks I[END_REF][START_REF] Behrend | Uniformization of Deligne-Mumford curves[END_REF]. We assume that the reader is familiar with the notions of category fibered in groupoids and of stack (cf. [24, Sect. 2 and 3]).

2.1. Topological stacks. Let Top be the category of topological spaces. We fix a final object * , the point, in the category Top of topological spaces. We endow Top with the usual Grothendieck topology (covers are simply topological open covers); so we can talk about (the 2-category of) stacks over Top: it is closed under fibre products and by Yoneda's lemma the category of topological spaces embeds as a full subcategory of the 2-category of stacks.

We say a morphism f : Y → X of stacks over Top is representable, if for any map U → X from a topological space U to X , the fiber product V := U × X Y is equivalent to a topological space.

Several properties P of maps of topological spaces are stable under base change (cf. [START_REF] Noohi | Foundations of Topological Stacks I[END_REF]Sect. 4.1]). For example: to be open maps, epimorphisms, surjective maps, embeddings, closed embeddings, open embeddings, local homeomorphisms, covering maps, maps with finite fibers, maps with discrete fibers. We say a representable morphism f : Y → X of stacks over Top satisfies a property P if for any map U → X from a topological space U to X , the base extension V → U of f satisfies P. Definition 2.1. A pre-Deligne-Mumford topological stack is a stack X for which there exists an epimorphism u : U → X from a topological space U , such that u is representable by local homeomorphisms. We call the pair (U, u) an atlas of X . A pre-Deligne-Mumford topological stack X is called a Deligne-Mumford topological stack if U is Hausdorff for some atlas (U, u) and the diagonal X → X × X is representable by proper maps with closed discrete fibers. ⊘ One can define a site S(X ) of a Deligne-Mumford topological stack X in the following way. The objects of the underlying category of S(X ) are the atlases (U, u) of X , the arrows are the morphisms (ϕ, α) : (U, u) → (V, v) of two atlases where ϕ : U → V is a local homeomorphism of topological spaces and α : u

∼ -→ v • ϕ is a 2-isomorphism.
The topology on this category is the one induced by the pre-topology, where the covering families are of the following form: for an atlas (U, u), we denote by Cov(U, u) the set of families of morphisms

{(ϕ i , α i ) : (U i , u i ) → (U, u)} i∈I such that the map i∈I ϕ i : i∈I U i → U
is open and surjective.

Remark 2.2. The previous definition of pre-Deligne-Mumford topological stacks comes from [START_REF] Behrend | Uniformization of Deligne-Mumford curves[END_REF]Sect. 3.1]. In [START_REF] Noohi | Foundations of Topological Stacks I[END_REF]Sect. 7], Noohi gives a more general definition of pre-topological stacks. In [START_REF] Noohi | Foundations of Topological Stacks I[END_REF], our pre-Deligne-Mumford topological stacks are called weak Deligne-Mumford topological stacks (cf. [START_REF] Noohi | Foundations of Topological Stacks I[END_REF]Def. 14.3]). On the other hand our definition of Deligne-Mumford topological stacks is more restrictive than the one given in [START_REF] Behrend | Uniformization of Deligne-Mumford curves[END_REF]: indeed, in [START_REF] Behrend | Uniformization of Deligne-Mumford curves[END_REF]Def. 3.1] the authors do not require existence of a Hausdorff atlas and require only that the diagonal is representable by closed maps with discrete finite fibers. As the authors point out in [START_REF] Behrend | Uniformization of Deligne-Mumford curves[END_REF]Sect. 3.1] closedness is a property invariant under base extension only via local homeomorphism. On the other hand, we impose properness instead of closeness because we need a property invariant under any base extension. △

For a stack X , we denote by π 0 X the sheaf associated to the presheaf of sets on Top defined by W → {isomorphim classes in X (W )}. To any Deligne-Mumford topological stack X one can associate a topological space X, called the coarse moduli space of X : as a set X is equal to π 0 X ( * ). For any open substack X ′ ⊆ X (i.e. any representable open embedding X ′ → X ), we have a natural inclusion of coarse moduli spaces X ′ ⊆ X. These are defined to be the open sets of X. Definition 2.3. Let X be a Deligne-Mumford topological stack. We say that X is connected if it has no proper open-closed substack. We say that X is compact if its coarse moduli space X is compact. ⊘

The natural constructions of homotopy theory of topological spaces can be extended to Deligne-Mumford topological stacks. For instance, one can define a reasonable notion of homotopy between maps, and this allows us to define the n-th homotopy group of a pointed topological stack (X , x) as pointed homotopy classes of maps [(S n , * ), (X , x)] (cf. [START_REF] Noohi | Foundations of Topological Stacks I[END_REF]Sect. 17]).

As explained in [START_REF] Behrend | Cohomology of stacks[END_REF], there is a well defined singular (co)homology theory with integral coefficients for Deligne-Mumford topological stacks. The (co)homology theory with integral coefficients depends on the stacky structure, unlike the (co)homology theory with rational coefficients. Indeed, one has the following result. Proposition 2.4. [START_REF] Behrend | Cohomology of stacks[END_REF]Prop. 36] Let X be a Deligne-Mumford topological stack and X its coarse moduli space. Then the coarse moduli space morphism π : X → X induces isomorphisms

H k (X ; Q) ≃ H k (X; Q) .

Analytic and differentiable stacks.

Let Comp be the category of complex manifolds2 , endowed with the usual Grothendieck topology (where covers are simply topological open covers). As in the case of topological stacks, we can construct the 2-category of stacks over Comp, and by Yoneda's lemma this category contains the category of complex manifolds as a full subcategory. We say a morphism f : Y → X of stacks over Comp is representable by local biholomorphisms, if for any map U → X from a complex manifold U to X , the fiber product V := U × X Y is equivalent to a complex manifold, and the map V → U is a local biholomorphism. Definition 2.5. A stack X over Comp is called a smooth pre-Deligne-Mumford analytic stack if there exists an epimorphism u : U → X from a complex manifold U such that u is representable by local biholomorphisms. We call the pair (U, u) an atlas of X . ⊘ Remark 2.6. Since a holomorphic map of complex manifolds is a local biholomorphism if and only if it is a local homeomorphism, the previous definition coincides with the one given in [START_REF] Behrend | Uniformization of Deligne-Mumford curves[END_REF]Sect. 3.2]. △

We say a morphism f : Y → X of smooth pre-Deligne-Mumford analytic stacks is representable, if for any map U → X from a complex manifold U to X that is representable by local biholomorphisms, the fiber product V := U × X Y is equivalent to a complex manifold. Let P be a property of morphisms of complex manifolds that is invariant under base change with respect to local biholomorphisms. For example we can take P=closedness, P=to have finite fibers, P=to have discrete fibers, P=properness, P=to be unramified, or P=to be a covering space. Then, we say a representable morphism f : Y → X of smooth pre-Deligne-Mumford analytic stacks is P, if for any map U → X from a complex manifold U to X that is representable by local biholomorphisms, the base extension V → U is P. Definition 2.7. A smooth pre-Deligne-Mumford analytic stack X is called a smooth Deligne-Mumford analytic stack if, for some atlas (U, u) of X , U is Hausdorff and the diagonal X → X × X is representable by proper unramified3 finite maps. An orbifold is a smooth Deligne-Mumford analytic stack with generically trivial stabilizer. ⊘ Also in this case, one can define a site S(X ) in a way similar than before.

