Journal of Lie Theory Volume 13 (2003) 519–534 \begin{matrix}c \\ ◯ \end{matrix} 2003 Heldermann Verlag

Tensor fields and connections on holomorphic orbit spaces of finite groups

Andreas Kriegl, Mark Losik, and Peter W. Michor ^{∗}

Communicated by E. B. Vinberg

Abstract. For a representation of a finite group G on a complex vector space V we determine when a holomorphic \binom{p}{q}-tensor field on the principal stratum of the orbit space V/G can be lifted to a holomorphic G-invariant tensor field on V . This extends also to connections. As a consequence we determine those holomorphic diffeomorphisms on V/G which can be lifted to orbit preserving holomorphic diffeomorphisms on V . This in turn is applied to characterize complex orbifolds. Keywords: complex orbifolds, orbit spaces of complex finite group actions Subject Classification: 32M17

1. Introduction

Locally, an orbifold Z can be identified with the orbit space B/G, where B is a G-invariant neighborhood of the origin in a vector space V with a finite group G ⊂ GL(V) and, using this identification, one can easily define local (and then global) tensor fields and other differential geometrical objects in Z as appropriate G-invariant tensor fields and objects on B ⊂ V . In particular, one can naturally define Riemannian orbifolds, Einstein orbifolds, symplectic orbifolds, Kähler-Einstein orbifolds etc.

We study complex orbifolds, that is, orbifolds modeled on orbit spaces V/G, where G is a finite subgroup of GL(V) for a complex vector space V . In particular, the orbit spaces Z = M/G of a discrete proper group G of holomorphic transformations of a complex manifold M are complex orbifolds.

An orbifold X has a structure defined by the sheaf 𝔉_{X} of local invariant holomorphic functions in a local uniformizing system. X has also a stratification by strata S which are glued from local isotropy type strata of local uniformizing systems. In particular, the regular stratum X_{0} is an open dense complex manifold in X.

Holomorphic geometric objects on X (e.g. tensor fields and connections) are locally defined as invariant objects on the uniformizing system. Their restrictions

ISSN 0949–5932 / $2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag to the regular stratum X_{0} are usual holomorphic geometric objects on the complex manifold X_{0}.

∗M.L. and P.W.M. were supported by ‘Fonds zur Förderung der wissenschaftlichen Forschung, Projekt P 14195 MAT’

A natural question is to characterize these restrictions, i.e. to describe tensor fields and connections on X_{0} which are extendible to X. We look at the lifting problem for connections because this allows a very elegant approach to the lifting problem for holomorphic diffeomorphisms. And the last problem has immediate consequences for characterizing complex orbifolds, i.e., for answering the following question: Which data does one need besides 𝔉_{X} and X_{0} to characterize a complex orbifold X? The main goal of the paper is to answer these questions.

We have first to investigate the local situation, thus we consider a finite subgroup G ⊂ GL(V) and the orbit space Z = V/G with the structure given by the sheaf 𝔉_{V/G} of invariant holomorphic functions on V , and the orbit type stratification. The prime role is played by strata of codimension 1 with the orders of the corresponding stabilizer groups, which are arranged in the reflection divisor D_{V/G} which keeps track of all complex reflections in G. It turns out that the union Z_{1} of Z_{0} and of all codimension 1 strata is a complex manifold, see 3.5. We characterize all G-invariant holomorphic tensor fields and connections on V in terms of the reflection divisor of the corresponding meromorphic tensor field and connection on Z_{1}, see 3.7 and 4.2. Our result gives a generalization 3.9 of Solomon’s theorem [10], see 3.10. Using the lifting property of connections we are able to prove that a holomorphic diffeomorphism Z = V/G → V/G′ = Z′ between two orbit spaces has a holomorphic lift to V which is equivariant over an isomorphism G → G′ if and only if f respects the regular strata and the reflection divisors, i.e. f(Z_{0}) ⊂ Z′_{0} and f_{∗}(D_{Z}) ⊂ D_{Z′}. In fact we give two proofs of this result, which in [4] is carried over to the algebraic geometry setting for algebraically closed ground fields of characteristic 0. The related problem of lifting (smooth) homotopies from (general) orbit spaces has been treated in [1] and [9].

Applying the local results we prove that a complex orbifold X is uniquely determined by the sheaf 𝔉_{X}, the regular stratum X_{0}, and the reflection divisor D_{X} alone, see 6.6.

2. Preliminaries

2.1. The orbit type stratification. Let V be an n-dimensional complex vector space, G a finite subgroup of GL(V), and π : V → V/G the quotient projection. The ring ℂ[V]^{G} has a minimal system of homogeneous generators σ^{1}, …, σ^{m}. We will use the map σ = (σ^{1}, …, σ^{m}) : V → ℂ^{m}. Denote by Z the affine algebraic variety in ℂ^{m} defined by the relations between σ^{1}, …, σ^{m}. It is known that σ(V) = Z.

We consider the orbit space V/G endowed with the quotient topology as a local ringed space defined by the following sheaf of rings 𝔉_{V/G}: if U is an open subset of V/G, 𝔉_{V/G}(U) is equal to the space of G-invariant holomorphic functions on π^{−1}(U). Clearly one may consider sections of 𝔉_{V/G} on U as functions on U. We call these functions holomorphic functions on U. It is known that the map of the orbit space V/G to Z induced by the map σ is a homeomorphism. Moreover, this homeomorphism induces an isomorphism of the sheaf 𝔉_{V/G}(U) and the structure sheaf of the complex algebraic variety Z (see [7]). Via the above isomorphism we identify the local ringed spaces V/G and Z. Under this identification the projection π is identified with the map σ. Let G and G′ be finite subgroups of GL(V) and let Z = V/G and Z′ = V/G′ be the corresponding orbit spaces. By definition a holomorphic diffeomorphism of the orbit space Z to the orbit space Z′ is an isomorphism of Z to Z′ as local ringed spaces.

Let K be a subgroup of G, (K) the conjugacy class of K. Denote by V_{(K)} the set of points of V whose isotropy groups belong to (K) and put Z_{(K)} = π(V_{(K)}). It is known that {Z_{(K)}} is a finite stratification of Z, called the isotropy type stratification, into locally closed irreducible smooth algebraic subvarieties (see [5]). Denote by Z^{i} the union of the strata of codimension greater than i and put Z_{i} = Z \Z^{i}. Then Z_{0} is the principal stratum of Z, i.e. Z_{0} = Z_{(K)} for K = {id}. It is known that Z_{0} is a Zariski open subset of Z and a complex manifold. It is clear that the restriction of the map σ to the set V_{reg} of regular points of V is an tale map onto Z_{0}.

