TAME CIRCLE ACTIONS
SUSAN TOLMAN AND JORDAN WATTS
1. Introduction
Abstract. In this paper, we consider Sjamaar’s holomorphic slice theorem [13], the birational equivalence theorem of Guillemin and Sternberg [7], and a number of important
standard constructions that work for Hamiltonian circle actions in both the symplectic category and the Kähler category: reduction, cutting, and blow-up. In each case, we show
that the theory extends to Hamiltonian circle actions on complex manifolds with tamed
symplectic forms. (At least, it does if the fixed points are isolated.)
Our main motivation for this paper is that the first author needs the machinery that we
develop here to construct a non-Hamiltonian symplectic circle action on a closed, connected
six-dimensional symplectic manifold with exactly 32 fixed points; this answers an open
question in symplectic geometry.
In this paper, we consider Sjamaar’s holomorphic slice theorem [13], the birational equivalence theorem of Guillemin and Sternberg [7], and a number of important standard constructions that work for Hamiltonian circle actions in both the symplectic category and the
Kähler category: reduction, cutting, and blow-up. In each case, we show that the theory
extends to Hamiltonian circle actions on complex manifolds with tamed symplectic forms.
(At least, it does if the fixed points are isolated.)
Our main motivation for this paper is the following question, which appears in McDuff
and Salamon [12], and is often referred to as the “McDuff conjecture”: Does there exist a
non-Hamiltonian symplectic circle action with isolated fixed points on a closed, connected
symplectic manifold? The first author needs the machinery that we develop here to answer
this question by constructing a non-Hamiltonian symplectic circle action on a closed, connected six-dimensional symplectic manifold with exactly 32 isolated fixed points in [14]. The
results in §7 (and also Proposition 3.1) play a key role in “adding” the isolated fixed points
in that paper.
Because of this motivation, we focus on the case that the fixed points are isolated, but
need to work with a slight generalisation of tamed forms. To explain concretely, we introduce
some notation. Let C× act holomorphically on a complex manifold (M, J). Denote by ξM
1
the vector field on M induced by the restricted action of S1 ⊂ C× . Let M S be the set of
1
points fixed by this action, and let Ωk (M )S be the set of S1 -invariant k-forms. Consider a
1
symplectic form ω ∈ Ω2 (M )S with a moment map Ψ : M → R. Recall that J tames ω if
Date: August 21, 2015.
Susan Tolman is partially supported by National Science Foundation Grant DMS-1206365.
Jordan Watts thanks the University of Illinois at Urbana-Champaign for their support.
1
ω(v, J(v)) > 0 for all non-zero tangent vectors v. In this paper, we will usually work with
1
the weaker condition that ω(ξM , J(ξM )) > 0 on M r M S .
Since ω(·, J(·)) is no longer a metric, some of the proofs that work for Kähler manifolds
do not work for this larger class without significant modification. However, the following
important fact remains true; we will use it repeatedly throught this paper. (Note that, in
the Kähler case the gradient flow ∇Ψ is equal to −J(ξM ), which is the vector field induced
by the R-action given by (t, x) 7→ et · x for all t ∈ R and x ∈ M .)
Lemma 1.1. Let (M, J) be a complex manifold with a holomorphic C× -action, a symplectic
1
1
form ω ∈ Ω2 (M )S satisfying ω(ξM , J(ξM )) > 0 on M rM S , and a moment map Ψ : M → R.
1
Then the function t 7→ Ψ(et · x) is strictly increasing for all x ∈ M r M S .
1
Proof. Given x ∈ M r M S , let γx (t) := Ψ(et · x). Then for all t ∈ R,
γ̇x (t) = dΨ(−J(ξM ))
= ω(ξM , J(ξM ))
> 0.
et ·x
et ·x
2. Tame Local Normal Form
The goal of this section is to prove a C× -equivariant holomorphic Bochner Linearisation
Theorem in our tame setting. (More precisely, we develop a C× -equivariant holomorphic
local normal form for a neighbourhood of a finite number of fixed points in the same moment
fibre.) Our proof is adapted from Sjamaar’s proof of the holomorphic slice theorem for Kähler
actions [13, Section 1].