Remark 2.8. Our definition of smooth Deligne-Mumford analytic stacks is more restrictive that the one given in [START_REF] Behrend | Uniformization of Deligne-Mumford curves[END_REF]: in [START_REF] Behrend | Uniformization of Deligne-Mumford curves[END_REF]Def. 3.3] the authors assume that the diagonal is representable only by closed maps with finite fibers. On the other hand, one can give a definition of Deligne-Mumford analytic stacks over the category Analytic of all analytic spaces (cf. [START_REF] Noohi | Foundations of Topological Stacks I[END_REF]). As pointed out in [START_REF] Behrend | Uniformization of Deligne-Mumford curves[END_REF]Rem. 3.4], the 2-category of smooth Deligne-Mumford analytic stacks defined above is equivalent to the sub 2-category of the 2-category of Deligne-Mumford analytic stacks of [START_REF] Noohi | Foundations of Topological Stacks I[END_REF] consisting of smooth Deligne-Mumford analytic stacks. Since the Deligne-Mumford analytic stacks we are interested in are smooth, we prefered not to work over Analytic since this category may be less familiar to the reader. △

By substituting in the previous definitions the term "local biholomorphism" with "local diffeomorphism", a definition of smooth (pre-)Deligne-Mumford differentiable stacks over the category Diff of differentiable manifolds can be given and was actually introduced in [START_REF] Behrend | Cohomology of stacks[END_REF].

One can define the coarse moduli space X of a smooth Deligne-Mumford analytic (resp. differentiable) stack X . It is an analytic space, but it may not in general be a smooth complex manifold. Proposition 2.9. [START_REF] Behrend | Cohomology of stacks[END_REF][START_REF] Noohi | Foundations of Topological Stacks I[END_REF] Let X be a Deligne-Mumford topological (resp. smooth differentiable, resp. smooth analytic) stack. Then there is a covering

{U i } of X by open substacks such that each U i is a quotient stack [Z/G],
where Z is a Hausdorff topological space (resp. a Hausdorff differentiable manifold, resp. Hausdorff complex manifold), and G a finite group acting continuously (resp. differentiably, resp. analytically) on Z.

As explained in [START_REF] Behrend | Cohomology of stacks[END_REF]Cor. 25], there is a well-defined de Rham cohomology theory for any smooth Deligne-Mumford differentiable stack X . Definition 2.10. Let X be a smooth Deligne-Mumford differentiable stack. We say that X is oriented if for any atlas (U, u) of X the associated groupoid

U × X U U . s t
is oriented, i.e., the manifolds U and U × X U and the maps s and t are oriented in a compatible way. In addition, we say that X is of finite type if U and U × X U have finite good covers compatible with s and t. ⊘

Let X be a finite type smooth Deligne-Mumford differentiable stack. Then there is a well-defined theory of cohomology with compact supports for X . In addition, there exists an integration map X :

H dim(X ) c (X ) → R , (1)
such that the induced pairing

H k dR (X ) ⊗ H dim(X )-k c (X ) → R is perfect. If X is compact, H k dR (X ) ≃ H k c (X )
for any k and there is a well defined map (1) which induces a perfect pairing.

Comparing algebraic, analytic, differentiable and topological stacks.

In [START_REF] Lerman | Orbifolds as stacks?[END_REF] a nice interpretation of differentiable stacks as Lie groupoids is thoroughly discussed, which helped us to understand how the different types of stacks are related.

Let X be a smooth separated Deligne-Mumford algebraic stack of finite type over C (cf. [START_REF] Laumon | Ergebnisse der Mathematik und ihrer Grenzgebiete[END_REF]Sect. 4]). Then by [START_REF] Keel | Quotients by groupoids[END_REF], there exists a coarse moduli space π : X → X. In general X is a separated algebraic space of finite type over C.

Let AlgDM be the 2-category of smooth separated Deligne-Mumford algebraic stacks of finite type over C. As explained in [START_REF] Behrend | Uniformization of Deligne-Mumford curves[END_REF]Sect. 3.3] (see also [START_REF] Lerman | Orbifolds as stacks?[END_REF]), it is equivalent to the weak 2-category of groupoids up to Morita equivalence

X 1 X 0 s t
in the category of separated schemes of finite type over C, where X 0 is smooth, s, t are étale morphisms and (s, t) :

X 1 → X 0 × X 0
is a proper unramified quasi-compact morphism (cf. [START_REF] Laumon | Ergebnisse der Mathematik und ihrer Grenzgebiete[END_REF]). Denote by AnDM (resp. DiffDM, resp. TopDM) the 2-category of smooth Deligne-Mumford analytic stacks (resp. differentible stacks, resp. topological stacks). The argument in [START_REF] Behrend | Uniformization of Deligne-Mumford curves[END_REF]Sect. 3.3] proves that the 2category of Deligne-Mumford topological stacks (resp. smooth Deligne-Mumford analytic stacks, resp. smooth Deligne-Mumford differentiable stacks) is equivalent to the weak 2category of groupoids up to Morita equivalence

X 1 X 0 s t
in the category of topological spaces (resp. complex manifolds, resp. differentiable manifolds), where the maps s, t are local homeomorphisms (resp. local biholomorphisms, resp. local diffeomorphisms) and (s, t) :

X 1 → X 0 × X 0
is a proper map with closed discrete fibers (resp. proper unramified finite map in the analytic and differential settings). Then one can define natural functors

AlgDM -an --→ AnDM -diff --→ DiffDM -top --→ TopDM .
Moreover, these functors respect the coarse moduli space construction. We shall also denote by top --→, resp.

diff --→ any composition of these functors ending with top --→, resp.

diff --→.

Remark 2.11. In the following, if X is a smooth Deligne-Mumford algebraic (analytic, differentiable) stack, its fundamental group is by definition the fundamental group of its underlying Deligne-Mumford topological stack. △

Differential geometry on smooth Deligne-Mumford analytic and differentiable stacks

In this section we shall sketch the generalization of some basic results of differential geometry.

