AN OKA PRINCIPLE FOR STEIN G-MANIFOLDS

arXiv:1608.05156v2 [math.CV] 9 Jan 2017

GERALD W. SCHWARZ
Abstract. Let G be a reductive complex Lie group acting holomorphically on Stein manifolds
X and Y . Let pX : X → QX and pY : Y → QY be the quotient mappings. Assume that we
have a biholomorphism Q := QX → QY and an open cover {Ui } of Q and G-biholomorphisms
−1
Φi : p−1
X (Ui ) → pY (Ui ) inducing the identity on Ui . There is a sheaf of groups A on Q such that
the isomorphism classes of all possible Y is the cohomology set H 1 (Q, A). The main question
we address is to what extent H 1 (Q, A) contains only topological information. For example, if
G acts freely on X and Y , then X and Y are principal G-bundles over Q, and Grauert’s Oka
principle says that the set of isomorphism classes of holomorphic principal G-bundles over Q is
canonically the same as the set of isomorphism classes of topological principal G-bundles over
Q. We investigate to what extent we have an Oka principle for H 1 (Q, A).

Contents
1. Introduction
2. Background
3. Strongly continuous homeomorphisms and vector fields
4. Logarithms in Ac
5. Homotopies in H 1 (Q, Ac )
6. H 1 (Q, A) → H 1 (Q, Ac ) is a bijection
References

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3
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1. Introduction
Let X be a Stein G-manifold where G is a complex reductive group. There is a quotient
space QX = X//G (or just Q if X is understood) and surjective morphism pX (or just p) from
X to Q. Then Q is a reduced normal Stein space and the fibers of p are canonically affine
G-varieties (generally, neither reduced nor irreducible) containing precisely one closed G-orbit.
For S a subset of Q we denote p−1 (S) by XS and we abbreviate X{q} as Xq , q ∈ Q. We have
a sheaf of groups AX (or just A) on Q where A(U) = AutU (XU )G is the group of holomorphic
G-automorphisms of XU which induce the identity map IdU on XU //G = U.
Let Y be another Stein G-manifold. In [KLS15, KLS] we determined sufficient conditions
for X and Y to be equivariantly G-biholomorphic. Clearly we need that QY is biholomorphic
to QX , so let us assume that we have fixed an isomorphism of QY with Q = QX . Let us also
suppose that there are no local obstructions to a G-biholomorphism of X and Y covering IdQ .
(See [KLS, Theorem 1.3] for sufficient conditions for vanishing of the local obstructions.) Then
there is an open cover Ui of Q and G-biholomorphisms Φi : XUi → YUi inducing IdUi . We say
that X and Y are locally G-biholomorphic over Q. Set Φij = Φ−1
i Φj . Then the Φij ∈ A(Ui ∩ Uj )
are a 1-cocycle, i.e., an element of Z 1 (Q, A) (we repress explicit mention of the open cover).
Conversely, given Ψij ∈ Z 1 (Q, A) (for the same open cover) we can construct a corresponding
complex G-manifold Y from the disjoint union of the XUi by identifying XUj and XUi over
2010 Mathematics Subject Classification. Primary 32M05. Secondary 14L24, 14L30, 32E10, 32M17, 32Q28.
Key words and phrases. Oka principle, Stein manifold, reductive complex group, categorical quotient.
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GERALD W. SCHWARZ

Ui ∩ Uj via Ψij . By [KLS, Theorem 5.11] the manifold Y is Stein, and it is obviously locally
G-biholomorphic to X over Q. Let Ψ′ij be another cocycle for {Ui } corresponding to the Stein
G-manifold Y ′ . If Y ′ is G-biholomorphic to Y (inducing IdQ ), then Ψij and Ψ′ij give the same
class in H 1 (Q, A). Thus H 1 (Q, A) is the set of G-isomorphism classes of Stein G-manifolds
Y which are locally G-biholomorphic to X over Q where the G-isomorphisms are required to
induce the identity on Q.
A fundamental question is whether or not H 1 (Q, A) contains more than topological information. For example, suppose that G acts freely on X so that X → Q is a principal G-bundle.
Then X corresponds to an element of H 1 (Q, E) where E is the sheaf of germs of holomorphic
mappings of Q to G. By Grauert’s famous Oka principle [Gra58], H 1 (Q, E) ≃ H 1 (Q, E c ) where
E c is the sheaf of germs of continuous mappings of Q to G. In other words, the set of isomorphism classes of holomorphic principal G-bundles over Q is the same as the set of isomorphism
classes of topological principal G-bundles over Q. The main point of this note is to establish a
similar Oka principle in our setting.
We define another sheaf of groups Ac on Q. For U open in Q, Ac (U) consists of “strongly
continuous” families σ = {σq } of G-automorphisms of the affine G-varieties Xq , q ∈ U. We
define the notion of strongly continuous family in §3. The sheaf A is a subsheaf of Ac .
Fix an open cover {Ui } of Q. Our main theorems are the following (the first of which is a
consequence of [KLS, Theorem 1.4]).
Theorem 1.1. Let Φij , Ψij ∈ Z 1 (Q, A) and suppose that there are ci ∈ Ac (Ui ) satisfying
′
Φij = ci Ψij c−1
j . Then there are ci ∈ A(Ui ) satisfying the same equation.
1
Theorem 1.2. Let Φij ∈ Z 1 (Q, Ac ). Then there are ci ∈ Ac (Ui ) such that ci Φij c−1
j ∈ Z (Q, A).