Proposition 2.1. Let (M, J) be a complex manifold with a holomorphic C× -action, a sym1
1
plectic form ω ∈ Ω2 (M )S satisfying ω(ξM , J(ξM )) > 0 on M r M S , and a moment map
1
Ψ : M → R. Given {p1 , . . . , pk } ∈ M S ∩ Ψ−1 (0), there exists a C× -invariant neighbourhood
`
of {p1 , . . . , pk } in M which is C× -equivariantly biholomorphic to a neighbourhood of kj=1 {0}
`
in kj=1 Cn , where C× acts linearly on each Cn .
To prove this, we need a holomorphic version of the Bochner Linearisation Theorem for
compact group actions on complex manifolds. While this is well-known (cf., for example,
[4]), we include the proof for completeness. Since we consider actions on orbifolds in Proposition 7.7, we will prove the orbifold version. Here, if a group G acts on a complex orbifold
(M, J), we say that the action respects J if every g ∈ G induces an automorphism of
(M, J).
Lemma 2.2. Let a compact Lie group G act on a complex orbifold (M, J); assume that
the action respects J. Given p ∈ M G , there exist a G-equivariant biholomorphism from a
G-invariant neighbourhood of 0 ∈ Tp M to a neighbourhood of p ∈ M .
e of G by Γ,
Proof. Let Γ be the (orbifold) isotropy group of p. There exists an extension G
e on an open set U
e ⊂ Cn that respects the complex structure and fixes pe ∈ U
e,
an action of G
e /Γ to M that sends [e
and a G-equivariant biholomorphism from U
p] to p. (See, for example,
2
e → TpeU
e be any holomorphic map whose differential at pe is the identity map
[11].) Let ψ : U
e . Since G
e is compact, we can average ψ to obtain a G-equivariant
e
e → TpeU
e
on TpeU
map ψ̄ : U
such that dψ̄|pe is equal to the identity, defined by
Z
ψ̄(q) :=
g∗ ψ(g −1 · q)dg
e
G
e , where dg is the Haar measure on G.
e Since the action respects the complex
for all q ∈ U
structure, the map ψ̄ is holomorphic. By the inverse function theorem, we can invert ψ̄ on
a neighbourhood of pe to construct the required biholomorphism.
Let (M1 , J1 ) and (M2 , J2 ) be complex manifolds with holomorphic C× -actions, A be an
1
S1 -invariant open subset of M1 , and ϕ : A → M2 be an
S -equivariant holomorphic map.
Then ϕ sends the vector J1 (ξM1 ) x to the vector J2 (ξM2 ) ϕ(x) for all x ∈ A. Hence, if (t− , t+ )
is the connected component of {t ∈ R | et · x ∈ A} containing 0, then
ϕ(et · x) = et · ϕ(x) for all t ∈ (t− , t+ ).
(1)
t
However, (1) need not hold for all t ∈ R such that e · x ∈ A. This motivates the following
definition.
Definition 2.3. Let (M, J) be a complex manifold with a holomorphic C× -action. Then a
subset A ⊆ M is orbitally convex (with respect to the C× -action) if it is S1 -invariant and
the set {t ∈ R | et · x ∈ A} is connected for all x ∈ A.
The following proposition, which is Proposition 1.4 in [13], is a consequence of the discussion above.
Proposition 2.4. Let (M1 , J1 ) and (M2 , J2 ) be complex manifolds with holomorphic C× actions. Assume that A ⊆ M1 is an orbitally convex open set, and ϕ : A → M2 is an
S1 -equivariant holomorphic map. Then, ϕ extends to a C× -equivariant holomorphic map
ϕ
e : C× ·A → M2 . Consequently, if ϕ(A) is open and orbitally convex in M2 and ϕ : A → ϕ(A)
is biholomorphic, then ϕ
e is biholomorphic onto the open set C× · ϕ(A).
Remark 2.5. The proof of Proposition 2.4 in [13] uses the definition stated here in Definition 2.3. Note, however, that A ⊂ M may not be orbitally convex even if the intersection
{et · x | t ∈ R} ∩ A is connected for all x ∈ A. To see this, let 1 ∈ Z act on C× ⊂ C by
z 7→ 2z, and let M := C× /Z with the natural C× action. Let B ⊂ M be the image of
S1 ⊂ C× , and let A = M r B. Then {et · x | t ∈ R} ∩ A is connected for all x ∈ M , but A is
not orbitally convex.
To complete the proof of Proposition 2.1, we show that we can choose the neighbourhoods
in Lemma 2.2 to be orbitally convex. For this, we need the following lemma.