In this paper we consider the orbit space Z = V/G with the above structure of local ringed space and the stratification {Z_{(K)}}.

2.2. The divisor of a tensor field. We shall use divisors of meromorphic functions on a complex manifold X. For technical reasons (see e.g. the last formula of this section) we define div(0) = ∑_{S} ∞.S, where the sum runs over all complex subspaces of X of codimension 1.

Let f and g be two meromorphic functions on X. Then we have

div(f + g) ≥ min{div(f), div(g)}, where div(f) denotes the divisor of f.

Taking the minimum means: For each irreducible complex subspace S of X of codimension 1 belonging to the support of f or g take the minimum of the coefficients in Z of S in div(f) and div(g).

Let P be a meromorphic tensor field (i.e., with meromorphic coefficient functions in local coordinates) on X. In local holomorphic coordinates y^{1}, …, y^{n} on an open subset U ⊂ X the tensor field P can be written as

P |_{U} = ∑_{i_{1},...,i_{p},j_{1},...,j_{q}} P_{j_{1}...j_{q}}^{i_{1}...i_{p}} \frac{∂}{∂y^{i_{1}}} ⊗ ⋯ ⊗ \frac{∂}{∂y^{i_{p}}} ⊗ dy^{j_{1}} ⊗ dy^{j_{q}}.

and we define the divisor of P on U as the minimum of all divisors div(P_{j_{1}...j_{q}}^{i_{1}...i_{p}}) ∈ Div(U) for all coefficient functions of P. The resulting coefficient of the complex subspace S of codimension 1 in div(P) ∈ Div(U) does not depend on the choice of the holomorphic coordinate system; e.g., for a vector field ∑_{i} X^{i} \frac{∂}{∂y^{i}} = ∑_{i,k} X^{i} \frac{∂u^{k}}{∂y^{i}} \frac{∂}{∂u^{k}} we have

div(∑_{i} X^{i} \frac{∂u^{k}}{∂y^{i}} ) ≥ min_{i} div(X^{i} \frac{∂u^{k}}{∂y^{i}} ) = min_{i}(div(X^{i})+div(\frac{∂u^{k}}{∂y^{i}} )) ≥ min_{i} div(X^{i}).

Finally we define the divisor of P on X by gluing the local divisors for any holomorphic atlas of X. Note that a tensor field P is holomorphic if and only if div(P) ≥ 0.

3. Invariant tensor fields

3.1. Let P be a G-invariant holomorphic tensor field of type \binom{p}{q} on V . Since σ is an tale map on V_{reg}, there is a unique holomorphic tensor field Q on Z_{0} of type \binom{p}{q} such that the pullback σ^{∗}(Q) coincides with the restriction of P to V_{reg}. It is clear that the tensor field P is uniquely defined by Q.

Consider a holomorphic tensor field Q of type \binom{p}{q} on Z_{0} and its pullback σ^{∗}(Q) which is a G-invariant holomorphic tensor field on V_{reg}. Then by the Hartogs extension theorem, σ^{∗}(Q) has a G-invariant holomorphic extension to V iff it has a holomorphic extension to σ^{−1}(Z_{1}).

Denote by ℌ the set of all reflection hyperplanes corresponding to all complex reflections in G and, for each H ∈ ℌ, by e_{H} the order of the cyclic subgroup of G fixing H. It is clear that σ(∪_{H∈ℌ}H) contains all strata of codimension 1. This implies immediately the following

3.2. Proposition. If ℌ = Ø, for each holomorphic tensor field P_{0} on Z_{0} the pullback σ^{∗}(P_{0}) has a G-invariant holomorphic extension to V .

3.3. The reflection divisor of the orbit space. Consider the set R_{Z} of all hyper surfaces σ(H) in Z, where H runs through all reflection hyperplanes in V . Note that σ(H) is a complex subspace of Z_{1} of codimension 1. We endow each S = σ(H) ∈ R_{Z} with the label e_{H} of the hyperplane H. It is easily seen that this label does not depend on the choice of H, we denote it by e_{S} and we consider e_{S}. S as an effective divisor on Z and we consider the effective divisor in Z_{1}

D = D_{V/G} = D_{Z} = ∑_{S∈R_{Z}} e_{S}. S,

which we call the reflection divisor.

3.4. Basic example. Let the cyclic group ℤ_{r} = ℤ/rℤ with generator ζ_{r} = e^{2πi/r} act on ℂ by z ↦ e^{2πik/r}z for r ≥ 2. The generating invariant is τ(z) = z^{r}.

We consider first a holomorphic tensor field P = f(z)(dz)^{⊗q} ⊗(\frac{∂}{∂z})^{⊗p} on ℂ. It is invariant, ζ_{r}^{∗}P = P, if and only if f(ζ_{r}z) = ζ_{r}^{p−q}f(z), so that in the expansion f(z) = ∑_{k≥0} f_{k}z^{k} at 0 of f the coefficient f_{k} ≠ 0 at most when k ≅ p−q mod r. Writing p − q = rs + t with s ∈ ℤ and 0 ≤ t < r we see that P is invariant if and only if f(z) = z^{t}g(z^{r}) for holomorphic g.

We use the coordinate y = τ(z) = z^{r} on ℂ/ℤ_{r} = ℂ, τ^{∗}dy = rz^{r−1}dz and τ^{∗}(\frac{∂}{∂y}|_{ℂ\0}) = \frac{1}{rz^{r−1}} \frac{∂}{∂z}|_{ℂ\0}, and we write

P |_{ℂ\0} = g(z^{r})z^{t}(dz)^{⊗q} ⊗ (\frac{∂}{∂z})^{⊗p}
= g(y)z^{t}(rz^{r−1})^{p−q}(dy)^{⊗q} ⊗ (\frac{∂}{∂y})^{⊗p}
= g(y)z^{−rs}(rz^{r})^{p−q}(dy)^{⊗q} ⊗ (\frac{∂}{∂y})^{⊗p}
= g(y)r^{p−q}y^{p−q−s}(dy)^{⊗q} ⊗ (\frac{∂}{∂y})^{⊗p}

(we omitted τ^{∗}). Thus a holomorphic tensor field P of type \binom{p}{q} on ℂ is ℤ_{r} - invariant if and only if P |_{ℂ\0} = τ^{∗}Q for a meromorphic tensor field

Q = g(y)y^{m}(dy)^{⊗q} ⊗ (\frac{∂}{∂y})^{⊗p}

on ℂ with g holomorphic with g(0) ≠ 0 and with

m ≥ p − q − s.