As a consequence we have the following Oka principle:
Corollary 1.3. The canonical map H 1 (Q, A) → H 1 (Q, Ac ) is a bijection.
Remark 1.4. Suppose that X is a smooth affine G-variety and that Z → Q is a morphism of affine
varieties. Then G acts on the fiber product Z ×Q X and we have the group AutZ,alg (Z ×Q X)G of
algebraic G-automorphisms of Z ×Q X which induce the identity on the quotient Z. A scheme
G with projection π : G → Q such that the fibers of G are groups whose structure depends
algebraically on q ∈ Q is called a group scheme over Q. (See [KS92, Ch, III] for a more precise
definition.) We say that the automorphism group scheme of X exists if there is a group scheme
G over Q together with a canonical isomorphism of Γ(Z, Z ×Q G) and AutZ,alg (Z ×Q X)G for all
Z → Q. The automorphism group scheme of X exists (and is an affine variety) if, for example,
p : X → Q is flat [KS92, Ch. III Proposition 2.2]. Assuming G exists, now consider X as a Stein
G-manifold and G as an analytic variety. Then for U open in Q, A(U) ≃ Γ(U, G) and one can
show that Ac (U) is the set of continuous sections of G over U. Thus, in this case, our theorems
reduce to the precise analogues of Grauert’s for the cohomology of G using holomorphic or
continuous sections.
For U an open subset of Q we have a topology on Ac (U) and A(U) and we define the notion of
a continuous path (or homotopy) in Ac (U) or A(U). We establish a result which is well-known
in the case of principal bundles but rather non-trivial in our situation.
Theorem 1.5. Let Φij (t) be a homotopy of elements in Z 1 (Q, Ac ), t ∈ [0, 1]. Then there are
homotopies ci (t) ∈ Ac (Ui ), t ∈ [0, 1], such that Φij (t) = ci (t)Φij (0)cj (t)−1 . Hence Φij (t) ∈
H 1 (Q, Ac ) is independent of t.
Theorem 1.6. Let Φij (t) ∈ Z 1 (Q, Ac ) be a homotopy, t ∈ [0, 1], where the Φij (0) and Φij (1) are
holomorphic. Then there is a homotopy Ψij (t) ∈ Z 1 (Q, A) with Ψij (0) = Φij (0) and Ψij (1) =
Φij (1).

AN OKA PRINCIPLE FOR STEIN G-MANIFOLDS

3

Here is an outline of this paper. In §2 we recall Luna’s slice theorem and related results. In
§3 we define the sheaf of groups Ac as well as a corresponding sheaf of Lie algebras LAc . In
§4 we show that sections of Ac sufficiently close to the identity are the exponentials of sections
of LAc . In §5 we establish our main technical result (Theorem 5.1) about homotopies in Ac .
We prove Theorem 1.1 and Theorem 1.6 as well as a preliminary version of Theorem 1.5. In
§6 we establish Theorem 1.2 and use it to prove Theorem 1.5. Finally, let X and Y be locally
G-biholomorphic over Q. We establish a theorem giving necessary and sufficient conditions for
a G-biholomorphism from XU → YU over IdU , where U ⊂ Q is Runge, to be the limit of the
restrictions to XU of G-biholomorphisms from X to Y over IdQ .
Remark 1.7. In [KLS15, KLS] we also consider G-diffeomorphisms Φ of X which induce the
identity over Q and are strict. This means that the restriction of Φ to Xq , q ∈ Q, induces an
algebraic G-automorphism of (Xq )red where “red” denotes reduced structure. One can adapt
the techniques developed here to prove the analogues of our main theorems for strong Ghomeomorphisms replaced by strict G-diffeomorphisms.
Acknowledgement. I thank F. Kutzschebauch and F. Lárusson for our collaboration on [KLS15]
and [KLS] which led to this paper.
2. Background
For details of what follows see [Lun73] and [Sno82, Section 6]. Let X be a Stein manifold with
a holomorphic action of a reductive complex Lie group G. The categorical quotient QX = X//G
of X by the action of G is the set of closed orbits in X with a reduced Stein structure that
makes the quotient map pX : X → QX the universal G-invariant holomorphic map from X to
a Stein space. The quotient QX is normal. When X is understood, we drop the subscript
X in pX and QX . If U is an open subset of Q, then p∗ induces isomorphisms of C-algebras
OX (XU )G ≃ OQ (U) and C 0 (XU )G ≃ C 0 (U). We say that a subset of X is G-saturated if it is a
union of fibers of p. If X is an affine G-variety, then Q is just the complex space corresponding
to the affine algebraic variety with coordinate ring Oalg (X)G .
Let H be a reductive subgroup of G and let B be an H-saturated neighborhood of the origin
of an H-module W . We always assume that B is Stein, in which case B//H is also Stein. Let
G×H B (or TB ) denote the quotient of G×B by the (free) H-action sending (g, w) to (gh−1, hw)
for h ∈ H, g ∈ G and w ∈ B. We denote the image of (g, w) in G ×H B by [g, w].
Let Gx be a closed orbit in X. Then the isotropy group Gx is reductive and the slice
representation at x is the action of H = Gx on W = Tx X/Tx (Gx). By the slice theorem, there
is a G-saturated neighborhood of Gx which is G-biholomorphic to TB where B is an H-saturated
neighborhood of 0 ∈ W .
3. Strongly continuous homeomorphisms and vector fields
The group G acts on O(X), f 7→ g · f , where (g · f )(x) = f (g −1x), x ∈ X, g ∈ G, f ∈ O(X).
Let Ofin (X) denote the set of holomorphic functions f such that the span of {g · f | g ∈ G} is
finite dimensional. They are called the G-finite holomorphic functions on X and obviously form
an O(Q) = O(X)G -algebra. If X is a smooth affine G-variety, then the techniques of [Sch80,
Proposition 6.8, Corollary 6.9] show that for U ⊂ Q open and Stein we have
Ofin (XU ) ≃ O(U) ⊗Oalg (Q) Oalg (X).
Let V be the direct sum of pairwise non-isomorphic non-trivial G-modules V1 , . . . , Vr . Let
O(X)V denote the elements of Ofin (X) contained in a copy of V . If H is a reductive subgroup of G and W an H-module, we similarly define Oalg (TW )V . Then for B an H-saturated
neighborhood of 0 ∈ W , Oalg (TW )V generates O(TB )V over O(B)H . By Nakayama’s Lemma,