3.1. Vector bundles on smooth Deligne-Mumford differentiable stacks. Let X be a smooth Deligne-Mumford differentiable stack. Definition 3.1. A C ∞ complex vector bundle of rank r on X is the following set of data:

• for any atlas (U, u) a C ∞ complex vector bundle E U,u of a rank r on U , • for any morphism (ϕ, α) : (U, u) → (V, v) of two atlases, where ϕ : U → V is a local diffeomorphism of differentiable manifolds and α : u

∼ -→ v • ϕ is a 2-isomorphism, an isomorphism of C ∞ complex vector bundles of rank r ϑ ϕ,α : ϕ * (E U,u ) ∼ -→ E V,v
such that for any composition (U, u)

(ϕ,α) ---→ (V, v) (ψ,β) ---→ (W, w), we have ϕ * ϑ ψ,β • ϑ ϕ,α = ϑ ψ•ϕ,ϕ * β•α . ⊘ Remark 3.2.
In [START_REF] Behrend | String topology for stacks[END_REF] the authors give another definition of complex vector bundles on X . Our definition is equivalent to their definition because of [START_REF] Behrend | String topology for stacks[END_REF]Prop. 3.2]. △

There are several operations on C ∞ complex vector bundles on X . We can indeed form the trivial vector bundle V X := V × X given any C-vector space V , resp. the complex conjugate Ē or the dual E ∨ of a vector bundle E, resp. the direct sum E ⊕ F , the tensor product E ⊗ F or the bundle of morphisms Hom(E, F ) of two vector bundles E and F , resp. the pull back f * E of a complex vector bundle E on X by a morphism f : Y → X . Furthermore, one can define a morphism ψ of C ∞ complex vector bundles on X from E to F as the data for any atlas (U, u) of a morphism ψ U,u : E U,u → F U,u commuting with ϑ ϕ,α . This enables to define C ∞ (X , E), the space of smooth sections of E, as the set of morphisms from C X to E.

A connection ∇ on E is the data of a connection ∇ U,u on E U,u for any atlas (U, u) which is compatible to the ϑ ϕ,α . In a similar vein, one can define riemannian metrics, hermitian metrics, Levi-Civita connection, principal bundles and connections, curvature tensors, . . . . The definitions are left to the reader, see however [START_REF] Dryden | Asymptotic expansion of the heat kernel for orbifolds[END_REF] for orbifolds, and [START_REF] Behrend | String topology for stacks[END_REF]. The theory of Chern-Weil forms of connections and the fact that, on smooth differentiable Deligne-Mumford stacks, they compute the rational Chern classes defined as in [START_REF] Behrend | Cohomology of stacks[END_REF] is established in [START_REF] Laurent-Gengoux | Chern-Weil map for principal bundles over groupoids[END_REF].

Holonomy.

Let M be a differentiable manifold. Denote by Conn M the groupoid whose objects are C ∞ complex vector bundles over M with a connection and whose arrows are gauge equivalences (vector bundle isomorphisms preserving the connection). Fix a C ∞ complex vector bundle of rank r with a connection (E, ∇) on a smooth connected Deligne-Mumford differentiable stack X . Then, see [START_REF] Laurent-Gengoux | Chern-Weil map for principal bundles over groupoids[END_REF], we can define the holonomy functor

Hom(M, X ) → Conn M , f → (f * E, f * ∇) .
When M = I is an open interval of the real numbers, x 0 , x 1 ∈ I and δ : I → X is a differentiable path we can define the holonomy h(δ, x 0 , x 1 ) along δ to be the natural linear transformation

δ * E x 0 → δ * E x 1 . If ∇ is flat then h(δ, x 0 , x 1 ) depends only on the differentiable homotopy class of f : (I, {x 0 , x 1 }) → (X , {f (x 0 ), f (x 1 )}).
It is easy to define differentiable paths [0, 1] → X as continuous paths [0, 1] → X top extending differentiably to a slightly larger interval. Now if γ : [0, 1] → X top is a continuous path we can approximate it in the C 0 -topology by differentiable paths. Since any homotopy of maps may be approximated by a differentiable homotopy in C 1 -topology and homotopy classes are open in in C 1 -topology, we conclude as in the classical case that the holonomy of the flat connection ∇ on E gives rise to a representation of the Poincaré groupoid of X hence to a group representation ρ : π 1 (X , x) → GL(r, C). If the flat connection in question preserves a hermitian metric h, then ρ takes values in the unitary group of (E x , h x ).

Conversely, let X be a universal cover 4 of X and consider an atlas (U, u) of X . Then the fibred product ( Ũ := U × X X , ũ) is an atlas of X and Ũ → U is topological Galois covering space of U such that its deck transformations group is π 1 (X , y), where y is point in X . Let ρ : π 1 (X , y) → GL(r, C) be a linear representation. Then π 1 (X , y) acts on Ũ × C r by ρ on the second factor and induces after passing to quotient a C ∞ complex vector bundle of rank r with a flat connection (E ρ U , ∇ ρ U ). This construction being functorial for morphisms of atlases, one obtains a C ∞ complex vector bundle on X of rank r with a flat connection (E ρ , ∇ ρ ) having ρ as its holonomy. This construction is an equivalence of categories.

Metric geometry of Riemannian Deligne-Mumford stacks.

Definition 3.3. A Riemannian Deligne-Mumford stack is a pair (X , g), where X is a smooth Deligne-Mumford differentiable stack and g a Riemannian metric on it. ⊘ Every smooth Deligne-Mumford differentiable stack admits a Riemannian metric since we are assuming second countability. One can define geodesics on X or more generally harmonic mappings from manifolds to (X , g) in the usual way thanks to Proposition 2.9. The infimal length of a path between two points of X top is the Riemannian distance d between these two points and, since with our definitions the stacks are separated, this gives a distance function on X top inducing its topological structure (which is Hausdorff). The extension of the basic theory of geodesics is carefully carried out in [START_REF] Guruprasad | Closed geodesics on orbifolds[END_REF].

For instance, the distance function permits to define the space of Lipschitz functions on X which can be characterized by Rademacher's Theorem as almost everywhere differentiable functions with g-bounded differential. The space of continuous functions on X coincides with the space of continuous functions on X top . However the space of Lipschitz (or of C ∞ ) functions on X does not depend only on the structure of the underlying topological stack X top , but it is a truly stacky invariant. For instance, if G ⊂ GL(n, C) is a finite complex reflection group (e.g. G = {±1} ∈ C * ) the distance function attached to a G-invariant hermitian form is just Hölder continuous with respect to a Riemannian metric in the usual sense on C n /G ≃ C n . In the example, d [Cw/{±1}] (0, z) = |z| 1/2 where z = w 2 is the natural coordinate on C w = [C w /{±1}] top .

3.4. Sheaves and vector bundles on smooth Deligne-Mumford analytic stacks. In this section we shall give definitions of (quasi)coherent analytic sheaves and vector bundles on a smooth Deligne-Mumford analytic stack which resembles the definition of (quasi)coherent sheaves given in [START_REF] Vistoli | Intersection theory on algebraic stacks and on their moduli spaces[END_REF] in the algebro-geometric setting.