It is easily checked that the above inequality is equivalent to the following one

mr + (q − p)(r − 1) ≥ 0.

3.5. Suppose ℌ ≠ Ø. Let z ∈ Z_{1} \ Z_{0} and v ∈ σ^{−1}(z). Then there is a unique hyperplane H ∈ ℌ such that v ∈ H and the isotropy group G_{v} is isomorphic to a cyclic group. It is evident that the order r_{z} = e_{H} of G_{v} depends only on z = σ(v) and is locally constant on Z_{1} \ Z_{0}.

By the holomorphic slice theorem (see [5], [6]) there is a G_{v} -invariant open neighborhood U_{v} of v in V such that the induced map U_{v}/G_{v} → V/G is a local biholomorphic map at v.

Choose orthonormal coordinates z^{1}, …, z^{n} in V with respect to a Ginvariant Hermitian inner product on V , so that H = {z^{n} = 0}. Then the ring ℂ[V]^{G_{v}} is generated by z^{1}, …, z^{n−1}, (z^{n})^{r}, where r = r_{z}.

Put τ^{1} = z^{1}, …, τ^{n−1} = z^{n−1}, τ^{n} = (z^{n})^{r}, and τ = (τ^{1}, …, τ^{n}) : U_{v} → ℂ^{n}. Then there are holomorphic functions f^{i} (i = 1, …, n) in an open neighborhood W_{z} of z ∈ ℂ^{m} such that τ^{a} = f^{a} ∘ σ|_{U_{v}}. On the other hand, we know that in an open neighborhood of v all σ^{a} for (a = 1, …, m) are holomorphic functions of the τ^{i}. We denote by y^{i} the holomorphic function on Z such that τ^{i} = y^{i} ∘ σ. Then we can use y^{i} as coordinates of Z defined in the open neighborhood W_{z} ⊆ ℂ^{m} of z. Note that we found holomorphic coordinates near each point of Z_{1}, so we have:

Corollary. The union Z_{1} of all codimension ≤ 1 strata, with the restriction of the sheaf 𝔉_{V/G}, is a complex manifold.

3.6. The reflection divisor of a meromorphic tensor field on Z_{1}. Let Γ_{M}(T_{q}^{p}(Z_{1})) be the space of meromorphic tensor fields (i.e. with meromorphic coefficient functions in local holomorphic coordinates on the complex manifold Z_{1}), and let P ∈ Γ_{M}(T_{q}^{p}(Z_{1})).

Let S be an irreducible component of Z_{1} \ Z_{0} and let z ∈ S. Local coordinates y^{1}, …, y^{n} on U ⊂ Z_{1}, centered at z, are called adapted to the stratification of Z_{1} if S = {y^{n} = 0} near z. By definition the coordinates y^{1}, …, y^{n} from 3.5 have this property. Denote by O_{z} the ring of germs of holomorphic functions and by M_{z} the field of germs of meromorphic functions, both at z ∈ Z_{1}.

Let y^{1}, …, y^{n} be local coordinates on U ⊂ Z_{1}, centered at z, adapted to the stratification of Z_{1} . Then on U the meromorphic tensor field P is given by

P |_{U} = ∑_{i_{1},...,i_{p},j_{1},...,j_{q}} P_{j_{1}...j_{q}}^{i_{1}...i_{p}} \frac{∂}{∂y^{i_{1}}} ⊗ ⋯ ⊗ \frac{∂}{∂y^{i_{p}}} ⊗ dy^{j_{1}} ⊗ dy^{j_{q}},

where the P_{j_{1}...j_{q}}^{i_{1}...i_{p}} are meromorphic on U. Let us fix one nonzero summand of the right hand side: for the coefficient function we have P_{j_{1}...j_{q}}^{i_{1}...i_{p}} = (y^{n})^{m}f for some integer m such that the germs at z of y^{n}, g, and h are pairwise relatively prime in O_{z} where f = g/h ∈ M_{z}. Suppose that the factor \frac{∂}{∂y^{n}} appears exactly p′ times and the factor dy^{n} appears exactly q′ times in this summand. The integer

μ = mr + (q′ − p′)(r − 1),

a priori depending on z, is constant along an open dense subset of S and it is called the reflection residuum of the summand at S. Finally let μ_{S}(P) be the minimum of the reflection residua at S of all summands of P in the representation of P.

Let ỹ^{1}, …, ỹ^{n} be arbitrary local coordinates on U ⊂ Z_{1}, centered at z, adapted to the stratification of Z_{1}. In a neighborhood of z we have y^{n} = fỹ^{n}, where f is a holomorphic function such that f(z) ≠ 0. Remark that ỹ^{n} divides \frac{∂y^{n}}{∂ỹ^{i}} and \frac{∂ỹ^{n}}{∂y^{i}} (i = 1, …, n) in O_{z}. A straightforward calculation using the above remark shows that the values of μ_{S}(P) calculated in the coordinates ỹ^{i} and in the coordinates y^{i} are the same. Then μ_{S}(P) does not depend on the choice of the system of local coordinates adapted to the stratification of Z_{1}. For details see [4]: there we checked this in the algebraic geometry setting where the use of tensor fields is less familiar.

We now can define the reflection divisor

div_{D}(P) = div_{D_{V/G}}(P) ∈ Div(U)

as follows: take the divisor div(P), and for each irreducible component S of Z_{1} \ Z_{0} do the following: if S appears in the support of div(P) ∈ Div(U), replace its coefficient by μ_{S}(P); if it does not appear, add μ_{S}(P).S to it. If S is not contained in Z_{1} \ Z_{0}, we keep its coefficient in div(P).

Finally we glue the global reflection divisor div_{D}(P) ∈ Div(Z_{1}) from the local ones, using a holomorphic atlas for Z_{1}.

3.7. Theorem. Let G ⊂ GL(V) be a finite group, with reflection divisor D = D_{V/G} = D_{Z}. Then we have:

• Let P be a holomorphic G-invariant tensor field on V . Then the reflection divisor div_{D}(π_{∗}P) ≥ 0.

• Let Q ∈ Γ_{M}(T_{q}^{p}(Z_{1})) be a meromorphic tensor field on Z_{1}. Then the G-invariant meromorphic tensor field π^{∗}Q extends to a holomorphic Ginvariant tensor field on V if and only if div_{D}(Q) ≥ 0.

The above remains true for G-invariant holomorphic tensor fields defined in a G-stable open subset of V .

Proof. This follows directly from Hartogs’ extension theorem, the basic example 3.4 using y^{1}, …, y^{n−1} as dummy variables, and the definition of the reflection divisor div_{D}(P) as explained in 3.6.