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GERALD W. SCHWARZ

f1 , . . . , fm ∈ O(X)V restrict to minimal generators of the O(U)-module O(XU )V for some
neighborhood U of q ∈ Q if and only if the restrictions of the fi to Xq form a basis of
O(Xq )V = Oalg (Xq )V . Thus by the slice theorem, the sheaf of algebras of G-finite holomorphic
functions is locally finitely generated as an algebra over OQ .
Definition 3.1. Let U ⊂ Q be relatively compact. Then there is a V as above such that
the O(XU )Vj are finitely generated over O(U) and generate Ofin (XU ) as O(U)-algebra. Let
f1 , . . . , fn be a generating set of ⊕O(XU )Vj with each fi in some O(XU )Vj . Then we call {fi }
a standard generating set of Ofin (XU ). When U = TB //G as before, we always assume that our
standard generators are the restrictions of homogeneous elements of Oalg (TW ).
Let U ⊂ Q, V and {f1 , . . . , fn } be as above. We say that a G-equivariant
homeomorphism
P
∗
Ψ : XU → XU is strong if it lies over the identity of U and Ψ fi = j aij fj where the aij are
in C 0 (XU )G ≃ C 0 (U). We also require that the aij (q) induce a G-isomorphism of O(Xq )V for
all q ∈ U. Then Ψ induces an algebraic isomorphism Ψq : Xq → Xq for all q ∈ U. It is easy
to see that the definition does not depend on our choice of V and the generators fi . We call
(aij ) a matrix associated to Ψ. Using a partition of unity on U it is clear that Ψ is strong if
and only if it is strong in a neighborhood of every q ∈ Q. In a neighborhood of any particular
q, we may assume that the fi restrict to a basis of O(Xq )V , in which case (aij ) is invertible in
a neighborhood of q. Then Φ−1 has matrix (aij )−1 near q. Thus if Φ is strong, so is Φ−1 . Let
Ac (U) denote the group of strong G-homeomorphisms of XU for U open in Q. Then Ac is a
sheaf of groups on Q.
We say that a vector field D on XU is formally holomorphic if it annihilates the antiholomorphic functions on XU . Let D be a continuous formally holomorphic vector field on XU , Ginvariant, annihilating O(XU )G . We say that D is strongly continuous (and write D ∈ LAc (U))
if for any q ∈ U there is a neighborhood
U ′ of q in U and a standard generating set f1 , . . . , fn
P
for Ofin (XU ′ ) such that D(fi ) = dij fj where the dij are in C 0 (U ′ ). We say that D has matrix
(dij ) over U ′ . The matrix is usually not unique. Clearly our definition of LAc (U) is independent
of the choices made. We denote the corresponding sheaf by LAc .
Remark 3.2. Let D ∈ LAc (U) and q ∈ Q. Then D is tangent to F = Xq and acts algebraically
on Oalg (F ), hence lies in the space of G-invariant derivations Deralg (F )G of Oalg (F ). Since
Deralg (F )G is the Lie algebra of the algebraic group Aut(F )G , the restriction of D to F can be
integrated for all time. It follows that D is a complete vector field.
Let U be open in Q, let ǫ > 0 and let K be a compact subset of U. Let f = {f1 , . . . , fn } be
a standard generating set of Ofin (XU ′ ) where U ′ is a neighborhood of K. Define
ΩK,ǫ,f = {Φ ∈ Ac (U) : ||(aij ) − I||K < ǫ}
where (aij ) is some matrix associated to Φ. Here ||(aij ) − I||K denotes the supremum of the
′
matrix norm of (aij ) − I over K. Let f ′ = {f1′ , . . . , fm
} be another standard generating set
defined on a neighborhood of K in U.
Lemma 3.3. Let ǫ′ > 0. Then there is an ǫ > 0 such that ΩK,ǫ,f ⊂ ΩK,ǫ′ ,f ′ .
Proof. We may assume that the fi and fj′ are standard generating sets of Ofin (U). There are
polynomials hi with coefficients in O(U) such that fi′ = hi (f1 , . . . , fn ), 1 ≤ i ≤ m. We may
assume that {f1′ , . . . , fs′ } are the fi′ corresponding to an irreducible G-module Vt . Let Φ ∈ ΩK,ǫ,f
with corresponding matrix (auv ) such that ||(buv )||K < ǫ where (buv ) = (auv ) − I. Let ri be the
degree of hi . Then for 1 ≤ i ≤ s we have
n
X
∗ ′
′
∗
∗
bkl pkl Mkl (f1 , . . . , fn )
(Φ fi ) − fi = hi (Φ f1 , . . . , Φ fn ) − hi (f1 , . . . , fn ) =
k,l=1

AN OKA PRINCIPLE FOR STEIN G-MANIFOLDS

5

where the pkl are polynomials in the auv of degree at most ri − 1 and the Mk,l are polynomials
in the fj with coefficients in O(U) which are independent of the auv and buv . Since Φ∗ fi′ is
a covariant corresponding to Vt , we can project the Mkl to O(XU )Vt in which case we get
P
s
′
j=1 Njkl fj where the Njkl are in O(U) and independent of the auv and buv . Hence
(Φ∗ fi′ )

−

fi′

=

s X
n
X

bkl Njkl pkl fj′ .

j=1 k,l=1

Since the Njkl pkl are bounded on K, choosing ǫ sufficiently small, we can force the terms
P
n