Let X be a smooth Deligne-Mumford analytic stack.

Definition 3.4.

A quasicoherent sheaf F on X is the following set of data:

• for any atlas (U, u) a quasicoherent analytic sheaf F U,u on U ,

• for any morphism (ϕ, α) : (U, u) → (V, v) of two atlases, where ϕ : U → V is a local biholomorphism of complex manifolds and α : u

∼ -→ v • ϕ is a 2-isomorphism, an isomorphism θ ϕ,α : ϕ * F U,u ∼ -→ F V,v
such that for any composition (U, u)

(ϕ,α) ---→ (V, v) (ψ,β)
---→ (W, w), we have

ϕ * θ ψ,β • θ ϕ,α = θ ψ•ϕ,ϕ * β•α .
A coherent sheaf is a quasicoherent sheaf F such that all F U,u are coherent. A locally free sheaf is a coherent sheaf F such that all F U,u are locally free. A morphism f :

F → F ′ is a collection of morphisms f U,u : F U,u → F ′ U,u such that for any morphism (ϕ, α) : (U, u) → (V, v) of two atlases we have ϕ * (f V,v ) • θ ϕ,α = θ ′ ϕ,α • f U,u . ⊘ Example 3.5.
The structure sheaf O X of X is defined by (O X ) U,u = O U for any atlas (U, u), with the obvious isomorphisms. The sheaf of differential Ω 1 X of X is defined by (Ω 1 X ) U,u = Ω 1 U . Since for any morphism (ϕ, α) : (U, u) → (V, v) of two atlases, the morphism ϕ is a local biholomorphism, there is a natural isomorphism Ω 1 U ≃ ϕ * (Ω 1 V ). A holomorphic vector bundle of rank r on X is a C ∞ -complex vector bundle E over X such that all E U,u are holomorphic vector bundles and all ϑ ϕ,α are isomorphisms of holomorphic vector bundles. As before in the differential setting, also in this setting one can perform the usual operations on vector bundles.

Let E be a rank r holomorphic vector bundle on X . For any atlas (U, u) we denote by E U,u the sheaf of holomorphic sections of the vector bundle E U,u . For any morphism (ϕ, α) : (U, u) → (V, v) of two atlases, the isomorphism ϑ ϕ,α induces an isomorphism

θ ϕ,α : ϕ * E U,u ∼ -→ E V,v .
In this way, we define a coherent sheaf E on X , the sheaf of sections of the vector bundle E.

Remark 3.6. The functor -an sends coherent sheaves to coherent analytic sheaves. The functor -diff sends holomorphic vector bundles to C ∞ complex vector bundles. △

Let X be a smooth Deligne-Mumford analytic stack. Then it is easy to see that there is a one-to-one correspondence between holomorphic vector bundles of rank r on X and locally free sheaves of rank r on it. Assume moreover that X is of the form [Z/G], where Z is a complex manifold and G a complex Lie group acting on it. By the same argument as in [START_REF] Laumon | Ergebnisse der Mathematik und ihrer Grenzgebiete[END_REF]Example 12.4.6], the category of coherent sheaves on X is equivalent to the category of G-equivariant coherent sheaves on Z. Similarly, the category of locally free sheaves on X is equivalent to the category of G-equivariant locally free sheaves on Z (this follows from the descent result for locally free sheaves with respect to fppf morphisms, [START_REF] Behrend | Algebraic stacks[END_REF]Prop. A.11]). Since the category of G-equivariant locally free sheaves on Z is equivalent to the category of G-equivariant holomorphic vector bundles on Z (see, e.g., [10, Sect. 5.1]), the category of holomorphic vector bundles of rank r on X is equivalent to the category of G-equivariant holomorphic vector bundles of rank r on Z.

In the following we shall call hermitian bundle a holomorphic vector bundle on X with a hermitian metric on it. Then if X is a smooth Deligne-Mumford analytic stack of the form [Z/G], the category of hermitian bundles on X is equivalent to the category of G-equivariant hermitian vector bundles on Z. Thus if E is a holomorphic vector bundle on a smooth Deligne-Mumford analytic stack, by Proposition 2.9 and a partition of unity argument on X top , one can always construct a hermitian metric on E.

The usual constructions of complex differential geometry -see [12, Chap. V] -carry over to smooth Deligne-Mumford analytic stacks 5 . A smooth Deligne-Mumford analytic stack carries a complexified tangent bundle T C X = T 1,0 X ⊕ T 0,1 X from which we may form the usual vector bundles Ω k X of k-forms and Ω p,q X of (p, q)-forms. A (1, 1) form ω on X is positive if so are its representatives ω U,u for any atlas (U, u) of X . A hermitian vector bundle (E, h) carries a canonical Chern connection ∇ h with a curvature form iΘ(E, h) ∈ C ∞ (X , Herm(E, h)⊗Ω 1,1 X ). The curvature of a hermitian line bundle (L, h) is just a real (1, 1) form on X .

Kähler Deligne-Mumford analytic stacks.

At a slightly deeper level, the De Rham, Dolbeault [12, Chap. IV, Sect. 6] and Hodge [12, Chap. VI, Thm. 3.16 and 3.17] isomorphism theorems are valid for smooth Deligne-Mumford analytic stacks. If we follow the proof of these basic results [START_REF] Demailly | Complex analytic and differential geometry[END_REF], we see that the only non obvious points are integration of top dimensional compactly supported forms on an oriented smooth Deligne-Mumford differentiable stack (so that we can define the L 2 norm on the space of sections of a metrized vector bundle on an oriented 6 Riemannian Deligne-Mumford differentiable stack), Stokes' theorem (so that we can perform integration by parts) and acyclicity of sheaves of (p, q)-forms (cf. Section 2.2 for the first two results, the latter descends from the analogous statement in the case of complex manifold simply by using a groupoid presentation of the stack).

The literature on the Grothendieck-Riemann-Roch theorem is not as complete as one would like, even for complex manifolds, see however [START_REF] Töen | Théorèmes de Riemann-Roch pour les champs de Deligne-Mumford[END_REF] for proper representable morphisms of smooth Deligne Mumford algebraic stacks over the complex numbers. Fortunately, this is not used here.