3.9. Corollary. The mapping σ establishes an injective correspondence between the space of holomorphic G-invariant tensor fields of type \binom{p}{q} on V which are skew-symmetric with respect to the covariant entries, and the space of holomorphic tensor fields on Z_{1} of the same type and the same skew-symmetry condition. If p = 0 the correspondence is bijective.

The above remains true for G-invariant holomorphic tensor fields defined in a G-stable open subset of V .

Proof. Let P be a holomorphic G-invariant tensor field on V satisfying the conditions of the corollary. For each nonzero decomposable summand of π_{∗}P take the integers m, p′, and q′ defined in 3.6. By skew symmetry of P we have q′ ≤ 1.

By Theorem 3.7 we get div_{D}(π_{∗}P) ≥ 0 and thus mr ≥ (p′ − q′)(r − 1) > −r. So m ≥ 0 and the summand is holomorphic on Z_{1}.

If Q is a holomorphic differential form on Z_{1} its pullback σ^{∗}Q is a Ginvariant holomorphic form on σ^{−1}(Z_{1}) and then has a holomorphic extension to the whole of V .

3.10. Remarks. Note that Corollary 3.9 is a generalization of Solomon’s theorem (see [10]): If G ⊂ GL(V) is a finite complex reflection group then every Ginvariant polynomial exterior q-form ω on V can be written as ω = σ^{∗}φ for a polynomial q-form φ on ℂ^{n}, where σ = (σ^{1}, …, σ^{n}) : V → ℂ^{n} is the mapping consisting of a minimal system of homogeneous generators of ℂ[V]^{G}.

Actually, in the case of a reflection group Z = ℂ^{n} and each holomorphic \binom{p}{q}-tensor field Q on Z_{1} has a holomorphic extension to Z by Hartogs’ extension theorem.

4. Invariant complex connections

4.1. Let Γ be a holomorphic G-invariant complex connection on V . Then the image σ_{∗}Γ of Γ under the map σ defines a holomorphic complex connection on Z_{0}.

Let z ∈ Z_{1} \ Z_{0}, v ∈ σ^{−1}(z), and r the order of G_{v}. Consider the coordinates z^{i} in V defined in 3.5. Denote by Γ_{jk}^{i} the components of the connection Γ with respect to these coordinates. By assumption, the Γ_{jk}^{i} are holomorphic functions on V . Recall the standard formula for the image γ of Γ under a holomorphic diffeomorphism f = (y^{a}(x^{i}))

γ_{bc}^{a} ∘ f = \frac{∂y^{a}}{∂x^{i}} \frac{∂x^{j}}{∂y^{b}} \frac{∂x^{k}}{∂y^{c}} Γ_{jk}^{i}(x^{l}) − \frac{∂^{2}y^{a}}{∂x^{i}∂x^{j}} \frac{∂x^{i}}{∂y^{b}} \frac{∂x^{j}}{∂y^{c}} .

Remark that the similar formula is true for the transformation of the components of connection under the change of coordinates.

Consider the generator g of the cyclic group G_{v} given by 3.5. Since g acts linearly, the connection reacts to it like a \binom{1}{2}-tensor field. Thus by the considerations of 3.4 we get in the notation of 3.5, where i, j, k = 1, …, n − 1:

Γ_{jk}^{i} = Γ̃_{jk}^{i} ∘ σ, Γ_{jk}^{n} = \frac{1}{r}z^{n}Γ̃_{jk}^{n} ∘ σ, Γ_{jn}^{i} = r(z^{n})^{r−1}Γ̃_{jn}^{i} ∘ σ,

Γ_{nk}^{i} = r(z^{n})^{r−1}Γ̃_{nk}^{i} ∘ σ, Γ_{jn}^{n} = Γ̃_{jn}^{n} ∘ σ, Γ_{nk}^{n} = Γ̃_{nk}^{n} ∘ σ,

Γ_{nn}^{i} = r^{2}(z^{n})^{r−2}Γ̃_{nn}^{i} ∘ σ, Γ_{nn}^{n} = r(z^{n})^{r−1}Γ̃_{nn}^{n} ∘ σ,

where the Γ̃_{bc}^{a} are holomorphic functions of the coordinates y^{a} (a = 1, …, n) introduced in 3.5.

Using the transformation formula for connections, we get the following formulas for the components γ_{bc}^{a} of the meromorphic connection σ_{∗}Γ with respect to the coordinates y^{a}

γ_{jk}^{i} = Γ̃_{jk}^{i}, γ_{jk}^{n} = y^{n}Γ̃_{jk}^{n}, γ_{jn}^{i} = Γ̃_{jn}^{i}, γ_{nk}^{i} = Γ̃_{nk}^{i},  (4.1.1)

γ_{jn}^{n} = Γ̃_{jn}^{n}, γ_{nk}^{n} = Γ̃_{nk}^{n}, γ_{nn}^{i} = \frac{1}{y^{n}} Γ̃_{nn}^{i}, γ_{nn}^{n} = Γ̃_{nn}^{n} − \frac{r − 1}{ry^{n}}.

Let ỹ^{a} for a = 1, …, n be other local coordinates centered at z and adapted to the stratification of Z_{1}. Then in a neighborhood of z we have

y^{n} = fỹ^{n}, ỹ^{n} = f̃y^{n},

where f and f̃ are holomorphic functions in a neighborhood of z and f̃f = 1. Then we have

\frac{∂y^{n}}{∂ỹ^{i}} = \frac{∂f}{∂ỹ^{i}} ỹ^{n}, \frac{∂ỹ^{n}}{∂y^{i}} = \frac{∂f}{∂y^{i}} y^{n} (i = 1, …, n − 1)

and on S = {y^{n} = 0}

\frac{∂y^{n}}{∂ỹ^{n}} = f, \frac{˜∂y^{n}}{∂y^{n}} = f̃.

Using these formulas one can check that in the coordinates ỹ^{a} the formulas 4.1.1 have the same form as in the coordinates y^{a}. For example, for the new component γ̃_{nn}^{n} we have

γ̃_{nn}^{n} + \frac{r − 1}{rỹ^{n}} = \frac{(r − 1)(1 − f̃\frac{∂ỹ^{n}}{∂y^{n}} (\frac{∂y^{n}}{∂ỹ^{n}} )^{2})}{ry^{n}f̃} + h,

where h is a holomorphic function near z. Since on S = {y^{n} = 0} we have

1 − f̃\frac{∂ỹ^{n}}{∂y^{n}} (\frac{∂y^{n}}{∂ỹ^{n}} )^{2} = 1 − f̃^{2}f^{2} = 0,

y^{n} divides in O_{z} the function

1 − f̃\frac{∂ỹ^{n}}{∂y^{n}} (\frac{∂y^{n}}{∂ỹ^{n}} )^{2}.