k,l=1 bkl Njkl pkl to be close to 0. Hence there is an ǫ > 0 such that ΩK,ǫ,f ⊂ ΩK,ǫ′ ,f ′ .
By the lemma, we get the same neighborhoods of the identity in Ac (U) from any standard
generating set of Ofin (XU ′ ) where U ′ is a neighborhood of K. Thus we can talk about neighborhoods of the identity without specifying the f in question. We then have a well-defined topology
on Ac (U) where Φ is close to Φ′ if Φ′ Φ−1 is close to the identity.
Let U, K and the fi be as above. Define
X
Ω′K,ǫ,f = {D ∈ LAc (U) | D(fi ) =
dij fj and ||(dij )||K < ǫ}
where D has (continuous) matrix (dij ) defined on a neighborhood of K. As before, the Ω′K,ǫ,f
give a basis of neighborhoods of 0 and define a topology on LAc (U), independent of the choice
of f.
Proposition 3.4. Let U be open in Q and let {f1 , . . . , fn } be a standard generating set for
Ofin (U).
(1) Let D be a G-invariant
P formally holomorphic vector field on XU which annihilates O(U)
such that D(fi ) = dij fj for dij ∈ C 0 (U). Then D is continuous, i.e., D ∈ LAc (U).
(2) LAc is a sheaf of Lie algebras and a module over the sheaf of germs of continuous
functions on Q.
(3) exp : LAc (U) → Ac (U) is continuous.
(4) LAc (U) is a Fréchet space.
Proof. Let D be as in (1) and let x0 ∈ XU . There is a subset, say f1 , . . . , fr , of the fi and
holomorphic invariant functions hr+1 , . . . , hs such that the zi = fi − fi (x0P
) and zj = hj − hj (x0 )
are local holomorphic coordinates at x0 . Then, near x0 , D has the form
ai ∂/∂zi where each
′
ai = D(fi ) is continuous. Hence D is continuous giving (1). Let D, D ∈ LAc (U) with matrices
′
(dij ) and (d′ij ). Let
P (eij ) be their matrix bracket. Then [D, D ] is G-invariant, annihilates O(U)
and sends fi to
eij fj . Hence we have (2). Part (3) is clear.
The topology on LAc (U), U open in Q, is defined by countably many seminorms, hence
LAc (U) is a metric space and it is Fréchet if it is complete. Let Dk be a Cauchy sequence in
LAc (U). Let K ⊂ U be a compact neighborhood of q ∈ U. There are matrices (dkij ) of elements
P k
of C 0 (K) such that Dk (fi ) =
dij fj over K. Since {Dk } is Cauchy, we may assume that
k
l
||(dij ) − (dij )||K < 1/m for k, l > Nm , m ∈ N. Then limk→∞ dkij = dij ∈ C 0 (K) for all i, j.
It follows that the pointwise limit of the Dk exists
Pand is a formally holomorphic vector field
D annihilating the invariants such that D(fi ) =
dij fj . By (1), D is of type LAc over the
interior of K. It follows that LF(U) is complete.

4. Logarithms in Ac
Let U be an open subset of Q isomorphic to TB = G ×H B where H is a reductive subgroup
of G and B is an H-saturated neighborhood of the origin in an H-module W . Let f1 , . . . , fn
be a standard generating set for Ofin (XU )G consisting of the restrictions to XU of homogeneous
polynomials in Oalg (TW ). Consider polynomial relations of the fi with coefficients in O(U).

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GERALD W. SCHWARZ

These are generated by the relations with coefficients in Oalg (TW )G . Let h1 , . . . , hm be generating
relations of this type. Let N be a bound for the degree of the hj . Now take the covariants which
correspond to all the irreducible G-representations occurring in the span of the monomials of
degree at most N in the fi . Let {fα } be a set
and let K ⊂ U
P of generators for these covariants
∗
0
be compact.
P Let Φ ∈ Ac (U). Then0 Φ fα = cα,β fβ where the cα,β ∈ C (U). We also have that
Φ∗ fi =
aij fj where the aij ∈ C (U). We fix a neighborhood Ω of the identity in Ac (U) such
that Φ ∈ Ω implies that ||(cα,β ) − I||K < 1/3 and that ||(aij ) − I||K < 1/2. For Φ ∈ Ω let Λ′
denote Id −Φ∗ . Then the formal power series S(Λ′ ) for log Φ∗ is −Λ′ −(1/2)(Λ′ )2 −1/3(Λ′ )3 −· · · .
Now we restrict to a fiber F = Xq , q ∈ K. Let M denote the span of the covariants fα
restricted to F . Then M is finite dimensional and we give it the usual euclidian topology. Let
Λ denote the restriction of Λ′ to M.
Lemma 4.1. Let m ∈ M. Then the series S(Λ)(m) converges in M.
P
P
Proof. We have m =
aα fα |F where the aα ∈ C. Then Λ(m) = α,β (δα,β − cα,β (q))aβ fβ |F .
P
Let C denote (cα,β (q)). Then ||I − C|| < 1/3. By induction, Λk acts on α aα fα |F via the
matrix (I − C)k , where ||(I − C)k || < (1/3)k . Let
C′ = −
Then S(Λ)(m) converges to

P

∞
X

(I − C)k .

k=1
α,β

′
Cα,β
aβ fβ |F ∈ M.



For f ∈ M, define D(f ) to be the limit of S(Λ)(f ). Then D is a G-equivariant linear
endomorphism of M.
Proposition 4.2. Suppose that m1 , m2 and m1 m2 are in M. Then
D(m1 m2 ) = D(m1 )m2 + m1 D(m2 ).
Proof. By [Pra86, Proof of Theorem 4]
k

Λ (m1 m2 ) =

2k X
ℓ
X

ckℓn Λn (m1 )Λℓ−n (m2 )

l=0 n=0

where ckℓn is the coefficient of xn y ℓ−n in (x+ y −xy)k . We
know that Λk is given by the action of
P
the matrix (I − C)k where ||I − C|| < 1/3. The series
1/k(x + y − xy)k converges absolutely
when x and y have absolute value at most 1/3. Thus we may make a change in the order of
summation:
D(m1 m2 ) =

∞ X
2k X
ℓ
X
1
k=1 l=0 n=0

k

ckℓn Λn (m1 )Λℓ−n (m2 ) =

∞ X
ℓ X
∞
X
1
ℓ=0 n=0 k=1

k

ckℓn Λn (m1 )Λℓ−n (m2 ).

P
By [Pra86, Proof of Theorem 4] the (actually finite) sum ∞
k=1 (1/k)ckℓn equals 0 unless we have
ℓ > 0 and (n = 0 or n = ℓ), in which case the value is 1/ℓ. Hence
D(m1 m2 ) = −

∞
X
1
ℓ=1

ℓ

(Λℓ (m1 )m2 + m1 Λℓ (m2 )) = D(m1 )m2 + m1 D(m2 ).


Proposition 4.3. Let D : M → M be as above. Then D extends to a G-equivariant derivation
of Oalg (F ).

AN OKA PRINCIPLE FOR STEIN G-MANIFOLDS

7

Proof. Let R = C[z1 , . . . , zn ]. We have a surjective morphism ρ : R → Oalg (F ) sending zi to
fi |F , i = 1, . . . , n. The kernel J of ρ is generated by polynomials of degree at most
(the hj
P N
′
considered as elements of R). Let E denote the derivation of R which sends zi to
dij zj where
(d′ij ) is the logarithm of (aij (q)). Recall that ||(aij ) − I||K < 1/2. By construction, E on the
span of the zi is the pull-back of D on the span of the fi |F . By Proposition 4.2, E restricted
to polynomials of degree at most N is the pull-back of D restricted to polynomials of degree
at most N in the fi |F . Hence E preserves the span of the elements of degree at most N in J.
Since these elements generate J, we see that E preserves J. Hence E induces a derivation of
R/J, i.e., D extends to a G-invariant derivation of Oalg (F ).