Say a smooth Deligne-Mumford analytic stack is Kähler if it carries a closed positive (1, 1)form. This is equivalent to require that the coarse moduli space is Kähler. The Lefschetz package, the ∂ ∂-lemma hold on compact Kähler Deligne-Mumford analytic stacks and so does the Kodaira-Akizuki-Nakano theorem. An application is an analytic version of Olsson-Matsuki's proof of the Kawamata-Viehweg vanishing theorem in characteristic zero [START_REF] Matsuki | Kawamata-Viehweg vanishing as Kodaira vanishing for stacks[END_REF]. Yau's solution of the Calabi conjecture [START_REF] Yau | On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation[END_REF] is also valid on stacks. For orbifolds, an early reference 5 One can mimic the arguments in [START_REF] Behrend | Cohomology of stacks[END_REF]: one can define everything and the level of groupoid presentations of the stack and then prove that it is invariant under Morita equivalence. 6 One needs to use densities for non-orientable stacks.

is [START_REF] Kobayashi | Einstein-Kähler V -metrics on open Satake V -surfaces with isolated quotient singularities[END_REF], see also [START_REF] Eyssidieux | Singular Kähler-Einstein metrics[END_REF]. Although we will not use this, we note that the characterization of the Kähler cone, extends to compact Kähler Deligne-Mumford analytic stacks.

Theorem 3.7. [START_REF] Demailly | Numerical characterization of the Kähler cone of a compact Kähler manifold[END_REF] Let X be a n-dimensional compact Kähler Deligne-Mumford analytic stack and let ω be Kähler form on X . A class {α} ∈ H 1,1 (X , R) is represented by a Kähler form iff for every closed irreducible analytic substack Z ⊂ X one has {α} dim(Z ) .[Z ] > 0 and X {α} n-p {ω} p > 0 for all 1 ≤ p ≤ n -1.

We will not need the full strength of this statement and will only sketch the proof of the consequence we shall use. Definition 3.8. Let L be a holomorphic line bundle on a smooth Deligne-Mumford analytic stack X . Let Z ⊆ X be a closed substack of X . We say L is positive on Z if there is an open analytic substack U such that Z ⊂ U and a hermitian metric h on L such that iΘ(L |U , h |U ) is positive. ⊘ Proposition 3.9. Let L be a holomorphic line bundle on a smooth Deligne-Mumford analytic stack X . Assume there exists n ∈ N \ {0} such that L ⊗n ≃ π * (M ) where π : X → X is the natural map to the coarse moduli space X of X and M is a holomorphic line bundle on X.

Let Z ⊆ X be a compact analytic subspace. Then L is positive on Z := Z × X X if and only if M is ample on Z (which is then a projective algebraic variety).

Proof. The corresponding statement for analytic spaces was proved in Pȃun's thesis, see [START_REF] Demailly | Numerical characterization of the Kähler cone of a compact Kähler manifold[END_REF]Prop. 3.3] for the required level of generality. By Proposition 2.9, it rests on gluing and regularization techniques for quasi-plurisubharmonic functions [START_REF]Cohomology of q-convex spaces in top degrees[END_REF] that are local in nature and compatible with the action of a finite group.

Remark 3.10. The previous proposition holds also for Deligne-Mumford analytic stacks that need not be smooth. △ Remark 3.11. A positive line bundle in the above sense needs not be ample in the sense of [START_REF] Ross | Weighted projective embeddings, stability of orbifolds, and constant scalar curvature Kähler metrics[END_REF] because this latter condition requires that the stack has cyclic isotropy groups and the isotropy representations of the line bundle are faithful. See [START_REF] Kresch | On the geometry of Deligne-Mumford stacks[END_REF] for related issues. △

Hermite-Einstein metrics on compact Kähler Deligne-Mumford analytic stacks

4.1. Uhlenbeck-Yau theorem. Let X be a n-dimensional connected compact Kähler Deligne-Mumford analytic stack and let ω be Kähler form on X . Let E be a coherent sheaf on X . We call rank of E the zero degree part of the Chern character of its K-theory class [E].

Definition 4.1. A holomorphic vector bundle E on X is ω-stable if every coherent subsheaf F ⊂ E satisfies µ(F) < µ(E), where µ(F) is the ω-slope of F defined as µ(F) := X c 1 (F).{ω} dim(X )-1 rk(F)
.

⊘

The proof of the celebrated Uhlenbeck-Yau theorem carries over to Kähler Deligne-Mumford analytic stacks. Theorem 4.2. [START_REF] Uhlenbeck | On the existence of Hermitian-Yang-Mills connections in stable vector bundles[END_REF] Let X be a n-dimensional connected compact Kähler Deligne-Mumford analytic stack and let ω be Kähler form on X . Let E be a ω-stable holomorphic vector bundle on X . Then it carries a smooth Hermite-Einstein metric h namely a hermitian metric satisfying the equation:

ΛiΘ(E, h) = 2πµ(E)Id E .

4.2.

Comparison with Nironi's stability condition. Let (P, O P (1)) be a n-dimensional polarized projective cyclic orbifold in the sense of [START_REF] Ross | Weighted projective embeddings, stability of orbifolds, and constant scalar curvature Kähler metrics[END_REF]. One may take P := P(a 0 , . . . , a n ) the n-dimensional weighted projective stack for some coprime positive integers a 0 , . . . , a n with n ≥ 1. Then for some m, G := ⊕ m i=1 O P (i) is a generating sheaf [START_REF] Kresch | On the geometry of Deligne-Mumford stacks[END_REF]Sect. 5.2]. In [START_REF] Ross | Weighted projective embeddings, stability of orbifolds, and constant scalar curvature Kähler metrics[END_REF] the authors define a Kahler form ω induced by the curvature of a metric on O P (1). Then for any n-dimensional coherent sheaf E on X we have

1 rk(E) P c 1 (E) • {ω} n-1 = 1 rk(E) P c 1 (E) • c 1 (O P (1)) n-1 .
By using [START_REF] Nironi | Moduli spaces of semistable sheaves on projective Deligne-Mumford stacks[END_REF]Prop. 3.18] one can prove

1 rk(E) P c 1 (E) • c 1 (O P (1)) n-1 = 1 m n µ G (E) ,
where µ G (E) is the G-slope introduced by Nironi. Therefore the ω-stability condition is equivalent to the µ G -stability condition of Nironi.

Hermite-Einstein metrics on some noncompact Kähler Deligne-Mumford analytic stacks

In this section we shall prove a variant of [START_REF] Bando | Einstein-Hermitian metrics on noncompact Kähler manifolds[END_REF] for stacks.