Thus

γ̃_{nn}^{n} + \frac{r − 1}{rỹ^{n}}

is holomorphic in a neighborhood of z.

4.2. Theorem. Let γ be a holomorphic complex linear connection on Z_{0} such that for each z ∈ Z_{1} \ Z_{0} it has an extension to a neighborhood of z whose components in the coordinates adapted to the stratification of Z_{1} are defined by the formulas 4.1.1 where Γ̃_{bc}^{a} are holomorphic. Then there is a unique G-invariant holomorphic complex linear connection Γ on V such that σ_{∗}Γ coincides with γ on Z_{0}. This remains true if we replace V by a G-open subset of G.

Proof. Since σ is tale on the principal stratum, there is a unique G-invariant complex linear connection Γ_{0} on σ^{−1}(Z_{0}) such that σ_{∗}Γ_{0} = γ. The condition of the theorem implies that the connection Γ_{0} has a holomorphic extension to σ^{−1}(Z_{1}). Then by Hartogs’ extension theorem the connection Γ_{0} has a unique holomorphic extension Γ to the whole of V .

5. Lifts of diffeomorphisms of orbit spaces

5.1. Let G and G′ be finite subgroups of GL(V) and GL(V′) and let F be a holomorphic diffeomorphism V → V′ which maps G-orbits to G′-orbits bijectively. Then the map F induces an isomorphism f of the sheaves 𝔉_{V/G} → 𝔉_{V′/G′}, i.e. a holomorphic diffeomorphism of orbit spaces V/G and V′/G′.

Lemma. There is a unique isomorphism a : G → G′ such that F ∘ g = a(g) ∘ F for every g ∈ G.

Note that a and its inverse a^{−1} map complex reflections to complex reflections. Proof. The cardinalities of the two groups are the same since F maps a generic regular orbit to a regular orbit. Consequently, it maps regular points to regular points and we have σ′ ∘ F = f ∘ σ : V → V′/G′ for a holomorphic diffeomorphism f : V/G → V′/G′, where σ : V → V/G and σ′ : V′ → V′/G′ are the quotient projections.

Fix some G-regular v ∈ V . Then F (v) and F (gv) for g ∈ G are regular points of V′ of the same orbit. Therefore, there is a unique a(g) ∈ G such that F (gv) = a(g)(F (v)). We have σ′ ∘ F ∘ g = f ∘ σ ∘ g = f ∘ σ = σ′ ∘ F = σ′ ∘ a(g) ∘ F. Since σ′ is tale on V′_{reg} we see that F ∘ g = a(g) ∘ F locally near v and thus globally. By uniqueness, the map g → a(g) is an isomorphism of G onto G′.

In this section we study when a diffeomorphism f of the orbit spaces Z → Z′ has a holomorphic lift F.

5.2. Corollary. Let F : V → V be a holomorphic diffeomorphism which maps G-orbits onto G′-orbits, and f : Z → Z′ the corresponding holomorphic diffeomorphism of the orbit spaces. Then f maps the isotropy type stratification of Z onto that of Z′ and, moreover, it maps D_{Z} to D_{Z′}.

Proof. This follows from Lemma 5.1 and the definition 3.3 of the reflection divisor.

5.3. Theorem. Let G and G′ be two finite subgroups of GL(V) and let f : Z → Z′ be a holomorphic diffeomorphism of the corresponding orbit spaces such that f(Z_{0}) = Z′_{0} and f_{∗}(D_{Z}) = D_{Z′}. If Q is a holomorphic tensor field of type \binom{p}{q} on Z_{0} which satisfies the conditions of Theorem 3.7, then f_{∗}(Q) also satisfies these conditions on Z′_{0} and thus there exists a unique G′-invariant holomorphic tensor field Q′ of type \binom{p}{q} such that σ′_{∗}Q′ coincides with f_{∗}Q on Z′_{0}.

This is also true for holomorphic connections if we replace Theorem 3.7 by Theorem 4.2. The theorem remains true if we replace V by invariant open subsets of V .

Proof. Since f(Z_{0}) = Z′_{0} the tensor field f_{∗}Q is also holomorphic on Z′_{0}. Let z ∈ Z_{1} \ Z_{0}. Then there is a complex space S ∈ R_{Z} of codimension 1 such that z ∈ S. By assumption f(z) ∈ Z′_{1} \ Z′_{0} and f(z) ∈ f(S) ∈ R_{Z′} and r_{z} = e_{S} = e_{f(S)} = r_{f(z)}. Now, obviously f_{∗}Q satisfies the conditions of Theorem 3.7 at f(x). Thus there exists a G′-invariant holomorphic tensor field Q′ on V with σ′_{∗}Q′ = f_{∗}Q.

A similar argument applies to connections.

5.4 Theorem. Let G and G′ be two finite subgroups of GL(V). Let f : Z → Z′ be a holomorphic diffeomorphism of the orbit spaces such that f(Z_{0}) = Z′_{0} and f_{∗}(D_{Z}) = D_{Z′}.

Then f lifts to a holomorphic diffeomorphism F : V → V , i.e. σ′ ∘ F = f ∘ σ.

The local version is also true. Namely, if B is a ball in the vector space V centered at 0 (for an invariant Hermitian metric), U = σ(B), and f : U → Z′ is a local holomorphic diffeomorphism of U onto a neighborhood U′ of σ′(0) such that f(U ∩ Z_{0}) = U′ ∩ Z′_{0} and f maps D_{Z} ∩ U to D_{Z′} ∩ U′, then there is a holomorphic lift F : B → V .

Proof. Let Γ be the natural flat connection on V . Then Γ is uniquely defined by the holomorphic connection σ_{∗}Γ on Z_{0} which satisfies the conditions of Theorem 4.2. By Theorem 5.3 there is a unique G-invariant holomorphic complex linear connection Γ′ on V such that σ′_{∗}Γ′ coincides with f_{∗}(σ_{∗}Γ) on Z′_{0}. It is evident that Γ′ is a torsion free flat connection, since Γ is it and Γ′ is locally isomorphic to Γ on an open dense subset.