Corollary 4.4. Let Φ ∈ Ω and let U ′ denote the interior of our compact subset K ⊂ U. There is
a D ∈ LAc (U ′ ) such that exp(D) = Φ|XU ′ . The mapping Ω ∋ Φ → D ∈ LAc (U ′ ) is continuous.
Proof. For q ∈ U ′ , let Dq be the G-equivariant derivation of Oalg (Xq ) constructed
P above. Let
dij fj where
D be the vector field on XU ′ whose value on Xq is Dq , q ∈ U ′ . Then D(fi ) =
′
(dij ) = log(aij ). By Proposition 3.4, D ∈ LAc (U ). By construction, exp(Dq ) = Φq for all
q ∈ U ′ . Hence exp D = Φ|XU ′ . The continuity of Φ 7→ D is clear since (dij ) = log(aij ).

Definition 4.5. Let U ⊂ Q be open and let f1 , . . . , fn be a standard generating set of Ofin (XU ).
′
Let U ′ ⊂ U be open with U ⊂ U. We say that Φ ∈ Ac (U) admits a logarithm in LAc (U ′ ) if
the following hold.
P
(1) Φ∗ fi = aij fj where ||(aij ) − I||U ′ < 1/2. P
(2) There is a D ∈ LAc (U ′ ) such that D(fi ) = dij fj on XU ′ where (dij ) = log(aij ).
Note that (aij ) is not unique. The condition is that some (aij ) corresponding to Φ satisfies (1)
and (2).
Remarks 4.6. The formal series log Φ∗ , when applied to any fi , converges to D(fi ). Hence D is
independent of the choice of (aij ). Properties (1) and (2) imply that exp D = Φ over U ′ . Note
that ||(dij )||U ′ < log 2 and (dij ) is the unique matrix satisfying this property whose exponential
is (aij ).
Corollary 4.4 and its proof imply the following result.
Theorem 4.7. Let K ⊂ U ⊂ Q where K is compact and U is open. Then there is a neighborhood Ω of the identity in Ac (U) and a neighborhood U ′ of K in U such that every Φ ∈ Ω admits
a logarithm D = log Φ in LAc (U ′ ). The mapping Φ → log Φ is continuous.
Corollary 4.8. Let Φn be a Cauchy sequence in Ac (U). Then Φn → Φ ∈ Ac (U).
Proof. Since this is a local question, we can assume that we have a standard generating set {fi }
for Ofin (U). Let q ∈ U and let U ′ be a relatively compact neighborhood of q in U. Then there is
a neighborhood Ω of the identity in Ac (U) such that any Ψ ∈ Ω admits a logarithm in LAc (U ′ ).
Let Ω0 be a smaller neighborhood of the identity with Ω0 ⊂ Ω. There is an N ∈ N such that
−1
′
n ≥ N implies that Φ−1
N Φn ∈ Ω0 , hence log(ΦN Φn ) = Dn ∈ LAc (U ), and Dn converges to an
element D ∈ LAc (U ′ ) by Proposition 3.4. Set Φ = ΦN exp D ∈ A(U ′ ). Since exp Dn = Φ−1
N Φn
over U ′ we have Φn → Φ in Ac (U ′ ).

5. Homotopies in H 1 (Q, Ac )
We establish our main technical result concerning homotopies in H 1 (Q, Ac ). We give proofs
of Theorems 1.1 and 1.6 and a special case of Theorem 1.5.
Let Φ(t) ∈ Ac (U), t ∈ C, where C is a topological space. We say that Φ(t) is continuous
if relative to a standard generating set, Φ(t) has corresponding matrices (aij (t, q)) where each
aij is continuous in t and q ∈ U. (It is probably false that every continuous map C → Ac (U)

8

GERALD W. SCHWARZ

is continuous in our sense.) Let Ac (U) denote the set of all continuous paths Φ(t) ∈ Ac (U),
t ∈ [0, 1], starting at the identity. We have a topology on Ac (U) as in §3 and Ac (U) is a
topological group. When we talk of homotopies in Ac (U) we mean that the corresponding
families with parameter space [0, 1]2 are continuous as above. We define continuous families
of elements of LAc (U) similarly. One defines A(U) similarly to Ac (U) where, of course, the
relevant aij (t, q) are required to be holomorphic in q and continuous in t.
Here is our main technical result about Ac .
Theorem 5.1.
(1) The topological group Ac (Q) is pathwise connected.
(2) If U ⊂ Q is open, then Ac (Q) is dense in Ac (U).
(3) H 1 (Q, Ac ) = 0.
Proof. Let Φ(t) be an element of Ac (Q). Since {0} is a deformation retract of [0, 1], there is a
homotopy Φ(s, t) with Φ(0, t) = Φ(t) and Φ(1, t) the identity automorphism. Hence we have
(1). For (2), let Φ ∈ Ac (U). Let K be a compact subset of U and U ′ a relatively compact
neighborhood of K in U. It follows from Theorem 4.7 that there are 0 = t0 < t1 < · · · < tm = 1
and continuous families Dj (s) in LAc (U ′ ) for s ∈ [0, tj+1 − tj ] such that, over U ′ , Φ(s + tj ) =
Φ(tj ) exp(Dj (s)), s ∈ [0, tj+1 − tj ], j = 0, . . . , m − 1. Multiplying by a cutoff function, we can
assume that the Dj (s) are in LAc (Q). Then our formula gives an element of Ac (Q) which
restricts to Φ on a neighborhood of K and we have (2).
Let K ⊂ Q be compact which is of the form K ′ ∪ K ′′ where K ′ and K ′′ are compact. Let
U = U ′ ∪ U ′′ be a neighborhood of K where K ′ ⊂ U ′ , K ′′ ⊂ U ′′ . Let Φ(t) be in Z 1 (U, Ac ) for
the open covering {U ′ , U ′′ } of U. Then Φ(t) is just an element in Ac (U ′ ∩ U ′′ ). By (2) we can
′
write Φ = Ψ1 Ψ−1
2 where Ψ1 is defined over Q (hence over U ) and Ψ2 is close to the identity
over K ′ ∩ K ′′ . Then Ψ2 (t) = exp D(t) where D(t) ∈ LAc (U ′ ∩ U ′′ ) is a continuous family and
D(0) = 0. Using a cutoff function again, we can find D0 (t) ∈ LAc (Q) which equals D(t) in
−1
a neighborhood of K ′ ∩ K ′′ and vanishes when t = 0. We have Φ = Ψ1 Ψ−1
2 where Ψ2 is the
exponential of D0 (t). Thus the cohomology class of Φ becomes trivial if we replace U ′ and
U ′′ by slightly smaller neighborhoods of K ′ and K ′′ . Let H 1 (K, Ac ) denote the direct limit of
H 1 (U, Ac ) for U a neighborhood of K. As in [Car58, §5], our result above shows that
S there
is a sequence of compact sets K1 ⊂ V2 ⊂ K2 . . . with Vn the interior of Kn , Q = Kn and
H 1 (Kn , Ac ) = 0 for all n.
Let {Ui } be an open cover of Q and Φij ∈ Ac (Ui ∩ Uj ) a cocycle. There are cni ∈ Ac (Ui ∩ Vn )
such that Φij = (cni )−1 cnj on Ui ∩ Uj ∩ Vn . Thus cn+1
(cni )−1 = cn+1
(cnj )−1 on Ui ∩ Uj ∩ Vn . The
i
j
cn+1
(cni )−1 define a section d ∈ Ac (Vn ). By (2) there is a section d′ of Ac (Q) which is arbitrarily
i
close to d on Kn−1 . Replace each cn+1
by (d′ )−1 cn+1
. Then cn+1
is very close to cni on Kn−1
i
i
i
n
and we can arrange that the limit as n → ∞ of the ci converges on every compact subset to
ci ∈ Ac (Ui ) such that Φij = c−1
i cj . We have used Corollary 4.8. This completes the proof of
(3).