5.1. Deleted neighborhoods of smooth divisors. Let (X , g) be a Riemannian Deligne-Mumford stack. Given an atlas (U, u) of X one can construct a groupoid action of

U × X U U s t
on the tangent space to U and the resulting quotient stack π : Tot(T X ) → X is a differentiable Deligne-Mumford stack over X which is the total space of the tangent vector bundle T X to X . One can generalize this construction to any (real) vector bundle E on X . The zero section of any (real) vector bundle E defines a substack Z(E) → Tot(E) of its total space. Moreover, Tot(E) \ Z(E) is representable (i.e.: it is a manifold) if and only if the isotropy action at all points of X has no non zero fixed vectors. Given a metric on E one can construct similarily the unit sphere bundle S(E) as a smooth substack of Tot(E) \ Z(E) which is a locally trivial sphere fibration over X . Moreover, the long exact sequence of homotopy groups holds true [START_REF] Noohi | Foundations of Topological Stacks I[END_REF]. In particular, Tot(T X ) \ X is an honest manifold (i.e.: it is representable) if and only if the tangent isotropy actions have no non zero fixed vectors. One also has an equivalence Tot(E)\Z(E) → S(E)×R >0 the second factor ρ : Tot(E) → R >0 being the norm function.

Given i : S → X a closed smooth substack, the normal bundle N S |X over S can be constructed as i * T X /T S . Then, using the exponential map in normal directions, one constructs an open differentiable substack V ⊂ X containing S as a closed substack such that the pair (V , S ) is equivalent to (Tot(N S |X ), S ).

If X is now a smooth Deligne-Mumford analytic stack and D ⊂ X be a smooth integral codimension one closed substack. Then N D diff |X diff is the smooth bundle underlying O X (D) and there is an open differentiable substack V ⊂ X diff containing D diff as a retract and a homotopy equivalence

U D := S(N D diff |X diff ) → V \ D diff such that the resulting map U D → D diff
is a locally trivial S 1 -fibration. In particular one has an exact sequence:

1 → C D → π 1 (U D ) → π 1 (D) → 1 , (2)
where C D is a normal cyclic subgroup whose order may take any non negative integer value or +∞. If O X (D) is ample on D, then U D is a complex manifold.

5.2.

Asymptotically flat metrics on deleted neighborhoods. Assume now (X , ω 0 ) is Kähler Deligne-Mumford stack and D ⊂ X is a smooth divisor. A riemannian metric g c on X \ D has cone-like singularities if there exists a metric ds 2 on U D such that the restriction of g c to V \ D diff is asymptotic to dr 2 + r 2 ds 2 in the sense made precise by [1, Sect. 1, Def. 1] on manifolds where r is the distance function to any given point (note that r ∼ ρ -1 ). The curvature of such metric g c decays to zero when approaching D.

If O X (D)| D is positive, there exists an hermitian metric h D on O X (D) such that its curvature form is positive definite on a neighbourhood of D. As in [1, Sect. 1], we can construct a cone-like complete metric ω c on X \ D by the following formula:

ω c := √ -1∂ ∂ 1 a exp(a log σ D -2 h D ) + Cω 0 , where σ D ∈ H 0 (X , O X (D))
is the tautological section of D, a is an arbitrary positive number and C is a big enough positive real number.

Definition 5.1. A holomorphic hermitian vector bundle (

E, h) on X \ D is asymptotically flat if |Θ(E |(X • ,ωc) , h)| gc = O(r -2-ǫ ) as r → ∞ for some ǫ > 0. ⊘
Restricting the Chern connection of (E, h) to {r = R} for R ≫ 1 we get a hermitian connection D R on the pull-back of E to U D .

The asymptotic flatness condition gives immediately (cf. [1, Sect. 1, Def. 3]):

Lemma 5.2. The holonomy of D R along any path has a limit when R → ∞ which is homotopy invariant.

This defines a unitary representation of π 1 (U D ) and we say (E, h) has trivial holonomy at infinity if the representation is trivial when restricted to the group C D defined by (2). 5.3. Bando's Instanton Theorem. We are now ready to state an analog of [1, Thm. 1]: Theorem 5.3. Let (X , ω 0 ) be a n-dimensional (n ≥ 2) connected compact Kähler Deligne-Mumford stack and let D ⊂ X be a smooth divisor such that O X (D) is ample and positive on D. Let ω c be a Kähler metric with cone-like singularities on X • := X \ D. Then a holomorphic vector bundle E on X is such that E |D can be endowed with a flat unitary connection ∇ iff there exists an Hermite-Einstein asymptotically flat vector bundle (E ′ , h ′ ) on (X • , ω c ) having trivial holonomy at infinity which induces ∇ and E is an extension of E ′ .

Proof. The previous section gives one direction of the first statement. Bando's proof of the much harder converse applies mutatis mutandis: extend the flat hermitian metric to a deleted neighborhood of D by using a retraction, then extend it to X and flow it to an asymptotically flat Hermite-Einstein metric. The extension theorem, being local, also follows in the same way.

Corollary 5.4 ([1, Cor. 2]

). If furthermore X • is an ALE manifold then we can replace the curvature decay condition by

X |Θ(E ′ , h ′ )| n < +∞; it is equivalent to |Θ(E ′ , h ′ )| = O(r -(2n-ǫ)
) for any ǫ > 0.

6. An application: orbifold compactifications of ALE spaces of type A k-1

In this section we describe an explicit application of Theorem 5.3.

6.1. Homotopy theory of spherical curves. In this section we shall recall some homotopical properties of spherical Deligne-Mumford curves from [START_REF] Behrend | Uniformization of Deligne-Mumford curves[END_REF].

For any pair of integers m, n ≥ 1, we call weighted projective line of type (m, n) the smooth Deligne-Mumford analytic stack P(m, n) := [C 2 \ {0}/C * ], where the action of C * on C 2 \ {0} is given by t • (x, y) := (t m x, t n y) for any t ∈ C * and (x, y) ∈ C 2 \ {0}. Note that P(1, 1) ≃ P 1 . The stack P(m, n) has at most two orbifold points and its coarse moduli space is P 1 . Obviously, P(m, n) ≃ P(n, m). Moreover, a weighted projective line is an orbifold if an only if m and n are relatively coprime. We call these orbifold weighted projective lines. The weighted projective line P(m, n) is a µ d -gerbe over P( m d , n d ), where d = gcd(m, n). As explained in [START_REF] Behrend | Uniformization of Deligne-Mumford curves[END_REF]Sect. 4.3], the fundamental group π 1 (P(m, n)) is trivial.

For any pair of integers m, n ≥ 1, a football of type (m, n) is an orbifold F (m, n) whose coarse moduli space is P 1 and has two orbifold points of order m and n at 0 and ∞, respectively. When m and n are relatively prime, F (m, n) is naturally isomorphic to P(m, n). Definition 6.1. Let D be a one-dimensional smooth Deligne-Mumford analytic stack. We say that D is a spherical Deligne-Mumford curve if its universal cover is P(m, n) for some positive integer numbers m, n.

⊘
1 → Z d → H → π 1 (D) → π 1 (F (m, n)) → 1 ,
where d = gcd(m, n). Moreover, the universal cover of D is isomorphic to P(m, n).