Let v ∈ V be G-regular and let v′ ∈ V be G′-regular, such that (f ∘σ)(v) = σ′(v′). Then there is a biholomorphic map F of a neighborhood W of v onto a neighborhood of v′ such that σ′ ∘ F = f ∘ σ on W and F (v) = v′. Moreover by construction F is a locally affine map of the affine space (V, Γ) into (V, Γ′) equipped with the above structures of locally affine spaces, thus we have

F = exp_{v′}^{Γ′} ∘T_{v}F ∘ (exp_{v}^{Γ})^{−1}  (1)

where exp_{v}^{Γ} : T_{v}V → V is the holomorphic geodesic exponential mapping centered at v given by the connection Γ and its induced spray. It is globally defined, thus complete and a holomorphic diffeomorphism since Γ is the standard flat connection. Likewise exp_{v′}^{Γ′} is the holomorphic exponential mapping of the flat connection Γ′. The formula above extends F to a globally defined holomorphic mapping if exp_{v′}^{Γ′} : T_{v}V → V is also globally defined (complete). Assume for contradiction that this is not the case. Let F be maximally extended by equation (1); it still projects to f : Z → Z′. We consider exp_{v′}^{Γ′} as a real exponential mapping, and then there is a real geodesic which reaches infinity in finite time and this is the image under F of a finite part exp_{v}^{Γ}([0, t_{0})w) of a real geodesic of Γ emanating at v. The sequence exp_{v}^{Γ}((t_{0} − 1/n)w) converges to exp_{v}^{Γ}(t_{0}w) in V , but its image under F diverges to infinity by assumption. On the other hand, the image under F is contained in the set (σ′)^{−1}(fσ(exp_{v}^{Γ}([0, t_{0}]w))) which is compact since σ′ is a proper mapping. Contradiction.

Any holomorphic lift F of a holomorphic diffeomorphism f is a holomorphic diffeomorphism of V which maps G-orbits onto G′ orbits, by the following argument: Let F′ be a holomorphic lift of f^{−1}. Evidently the map F′ ∘ F preserves each G-orbit. Then, for a G-regular point v ∈ V , there is a g ∈ G such that F′ ∘ F = g in a neighborhood of v and, then, on the whole of V . Similarly F ∘ F′ = g′ ∈ G′. This implies that F is a holomorphic diffeomorphism of V . By definition the lift F respects the partitions of V into orbits.

We give a second proof of Theorem 5.4 based on the known results about the fundamental groups of V_{reg} and Z_{0} for finite complex reflection groups. It is an extension of the proof of [8], using results of [2].

5.5. Lemma. Let G and G′ be two finite subgroups of GL(V) and let f : Z → Z′ be a holomorphic diffeomorphism of the corresponding orbit spaces. Suppose v_{0} ∈ V_{reg}, v′_{0} ∈ V′_{reg}, and f ∘ σ(v_{0}) = σ′(v′_{0}). If the image of the fundamental group π_{1}(V_{reg}, v_{0}) under f ∘ σ is contained in the subgroup σ′_{∗}(π_{1}(V_{reg}), v′_{0}) of π_{1}(Z′_{0}, σ′(v′_{0})), the holomorphic lift of f ∘ σ mapping v_{0} to v′_{0} exists.

Proof. Consider the restriction φ of the map f ∘ σ to V_{reg}. Since the restriction of σ to V_{reg} is a covering map onto Z_{0}, the condition of the lemma implies that there is a holomorphic lift F_{0} of the map φ to V_{reg}. The map F_{0} is bounded on B ∩ V_{reg} for each compact ball B in V since its image is contained in the compact set (σ′)^{−1}(f(σ(B))). Then by the Riemann extension theorem F_{0} has a holomorphic extension F to V which is the required holomorphic lift of f.

5.6. Next we prove Theorem 5.4 in the case when the group G is generated by complex reflections. Put

B := π_{1}(Z_{0}) and P := π_{1}(V_{reg}).

The groups B and P are called the braid group and the pure braid group associated to G, respectively. It is clear that the map σ induces an isomorphism of P onto a subgroup of B.

The following results about the groups B and P are well known (see, for example, [2]). The braid group B is generated by those elements which are represented by loops around the hypersurfaces σ(H) for H ∈ ℌ. The pure braid group P is generated by the elements of B of the type s^{e_{H}}, where s is any of the above generators of B represented by a loop around the hypersurface σ(H). This implies the following

Proposition. Suppose the group G is generated by complex reflections. Let f be a holomorphic diffeomorphism of the orbit space Z = ℂ^{n} with f(Z_{0}) = Z_{0} which also preserves D_{Z}. Then f|_{Z_{0}} preserves the subgroup P of B.

The following proposition is an immediate consequence of Lemma 5.5 and Proposition 5.6.

5.7. Proposition. Suppose the groups G and G′ are generated by complex reflections. Let f : Z → Z′ be a holomorphic diffeomorphism between the corresponding orbit spaces, such that f(Z_{0}) = Z′_{0} and f_{∗}(D_{Z}) = D_{Z′}.

Then f has a holomorphic lift F to V .

Second proof of 5.4. Now let G ⊂ GL(V) be a finite group and let G_{1} be the subgroup generated by all complex reflections in G. Clearly G_{1} is a normal subgroup of G. Let G_{2} = G/G_{1}. Let σ_{1}^{1}, …, σ_{1}^{n} be a system of homogeneous generators of ℂ[V]^{G_{1}} and σ_{1} : V → ℂ^{n} the corresponding orbit map. Then the action of G on V induces the action of the group G_{2} on V_{1} := ℂ^{n} = σ_{1}(V). Since each representation of the group G_{2} is completely reducible, by standard arguments of invariant theory, we may assume that the generators σ_{1}^{i}’s are chosen in such a way that the above action of G_{2} on V_{1} = ℂ^{n} is linear. Then the representation of G_{2} on V_{1} contains no complex reflections. Let σ_{2}^{1}, …, σ_{2}^{m} be a system of homogeneous generators of ℂ[V_{1}]^{G_{2}} and σ_{2} : V_{1} → ℂ^{m} the corresponding orbit map. Then σ^{i} = σ_{2}^{i} ∘ σ_{1} (i = 1, …, m) is a system of generators of ℂ[V]^{G} with orbit map σ = σ_{2} ∘ σ_{1}. Similarly for G′.

Let f : Z → Z′ be a holomorphic diffeomorphism, such that f(Z_{0}) = Z′_{0} and f_{∗}(D_{Z}) = D_{Z′}. Since the group G_{2} contains no complex reflections the set V_{1,reg} of regular points of the action of G_{2} on V_{1} is obtained from V_{1} by removing some subsets of codimension ≥ 2. And similarly for G′. Then the fundamental group π_{1}(V_{1,reg}) = π_{1}(V_{1}) = 0 is trivial and by lemma 5.5 the diffeomorphism f has a holomorphic lift F_{1} : V_{1} → V′_{1} which is a holomorphic diffeomorphism mapping the principal stratum to the principal stratum, and the reflection divisor to the reflection divisor, since G_{2} contains no complex reflections on V_{1}. Thus the diffeomorphism F_{1} has a holomorphic lift to V by Proposition 5.7, which is a holomorphic lift of f.