Note that (3) says that for any homotopy of a cocycle Φij (t) starting at the identity there
are ci (t) ∈ Ac (Ui ) such that Φij (t) = ci (t)−1 cj (t) for all t ∈ [0, 1]. Hence Φij (t) is the trivial
element in H 1 (Q, Ac ) for all t. We now use a trick to show a similar result if we only assume
that Φij (0) ∈ Z 1 (Q, A).
Let Ψij ∈ Z 1 (Q, A) for some open cover {Ui } of Q. By [KLS, Theorem 5.11], there is a Stein
G-manifold Y with quotient Q corresponding to the Ψij . Let Xi = XUi and Yi = YUi . Then
there are G-biholomorphisms Ψi : Xi → Yi over the identity of Ui such that Ψ−1
i Ψj = Ψij .
Here is an analogue of the twist construction in Galois cohomology. We leave the proof to
the reader.

AN OKA PRINCIPLE FOR STEIN G-MANIFOLDS

9

Lemma 5.2. Let Ψij ∈ Z 1 (Q, A) and Φij ∈ Z 1 (Q, Ac ) be cocycles for the open cover {Ui } of
Q. Let Y and Ψi : Xi → Yi be as above. The mapping Φij 7→ Ψi Φij Ψ−1
j induces an isomorphism
1
1
Y
of H (Q, Ac ) and H (Q, Ac ) which sends the class Ψij to the trivial class of H 1 (Q, AYc ).
Corollary 5.3. Let Φij (t) be a homotopy of cocycles with values in Ac (Ui ∩ Uj ) where {Ui } is
an open cover of Q. Suppose that Φij (0) is holomorphic. Then there are ci ∈ Ac (Ui ) such that
Φij (t) = ci (t)−1 Φij (0)cj (t) for all t.
Proof. By Lemma 5.2 we may reduce to the case that Φij (0) is the identity, so we can apply
Theorem 5.1.

Let X, Y and the Ψi be as above. We say that a G-homeomorphism Φ : X → Y is strong if
◦ Φ : Xi → Xi is strong for all i, i.e., in Ac (Ui ). It is easy to see that this does not depend
upon the particular choice of the Ψi . Similarly one can define what it means for a family Φ(t) of
strong G-homeomorphisms to be continuous, t ∈ [0, 1]. Then we have the following nice result
[KLS, Theorem 1.4].

Ψ−1
i

Theorem 5.4. Let Φ : X → Y be strongly continuous. Then there is a continuous family Φ(t)
of strong G-homeomorphisms from X to Y with Φ(0) = Φ and Φ(1) holomorphiic.
Proof of Theorem 1.1. We have Φij , Ψij ∈ Z 1 (Q, A) and ci ∈ Ac (Ui ) satisfying Φij = ci Ψij c−1
j .
Using Lemma 5.2 we may assume that Φij is the trivial class. Then the ci are the same thing as
a strong G-homeomorphism Θ : X → Y where Y is the Stein G-manifold corresponding to the
Ψij (after our twisting). By Theorem 5.4 not only are there di ∈ A(Ui ) such that Ψij = di d−1
j ,
but the di are ei (1) where ei (t) is a path in Ac (Ui ) starting at ci and ending at di . The di
correspond to a G-biholomorphism of X and Y over Q.