Proof. By [START_REF] Behrend | Uniformization of Deligne-Mumford curves[END_REF]], the fundamental group of D fits into the exact sequence

1 → ker(H → π 1 (D)) → H → π 1 (D) → π 1 (F (m, n)) → 1 .
On the other hand, as explained in [START_REF] Behrend | Uniformization of Deligne-Mumford curves[END_REF]Sect. 9], ker(H → π 1 (D)) ≃ π 2 (PGL(m, n)) (for a definition of PGL(m, n), see [START_REF] Behrend | Uniformization of Deligne-Mumford curves[END_REF]Sect. 8]). Finally, by [START_REF] Behrend | Uniformization of Deligne-Mumford curves[END_REF]Prop. 8.5 and 8.6], we get

π 2 (PGL(m, n)) = Z d if m = n and d = gcd(m, n) , Z m if m = n .
The second statement follows from [5, Prop.

≃ Z k on C 2 as ω • (z 1 , z 2 ) := (ωz 1 , ω -1 z 2 ) .
The quotient C 2 /Z k is a normal toric affine surface. To describe its fan we need to introduce some notation. Let N ≃ Z 2 be the lattice of 1-parameter subgroups of the torus T := C * ×C * . Fix a Z-basis {e 1 , e 2 } of N and define the vector v i := ie 1 + (1 -i)e 2 ∈ N for any integer i ≥ 0. Then the fan of C 2 /Z k consists of the two-dimensional cone σ := Cone(v 0 , v k ) ⊂ N Q and its subcones. The origin is the unique singular point of C 2 /Z k , and is a particular case of the so-called rational double point or Du Val singularity (see [START_REF] Cox | Toric varieties[END_REF]Def. 10.4.10]). By [11, Example 10.1.9 and Cor. 10.4.9], the minimal resolution of singularities of

C 2 /Z k is the smooth toric surface ϕ k : X k → C 2 /Z k defined by the fan Σ k ⊂ N Q , where Σ k (0) := {0} , Σ k (1) := ρ i := Cone(v i ) i = 0, 1, 2, . . . , k , Σ k (2) := σ i := Cone(v i-1 , v i ) i = 1, 2, . . . , k .
Note that the vectors v i are the minimal generators of the rays ρ i for i = 0, 1, . . . , k.

Let us consider the vector b

∞ := -v 0 -v k = -ke 1 + (k -2)e 2 in N. Denote by ρ ∞ the ray Cone(b ∞ ) ⊂ N R and by v ∞ its minimal generator. For even k, v ∞ = 1 2 b ∞ ; for odd k, v ∞ = b ∞ . Let σ ∞,k and σ ∞,0 be the two-dimensional cones Cone(v k , v ∞ ) ⊂ N R and Cone(v 0 , v ∞ ) ⊂ N R respectively.
Let Xk be the normal projective toric surface defined by the fan Σk ⊂ N R :

Σk (0) := {0} = Σ k (0) , Σk (1) := {ρ i | i = 0, 1, 2, . . . , k} ∪ {ρ ∞ } = Σ k (1) ∪ {ρ ∞ } , Σk (2) := {σ i | i = 1, 2, . . . , k} ∪ {σ ∞,k , σ ∞,0 } = Σ k (2) ∪ {σ ∞,k , σ ∞,0 } . First note that i : X k ֒→ Xk
Z σ := x ∈ C k+2 x i = 0 if ρ i / ∈ σ ⊂ C k+2 ,
and the action of (C * ) k is given by

(t 1 , . . . , t k )•(z 1 , . . . , z k+2 ) =          k-1 i=1 t i i t 2-k k z 1 , k-1 i=1 t -(i+1) i t k k z 2 , t 1 z 3 , . . . , t k z k+2 , k odd k-1 i=1 t i i t 1- k k z 1 , k-1 i=1 t -(i+1) i t k k z 2 ,
\ {0}/C * × µ k],
where the action is given by (t, ω) • (z 1 , z 2 ) = (tω z 1 , t z 2 ) for (t, ω) ∈ C * × µ k and (z 1 , z 2 ) ∈ C 2 \ {0}. Moreover, D ∞ ≃ P 1 is the coarse moduli space of D∞ and the line bundle O X can k ( D∞ ) is ample (cf. [START_REF] Bruzzo | Framed sheaves and SUSY gauge theories on stacky ALE spaces[END_REF]Rem. 3.18]). Remark 6.6. As explained in [START_REF] Bruzzo | Framed sheaves and SUSY gauge theories on stacky ALE spaces[END_REF]Sect. 3.3], D∞ is an orbifold of type ( k, k) with two orbifold points p0 and p∞ , which are (π can k ) -1 (0) red and (π can k ) -1 (∞) red respectively, where 0, ∞ ∈ D ∞ are the two torus fixed points. By Proposition 6.3, the fundamental group of D is Z k. Moreover, its universal cover is P( k k , k k ) = P(1, 1) ≃ P 1 . △

7 Following [START_REF] Fantechi | Smooth toric Deligne-Mumford stacks[END_REF], a Deligne-Mumford torus T is a product T × BG where T is a ordinary torus and G a finite abelian group. A smooth toric Deligne-Mumford stack is a smooth separated Deligne-Mumford algebraic stack X of finite type over C, with as a coarse moduli space π : X → X a scheme, together with an open immersion of a Deligne-Mumford torus ı : T ֒→ X with dense image such that the action of T on itself extends to an action a : T × X → X .

Root stack over the canonical orbifold. Let φ k : X k → X can k be the stack obtained from X can k by performing a k-root construction 8 along the divisor D∞ . As explained in [START_REF] Bruzzo | Framed sheaves and SUSY gauge theories on stacky ALE spaces[END_REF]Sect. 3.4 Under this description of D ∞ , the restriction of the line bundle O X k (D ∞ ) on D ∞ is associated with the character of C * ×µ k given by the projection to the first factor (cf. [START_REF] Bruzzo | Framed sheaves and SUSY gauge theories on stacky ALE spaces[END_REF]Lem. 3.35]). So it is easy to see that O X k (D ∞ ) is ample on D ∞ (see also [START_REF] Bruzzo | Framed sheaves on projective stacks[END_REF]Sect. 6.1]). Moreover, (cf. 8 For the theory of root stacks we refer to [START_REF] Cadman | Using stacks to impose tangency conditions on curves[END_REF]. Theorem 6.9. Given a holomorphic vector bundle E on X k , its restriction E |D∞ is isomorphic to a fixed vector bundle F ∞ endowed with a flat unitary connection ∇ iff there exists a Hermite-Einstein vector bundle (E ′ , h ′ ) on (X k , ω k ) such that

X k |Θ(E ′ , h ′ )| 2 < +∞ and
(E ′ , h ′ ) has at infinity the holonomy given by the holonomy of ∇, where ω k is the ALE metric on X k , and E is an extension of E ′ . Remark 6.10. The original result of Bando permits only to treat the case where k = 2 and the holonomy is trivial on C D∞ /C D∞ = Z 2 . △