6. An intrinsic characterization of a complex orbifold

We recall the definition of orbifold.

6.1. Definition. [11] Let X be a Hausdorff space. An atlas of a smooth ndimensional orbifold on X is a family {U_{i}}_{i∈I} of open sets that satisfy:

1. {U_{i}}_{i∈I} is an open cover of X.

2. For each i ∈ I we have a local uniformizing system consisting of a triple (Ũ_{i}, G_{i}, φ_{i}), where Ũ_{i} is a connected open subset of ℝ^{n} containing the origin, G_{i} is a finite group of diffeomorphisms acting effectively and properly on Ũ_{i}, and φ_{i} : Ũ_{i} → U_{i} is a continuous map of Ũ_{i} onto U_{i} such that φ_{i} ∘ g = φ_{i} for all g ∈ G_{i} and the induced map of Ũ_{i}/G_{i} onto U_{i} is a homeomorphism. The finite group G_{i} is called a local uniformizing group.

3. Given x̃_{i} ∈ Ũ_{i} and x̃_{j} ∈ Ũ_{j} such that φ_{i}(x̃_{i}) = φ_{j}(x̃_{j}), there is a diffeomorphism g_{ij} : Ṽ_{j} → Ṽ_{i} from a neighborhood Ṽ_{j} ⊆ Ũ_{j} of x̃_{j} onto a neighborhood Ṽ_{i} ⊆ Ũ_{i} of x̃_{i} such that φ_{j} = φ_{i} ∘ g_{ij}.

Two atlases are equivalent if their union is again an atlas of a smooth orbifold on X. An orbifold is the space X with an equivalence class of atlases of smooth orbifolds on X.

If we take in the definition of orbifold ℂ^{n} instead of ℝ^{n} and require that G_{i} is a finite group of holomorphic diffeomorphisms acting effectively and properly on Ũ_{i} and the maps g_{ij} are biholomorphic, we get the definition of complex analytic n-dimensional orbifold.

6.2. Theorem. [11] Let M be a smooth manifold and G a proper discontinuous group of diffeomorphisms of M. Then the orbit space M/G has a natural structure of smooth n-dimensional orbifold. If M is a complex n-dimensional manifold and G is a group of holomorphic diffeomorphisms of M, the orbit space M/G is a complex n-dimensional orbifold.

6.3 Definitions. In the definition of atlas of a complex orbifold on X we can always take Ũ_{i} to be balls of the space ℂ^{n} (with respect to some Hermitian metric) centered at the origin and the finite subgroups G_{i} to be subgroups of the GL(n) acting naturally on ℂ^{n}. In the sequel we consider atlases of complex orbifolds satisfying these conditions.

Let X be a complex orbifold with an atlas (Ũ_{i}, G_{i}, φ_{i}). A function f : U_{i} → ℂ is called holomorphic if f ∘ φ_{i} is a holomorphic function on Ũ_{i}. The germs of holomorphic functions on X define a sheaf 𝔉_{X} on X. It is evident that the sheaf 𝔉_{X} depends only on the structure of complex orbifold on X.

Consider a uniformizing system (Ũ_{i}, G_{i}, φ_{i}) of the above atlas and the corresponding action of G_{i} on ℂ^{n}. Then we have the isotropy type stratification of the orbit space ℂ^{n}/G_{i}, the induced stratification of U_{i}, and the divisor D_{U_{i}}.

By corollary 5.2 we get the stratification on X by gluing the strata on the U_{i}’s. Denote by X_{0} the principal stratum of this stratification. By definition, for each x ∈ X_{0}, for each uniformizing system (Ũ_{i}, G_{i}, φ_{i}), and for each y ∈ Ũ_{i} such that φ_{i}(y) = x, the isotropy group G_{y} of y is trivial. Note that X_{0} is a complex manifold. Note that X_{1} is also a complex manifold since this holds locally as noted in 3.5.

Denote by R_{X} the set of all strata of codimension 1 of X. Since the pullbacks of the reflection divisors D_{U_{i}} to U_{i} ∩ U_{j} agree by 5.2 we may glue them into the reflection divisor D_{X} on X_{1}.

6.4. Definition. Let X and X̃ be two smooth orbifolds. The orbifold X̃ is called a covering orbifold for X with a projection p : X̃ → X if p is a continuous map of underlying topological spaces and each point x ∈ X has a neighborhood U = Ũ/G (where Ũ is an open subset of ℝ^{n}) for which each component V_{i} of p^{−1}(U) is isomorphic to Ũ/G_{i}, where G_{i} ⊆ G is some subgroup. The above isomorphisms U = Ũ/G and V_{i} = Ũ/G_{i} must respect the projections.

Note that the projection p in the above definition is not necessarily a covering of the underlying topological spaces. It is clear that a covering orbifold for a complex orbifold is a complex orbifold. Hereafter we suppose that all orbifolds and their covering orbifolds are connected.

6.5. Theorem. [11] An orbifold X has a universal covering orbifold p : X̃ → X. More precisely, if x ∈ X_{0}, x̃ ∈ X̃_{0} and p(x̃) = x, for any other covering orbifold p′ : X̃′ → X and x̃′ ∈ X̃′ such that p′(x̃′) = x there is a cover q : X̃ → X̃′ such that p = p′ ∘ q and q(x̃) = x̃′. For any points x̃, x̃′ ∈ p^{−1}(x) there is a deck transformation of X̃ taking x̃ to x̃′.

Now we prove the main theorem of this section.

6.6. Theorem. An n-dimensional complex orbifold X is uniquely determined by the sheaf of holomorphic functions 𝔉_{X}, the principal stratum X_{0}, and the reflection divisor D_{X}.

Proof. For each x ∈ X, there exists V = ℂ^{m}, a finite group G ⊂ GL(m), a ball B in V centered at 0, an open subset U of X containing x, and an isomorphism ψ : π(B) → U between the sheaves 𝔉_{Z}|_{π(B)} and 𝔉_{X}|_{U}. Consider the map π : V → Z = V/G, the stratum Z_{0} and the reflection divisor D_{Z}. We suppose also that ψ(Z_{0} ∩ B/G) ⊆ X_{0} and ψ_{∗}(D_{π(B)}) = D_{U}. It suffices to prove that the germ of the uniformizing system {B, G, ψ ∘π|B} at x is the germ of some uniformizing system of the orbifold X.