We now prepare to prove Theorem 1.6.
Lemma 5.5. Let Φ ∈ Ac (Q) such that Φ(1) is holomorphic. Then Φ is homotopic to Φ′ ∈ A(Q)
where Φ′ (1) = Φ(1).
Proof. We have to make use of a sheaf of groups F on Q which is a subsheaf of the sheaf of
G-diffeomorphisms of X which induce the identity on Q and are algebraic isomorphisms on the
fibers of p. See [KLS, Ch. 6]. We give F (U) the usual C ∞ -topology. Let F(U) denote the
sheaf of homotopies Ψ(t) of elements of F (U), t ∈ [0, 1], where Ψ(0) is the identity and Ψ(1)
is holomorphic. Then [KLS, Theorem 10.1] tells us that F(Q) is pathwise connected. Hence
for Φ ∈ F(Q) there is a homotopy Ψ(s) ∈ F(Q) such that Ψ(0) = Φ and Ψ(1) is the identity.
Then Ψ(s) evaluated at t = 1 is a homotopy from Φ(1) to the identity in A(Q), establishing
the lemma when Φ ∈ F(Q).
We now use a standard trick. Let ∆ denote a disk in C containing [0, 1] with trivial G-action.
Then ∆ × X has quotient ∆ × Q with the obvious quotient mapping. Let ρ : ∆ → [0, 1] be
continuous such that ρ sends a neighborhood of 0 to 0 and a neighborhood of 1 to 1. For
(z, x) ∈ ∆ × X, define Ψ(z, x) = (z, Φ(ρ(z), x)). Then Ψ ∈ Ãc (∆ × Q) where Ãc = Ac∆×X .
Moreover, Ψ is the identity on the inverse image of a neighborhood of {0}×Q and is holomorphic
on the inverse image of a neighborhood of {1} × Q. By [KLS, Theorem 8.7] we can find a
homotopy Ψ(s) which starts at Ψ and ends up in F (∆ × Q). Moreover, the proof shows that we
can assume that the elements of the homotopy are unchanged over a neighborhood of {0, 1}×Q.
Restricting Ψ(1) to [0, 1] ⊂ ∆ we have an element in F(Q) which at time 1 is still Φ(1). Then
we can apply the argument above.

Proof of Theorem 1.6. By Lemma 5.2 we may assume that Φij (0) is the identity cocycle. Since
H 1 (Q, Ac ) is trivial, there are ci ∈ Ac (Ui ) such that Φij (t) = ci (t)cj (t)−1 for t ∈ [0, 1]. Now
the ci (1) define a strongly continuous G-homeomorphism from X to the Stein G-manifold Y

10

GERALD W. SCHWARZ

corresponding to Φij (1). By Theorem 5.4 there is a homotopy ci (t), 1 ≤ t ≤ 2, such that the
ci (2) are holomorphic and split Φij (1). Reparameterizing, we may reduce to the case that the
original ci (t) are holomorphic for t = 1. Now apply Lemma 5.5 to Ψij (t) = ci (t)cj (t)−1 .

6. H 1 (Q, A) → H 1 (Q, Ac ) is a bijection
We give a proof Theorem 1.2. We are given an open cover {Ui } of Q and Φij ∈ Z 1 (Q, Ac ).
We want to find ci ∈ Ac (Ui ) such that c−1
i Φij cj is holomorphic. We may assume that the Ui are
relatively compact, locally finite and Runge. We say that an open set U ⊂ Q is good if there
are sections ci ∈ Ac (Ui ∩ U) such that c−1
i Φij cj is holomorphic on Uij ∩ U for all i and j where
Uij denotes Ui ∩ Uj . This says that {Φij } is cohomologous to a holomorphic cocycle on U. The
goal is to show that Q is good. It is obvious that small open subsets of Q are good.
Lemma 6.1. Suppose that Q = Q′ ∪ Q′′ where Q′ and Q′′ are good and Q′ ∩ Q′′ is Runge in Q.
Then Q is good.
Proof. By hypothesis, we have c′i ∈ Ac (Q′ ∩ Ui ) and c′′i ∈ Ac (Q′′ ∩ Ui ) such that
Ψ′ij = (c′i )−1 Φij c′j , and Ψ′′ij = (c′′i )−1 Φij c′′j are holomorphic.
Then on Uij ∩ Q′ ∩ Q′′ we have
′
′ −1 ′′
Ψ′′ij = h−1
i Ψij hj where hi = (ci ) ci .

The Ψ′ij are a holomorphic cocycle for the covering Ui ∩ Q′ of Q′ , hence they correspond to a
Stein G-manifold X ′ with quotient Q′ . Similarly the Ψ′′ij give us X ′′ , and X ′ and X ′′ are locally
G-biholomorphic to X over Q′ and Q′′ , respectively. The hi give us a strong G-homeomorphism
h : X ′ → X ′′ , everything being taken over Q′ ∩ Q′′ . By Theorem 5.4 there is a homotopy h(t, x)
with h(0, x) = h(x) and h(1, x) holomorphic. Let k(x) denote h(1, x). Then k corresponds to a
family ki homotopic to the family hi .
Now just consider the space Ui covered by the two open sets Ui ∩ Q′ and Ui ∩ Q′′ . Then
hi and ki are defined on the intersection of the two open sets and are homotopic where hi is
cohomologous to the trivial cocycle since hi = (c′i )−1 c′′i . By Corollary 5.3 and Theorem 1.1 the
cohomology class represented by ki (x) is holomorphically trivial. Hence there are holomorphic
sections h′i and h′′i such that ki = (h′i )−1 h′′i on Ui ∩ Q′ ∩ Q′′ . Then h′i Ψ′ij (h′j )−1 = h′′i Ψ′′ij (h′′j )−1 on
Uij ∩ Q′ ∩ Q′′ . We construct a holomorphic cocycle Ψij on Uij by Ψij = h′i Ψ′ij (h′j )−1 on Uij ∩ Q′
and h′′i Ψ′′ij (h′′j )−1 on Uij ∩ Q′′ .
Using Lemma 5.2 we may reduce to the case that Ψij is the trivial cocycle. As in the beginning
of the proof there are c′i ∈ Ac (Q′ ∩ Ui ) and c′′i ∈ Ac (Q′′ ∩ Ui ) such that
Φij |X ′ = c′i (c′j )−1 and Φij |X ′′ = c′′i (c′′j )−1
where X ′ = XQ′ and X ′′ = XQ′′ . Let hi = (c′i )−1 c′′i . Then hi = hj on Uij ∩ Q′ ∩ Q′′ , hence we
have a section h ∈ Ac (Q′ ∩ Q′′ ), and this section gives the same cohomology class as Φij (use the
open cover {Q′ ∩ Ui , Q′′ ∩ Ui }). By Theorem 5.4, h is homotopic to an element h̃ ∈ A(Q′ ∩ Q′′ ),
and this holomorphic section gives the same cohomology class by Corollary 5.3. Since going
to a refinement of an open cover is injective on H 1 , we see that our original Φij differs from a
holomorphic cocycle by a coboundary. Thus Q is good.