1

 1

[ 6 ,

 6 Sect. 3.4]) O X k (D ∞ ) ⊗ kk |D∞ ≃ π * k O Xk ( kD ∞ ) |D∞ ≃ π k * |D∞ O D∞ (k/ k) . So by Proposition 3.9 the line bundle O X k (D ∞ ) is positive on D ∞ . Remark 6.7. Since D top ∞ is a µ k -gerbe over Dtop ∞ , its universal cover is P( k, k) by Proposition 6.4. The group C * × µ k fits into a triangle as (3) where the homomorphism ρ : C * × µ k → GL(2, C) is given by the action (4) and the homomorphism C * → C * × µ k is simply t → (t, 1). Thus by the arguments in Section 6.1.1, we obtain that π 1 (D top ∞ ) ≃ Z k . △ Remark 6.8. It is easily seen that U D∞ and U D∞ are actually diffeomorphic manifolds. Their homotopy exact sequences fit into the diagram: vertical arrow is a surjection and the leftmost is an injection. Since π 1 (D top ∞ ) ≃ Z k and π 1 ( Dtop ∞ ) ≃ Z k, then C D∞ /C D∞ = Z k/ k. △ By Theorem 5.3 and Corollary 5.4 we obtain the following.

  Remark 6.2. Let D be one-dimensional smooth Deligne-Mumford analytic stack. By[START_REF] Behrend | Uniformization of Deligne-Mumford curves[END_REF] Prop. 4.6] there exists a one-dimensional analytic orbifold D and a finite group H such that D is a H-gerbe over D. By[START_REF] Behrend | Uniformization of Deligne-Mumford curves[END_REF] Prop. 7.4] D is spherical if and only if D is spherical. △ Let D be a spherical Deligne-Mumford curve which is an orbifold with two orbifold points. Then D is isomorphic to a football F (m, n) of type (m, n) for some positive integers m, n. The fundamental group of D is Z d , where d = gcd(m, n) and its universal cover is P( m d , n d ) on which Z d acts by rotations. Proposition 6.4. Let D be a one-dimensional smooth Deligne-Mumford analytic stack, which is an H-gerbe over a football F (m, n). Then we have the exact sequence

	Proposition 6.3. [5, Prop. 5.5].

  D is a global quotient stack of the form [C 2 \ {0}/E] where E is a central extension of a discrete group Γ by C * and the action of E on C 2 \ {0} is given by a C-representation ρ : E → GL(2, C) of E, which fits into the commutative triangle Moreover, L ρ is endowed by an hermitian metric and a unitary flat connection associated with ρ (cf. Section 3.2).

		7.6-(ii)].
	6.1.1. Spherical Deligne-Mumford curves with universal cover P(d, d). Let D be a one-
	dimensional smooth Deligne-Mumford analytic stack with universal cover P(d, d). By [5,
	Cor. 9.8], C *	(t d ,t d )
	E	ρ	GL(2, C)	(3)
	As explained in [5, Sect. 9.2], the fundamental group of D is Γ.

Remark 6.5. The Picard group Pic(D) of D consists of E-equivariant line bundles on C 2 \{0}, i.e., characters E → C * (cf. Section 3.4). Let ρ : Γ → C * be a one-dimensional representation of Γ ≃ π 1 (D). Then the composition of morphisms E → Γ ρ -→ C * defines a line bundle L ρ on D. △ 6.2. Orbifold compactification. In this section we describe a construction in

[START_REF] Bruzzo | Framed sheaves and SUSY gauge theories on stacky ALE spaces[END_REF]

of a compactification of the minimal resolution of the A k-1 toric singularity of C 2 /Z k , which turns out to be a projective toric orbifold.

Normal compactification. Let k ≥ 2 be an integer and denote by µ k the group of k-th roots of unity in C. A choice of a primitive k-th root of unity ω defines an isomorphism of groups µ k ≃ Z k . We define an action of µ k

  as an open dense subset. We denote by D ∞ the T -invariant divisor associated to the ray ρ ∞ . Canonical orbifold. Let k be k for odd k, otherwise k/2. Let π can k : X can k → Xk be the socalled canonical toric orbifold over Xk with torus T . It is the unique (up to isomorphism) smooth two-dimensional separated toric 7 Deligne-Mumford algebraic stack of finite type over C such that the locus over which π can Σk /(C * ) k , where Z Σk is the union over all cones σ ∈ Σk of the open subsets

	k	is not an isomorphism has a nonpositive dimension.
	As a global quotient stack, X can k	is isomorphic to Z

  t 1 z 3 , . . . , t k z k+2 , k even for (t 1 , . . . , t k ) ∈ (C * ) k and (z 1 , . . . , z k+2 ) ∈ Z Σk . The effective Cartier divisor D∞ := (π can k ) -1 (D ∞ ) red is a toric orbifold with torus C * . By [6, Prop. 3.10] D∞ is isomorphic as a quotient stack to [C 2

  ], X k is a two-dimensional toric orbifold with torus T and with coarse moduli spaceπ k := π can k • φ k : X k → Xk . As a global quotient stack, X k is isomorphic to [Z Σk /(C * ) k ],where the action of (C * ) k on Z Σk is (t 1 , . . . , t k )•(z 1 , . . . , z k+2 ) = z 2 ,t 1 z 3 , . . . , t k z k for odd k , , t 1 z 3 , . . . , t k z k for even k . for (t 1 , . . . , t k ) ∈ (C * ) k and (z 1 , . . . , z k+2 ) ∈ Z Σk . Let D ∞ be the effective Cartier divisor π -1 k (D ∞ ) red ; it is a smooth toric Deligne-Mumford stack with Deligne-Mumford torus T ≃ C * × Bµ k . By [6, Prop. 3.30] D ∞ is isomorphic as a global quotient stack toC 2 \ {0} C * × µ k ,where the action is given by(t, ω) • (z 1 , z 2 ) = t k ω z 1 , t k ω -1 z 2 for even k , t k ω k+1 2 z 1 , t k ω

	    	k-1 i=1	t i i t 2k-k 2 k	z 1 ,	k-1 i=1	t	-(i+1) i	t k 2
	   	k-1 i=1	t i i t k-k k	k	z 1 ,	k-1 i=1	t	-(i+1) i k z 2 k-1 t k k 2 z 2 for odd k ,	(4)

k for (t, ω) ∈ C * × µ k and (z 1 , z 2 ) ∈ C 2 \ {0}. Moreover, D ∞ is a µ k -gerbe over D∞ .

We are assuming that manifolds have a countable basis for their topology.

The attribute "unramified" means "injective differential" from the differential geometric point of view.

The theory of covering stacks is developed in[START_REF] Noohi | Foundations of Topological Stacks I[END_REF] Sect. 18].