Let y ∈ V_{reg} ∩ B. Then the ring 𝔉_{Z}(π(y)) of germs of 𝔉_{Z} at π(y) is isomorphic to the ring of germs of holomorphic functions on ℂ^{n} at 0 and thus we have m = n.

Consider the uniformizing system (Ũ_{i}, G_{i}, φ_{i}) of the orbifold X, where Ũ_{i} is a ball in ℂ^{n} centered at the origin, G_{i} is a finite subgroup of the group GL(n) acting naturally on V = ℂ^{n}, and where φ_{i}(0) = x. Consider the map π_{i} : V → V/G_{i} given by some system of generators of ℂ[V]^{G_{i}}. We may assume that φ_{i} = ψ_{i} ∘ π_{i}|Ũ_{i}, where ψ_{i} : 𝔉_{Ũ_{i}/G_{i}} → 𝔉_{U_{i}} is an isomorphism of sheaves.

\xymatrix{ℂ^{n} & B \ar[d]_{π} \ar@{^{(}->}[l] \ar@{.>}[rr]_{F} &  & Ũ_{i} \ar[d]^{π_{i}} \ar@{^{(}->}[r] & ℂ^{n} \\  & B/G \ar[dr]_{ψ} \ar@{.>}[rr]_{f} &  & Ũ_{i}/G_{i} \ar[dl]^{ψ_{i}} \\  &  & U}

Then the maps ψ and ψ_{i} define a map (germ) f of a holomorphic diffeomorphism B/G to U_{i}/G_{i} at 0 := π(0) such that f(0) = 0 := π_{i}(0). Then f induces an isomorphism 𝔉_{V/G}(0) → 𝔉_{V/G_{i}}(0), it maps (B/G)_{0} to (Ũ_{i}/G_{i})_{0} and f_{∗}(D_{B/G}) = D_{Ũ_{i}/G_{i}}. Thus by theorem 5.4 there is a germ of a holomorphic diffeomorphism F : B → Ũ_{i} which is equivariant for a suitable isomorphism G → G_{i}.

6.7. Corollary. Let M be a complex simply connected manifold, G a proper discontinuous group of holomorphic diffeomorphisms of M, and 𝔉_{X} the corresponding sheaf on the orbifold X = M/G. The G-manifold M is a universal covering orbifold for the orbifold X and it is defined uniquely up to a natural isomorphism of universal coverings by the sheaf 𝔉_{X}, the principal stratum X_{0}, and by the reflection divisor D_{X}.

Proof. Evidently the manifold M is a covering orbifold for X. If X̃ is a universal covering orbifold for X, by definition 6.4 there is a cover q : X̃ → M. By definition X̃ should be a manifold and q a cover of manifolds. Therefore, q is a diffeomorphism. Then the statement of the corollary follows from theorem 6.6.

An automorphism of the sheaf 𝔉_{X} is called a holomorphic diffeomorphism of the orbit space X. Theorem 6.5 and corollary 6.7 imply the following analogue of Theorem 5.4.

6.8. Theorem. Let M be a complex simply connected manifold, G a proper discontinuous group of holomorphic diffeomorphisms of M, and 𝔉_{X} the corresponding sheaf on the orbifold X = M/G. Each holomorphic diffeomorphism f of the orbit space X preserving X_{0} and D_{X} has a holomorphic lift F to M, which is G-equivariant with respect to an automorphism of G. The lift F is unique up to composition by an element of G.

Proof. By theorem 6.6 and corollary 6.7 the manifold M with the map f ∘ p : M → X, where p : M → X is the projection, is a universal covering orbifold for X. Then there is a holomorphic diffeomorphism F : M → M such that p ∘ F = f ∘ p. The equivariance property holds locally by 5.1, thus globally. The lift is uniquely given by choosing F (x) for a regular point x in the orbit f(p(x)).

6.9. Let V be a complex vector space with a linear action of a finite group G. The group ℂ^{∗} acts on V by homotheties and induces an action on Z = V/G.

Corollary. In this situation, the G-module V is uniquely defined up to a linear isomorphism by the sheaf 𝔉_{V/G} with the action of ℂ^{∗}, by Z_{0}, and the reflection divisor D_{Z}.

Proof. Consider the orbit space Z = V/G of a G-module V with the sheaf 𝔉_{V/G}, regular stratum Z_{0}, reflection divisor D_{Z}, and the action of ℂ^{∗} induced by the action of ℂ^{∗} on V by homotheties. Suppose that we have another G′-module V′ with the same data on Z′ = V′/G′ such that there is a biholomorphic map f : Z → Z′ preserving these data. By Theorem 4.5 there is a biholomorphic lift F : V → V′, and by lemma 5.1 there is an isomorphism a : G → G′ such that F ∘ g = a(g) ∘ F. Thus we may assume that G = G′, V = V′, Z = Z′, and a is the identity map. By definition the pullback A of the vector field on the orbit space V/G defined by the action of the group ℂ^{∗} on V/G coincides with the vector field on V defined by the above action of the group ℂ^{∗} on V . By construction F^{∗}A = A and then the map F commutes with the action of ℂ^{∗} on V , i.e. for each t ∈ ℂ^{∗} and v ∈ V we have F (tv) = tF (v). Since F is biholomorphic it is a linear automorphism of the vector space V . By definition F is then an automorphism of the G-module V .

6.10. Tensor fields and connections on orbifolds. The local results in section 3 show that the correct definition of a \binom{p}{q}-tensor field Q on an orbifold X is as follows: Q is a meromorphic \binom{p}{q}-tensor field on X_{1} such that div_{D_{X}}(Q) ≥ 0.

Likewise, we can define connections on orbifolds by requiring the local conditions of section 4.

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534 Kriegl, Losik, and Michor

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A. Kriegl Institut für Mathematik, Universität Wien, Strudlhofgasse 4, A-1090 Wien, Austria. Andreas.Kriegl@univie.ac.at

P. W. Michor Institut für Mathematik, Universität Wien, Strudlhofgasse 4, A-1090 Wien, Austria; and Erwin Schrödinger Institut für Mathematische Physik, Boltzmanngasse 9, A-1090 Wien, Austria Peter.Michor@esi.ac.at M. Losik Saratov State University, ul. Astrakhanskaya, 83, 410026 Saratov, Russia LosikMV@info.sgu.ru

Received July 15, 2002 and in final form October 29, 2002