Proof of Theorem 1.2. Using Lemma 6.1 as in [Car58, §5] we can show that there is a cover of Q
by compact subsets Kn such that K1 ⊂ V2 ⊂ K2 . . . where Vj is the interior of Kj and such that
a neighborhood of every Kn is good. We can assume that Ui ∩ Vn 6= ∅ implies that Ui ⊂ Vn+1 .
This is possible by replacing {Kn } by a subsequence. For each n we choose cni ∈ Ac (Ui ∩ Vn )
such that
(cni )−1 Φij cnj = Ψnij is holomorphic on Uij ∩ Vn .

AN OKA PRINCIPLE FOR STEIN G-MANIFOLDS

11

Then
n
Ψnij = (dni )−1 Ψn+1
ij dj on Uij ∩ Vn

where dni = (cn+1
)−1 cni gives a strongly continuous map from the Stein G-manifold Yn over Vn
i
obtained using the Ψnij to the Stein G-manifold Yn+1 obtained using the Ψn+1
ij . We know that
n
the map is homotopic to a holomorphic one. Hence there are homotopies di (t) on Ui ∩ Vn such
that
n
(1) Ψnij = (dni (t))−1 Ψn+1
ij dj (t) on Ui ∩ Uj ∩ Vn , for all t.
(2) dni (0) = (cn+1
)−1 cni .
i
(3) The dni (1) give a G-equivariant biholomorphic map from Yn to Yn+1 over IdVn .
Without changing the cni we may replace the cn+1
by sections c̃n+1
such that
i
i
n+1
n+1
n+1
e
(4) Ψ
= (c̃ )−1 Φij c̃
is holomorphic on Ui ∩ Uj ∩ Vn+1 .
ij
n+1
c̃i

i

j

cni

(5)
= on Ui ∩ Vn−2 .
It suffices to set c̃n+1
= cn+1
if Ui ∩ Vn−1 = ∅ and if not, then Ui ⊂ Vn , and we can set
i
i
c̃n+1
= cn+1
· dni (λ(x)),
i
i

where λ : Vn → [0, 1] is continuous, 0 for x ∈ Vn−2 and 1 for x 6∈ Vn−1 . Then one has (4) and
(5). Thus we can arrange that cn+1
= cni in Ui ∩ Vn−2 , hence we obviously have convergence of
i
n
the ci to a continuous section ci such that (ci )−1 Φij cj is holomorphic.

We now have Theorems 1.1 and 1.2 which imply Corollary 1.3, i.e., that H 1 (Q, A) →
H 1 (Q, Ac ) is a bijection.
Proof of Theorem 1.5. This is immediate from Corollary 5.3 and Theorem 1.2



We end with the analogue of an approximation theorem of Grauert.
Theorem 6.2. Let U ⊂ Q be Runge. Suppose that Φ : XU → YU is biholomorphic and Gequivariant inducing IdU . Here X and Y are locally G-biholomorphic over Q. Then Φ can be
arbitrarily closely approximated by G-biholomorphisms of X and Y over IdQ if and only if this
is true for strong G-homeomorphisms of X and Y .
Proof. Let K ⊂ U be compact and let Φ ∈ Mor(XU , YU )G be our holomorphic G-equivariant
map inducing IdU . We can find a relatively compact open subset U ′ of U which contains K
and is Runge in Q. By hypothesis, there is a strong G-homeomorphism Ψ : X → Y which is
arbitrarily close to Φ over U ′ . Then Ψ−1 Φ = exp D where D ∈ LAc (U ′ ), hence Ψ′ and Φ′ are
homotopic, where Φ′ is the restriction of Φ to U ′ and similarly for Ψ′ . Now Ψ is homotopic
to a biholomorphic G-equivariant map Θ : X → Y inducing IdQ , and Ψ′ is homotopic to the
restriction Θ′ of Θ to U ′ . Then (Φ′ )−1 Θ′ is holomorphic and homotopic to the identity section
over U ′ . Since the end points of the homotopy are holomorphic, by Theorem 1.6 we can find
a homotopy all of whose elements are holomorphic. By [KLS, Theorem 10.1] there is a section
∆ ∈ A(Q) which is arbitrarily close to (Φ′ )−1 Θ′ on U ′ . Then Θ∆−1 , restricted to U ′ , is arbitrarily
close to Φ′ , hence this is true over K. This establishes the theorem.

References
[Car58] Henri Cartan, Espaces fibrés analytiques, Symposium internacional de topologı́a algebraica, Universidad
Nacional Autónoma de México and UNESCO, Mexico City, 1958, pp. 97–121.
[Gra58] Hans Grauert, Analytische Faserungen über holomorph-vollständigen Räumen, Math. Ann. 135 (1958),
263–273.
[KLS] Frank Kutzschebauch, Finnur Lárusson, and Gerald W. Schwarz, Homotopy principles for equivariant
isomorphisms, preprint, arXiv:1503.00797.
[KLS15] Frank Kutzschebauch, Finnur Lárusson, and Gerald W. Schwarz, An Oka principle for equivariant
isomorphisms, J. reine angew. Math. 706 (2015), 193–214.

12

[KS92]
[Lun73]
[Pra86]
[Sch80]
[Sno82]

GERALD W. SCHWARZ

Hanspeter Kraft and Gerald W. Schwarz, Reductive group actions with one-dimensional quotient, Inst.
Hautes Études Sci. Publ. Math. (1992), no. 76, 1–97.
Domingo Luna, Slices étales, Sur les groupes algébriques, Soc. Math. France, Paris, 1973, pp. 81–105.
Bull. Soc. Math. France, Paris, Mémoire 33.
C. Praagman, Iterations and logarithms of formal automorphisms, Aequationes Math. 30 (1986), no. 23, 151–160.
Gerald W. Schwarz, Lifting smooth homotopies of orbit spaces, Inst. Hautes Études Sci. Publ. Math.
(1980), no. 51, 37–135.
Dennis M. Snow, Reductive group actions on Stein spaces, Math. Ann. 259 (1982), no. 1, 79–97.

Gerald W. Schwarz, Department of Mathematics, Brandeis University, Waltham MA 024549110, USA
E-mail address: schwarz@brandeis.edu