Lemma 2.6. Let C× act linearly on Cn , and let J be the standard complex structure on
1
Cn . Let U be an S1 -invariant neighbourhood of 0 in Cn , with a symplectic form ω ∈ Ω2 (U )S
satisfying ω(ξCn , J(ξCn )) > 0 on U r {0}, and a moment map Ψ : U → R sending 0 to 0.
There exist δ > 0 and an orbitally convex neighbourhood V ⊂ U of 0 so that the following
holds: Given z ∈ V , if (t− , t+ ) = {t ∈ R | et · z ∈ V } then
3
(1) either t+ = ∞, or there exists t ∈ (t− , t+ ) with Ψ(et · z) > δ; and
(2) either t− = −∞ or there exists t ∈ (t− , t+ ) with Ψ(et · z) < −δ.
Proof. We may assume Cn = Ck− × Ck+ × Cl , where C× acts on Ck− with negative weights
−α1− , . . . , −αk−− ∈ −N, on Ck+ with positive weights α1+ , . . . , αk++ ∈ N, and on Cl trivially.
Define continuous S1 -invariant functions N± : Ck± → R by
X
21
k±
2/α±
N± (z) =
|zj | j
,
j=1
and define N : Ck− × Ck+ × Cl → R by N (z− , z+ , w) = N− (z− )N+ (z+ ). Note that
N± (et · z± ) = e±t N± (z± ), and so
t
N (e · (z− , z+ , w)) = N (z− , z+ , w)
(2)
(3)
for all z± ∈ Ck± , w ∈ Cl , and t ∈ R.
Given ε > 0, define
Dε± = {z ∈ Ck± | N± (z) < ε} and Sε± = {z ∈ Ck± | N± (z) = ε} ( Dε± .
1
P
2 2
= |z|.
If ε < 1, then |zj | < 1 for all z ∈ Dε± and j ∈ {1, . . . , k± }, and so N± (z) ≥
j |zj |
l
Thus there exists ε > 0 and a compact, connected neighbourhood K of 0 ∈ C such that
Dε− × Dε+ × K ⊂ U .
If z ∈ Dε± , then lim et · z = 0. Moreover, et · z ∈ Dε± for all t ∈ ∓(0, ∞) by (2).
t→∓∞
Additionally, Ψ(0, 0, w) = 0 for any w ∈ K; therefore, by Lemma 1.1, Ψ(z− , 0, w) < 0 for
every nonzero z− ∈ Dε− , and Ψ(0, z+ , w) > 0 for every non-zero z+ ∈ Dε+ . Hence, since Sε±
and K are compact, there exist δ > 0 and ε0 ∈ (0, ε) such that
Ψ(Sε− × Dε+0 × K) ( (−∞, −δ) and Ψ(Dε−0 × Sε+ × K) ( (δ, ∞).
(4)
Define V := {(z− , z+ , w) ∈ Dε− × Dε+ × K | N (z− , z+ ) < εε0 } ⊂ U . Then V is an
orbitally convex neighbourhood of 0 by (2) and (3). Fix (z− , z+ , w) ∈ V . If z+ = 0, then
N (et · (z− , z+ , w)) = 0 for all t ∈ R, and so et · (z− , z+ , w) lies in V for all t ≥ 0. On the other
hand, if z+ 6= 0, then t+ := ln N+ ε(z+ ) > 0 because N+ (z+ ) < ε. A straightforward calculation
using (2) shows that et · (z− , z+ , w) ∈ Dε− × Dε+ × K for all t ∈ [0, t+ ). By (3), this implies
that et · (z− , z+ , w) ∈ V for all t ∈ [0, t+ ). Since et+ · (z− , z+ , w) ∈ Dε−0 × Sε+ × K, Claim 1
follows from (4). The proof of Claim 2 is nearly identical.
Proof of Proposition 2.1. By Lemma 2.2,`there exists
S1 -equivariant biholomorphism ϕ
` an
1
n
from an S -invariant neighbourhood U of j {0} in j C to a neighbourhood of {p1 , . . . , pk }
n
in M , where C× acts linearly on each
` C . Let δ be the positive real and V ⊂ U be the
orbitally convex neighbourhood of j {0} given in Lemma 2.6. Fix x ∈ ϕ(V ). Assume that
0
ϕ(z) = et ·x, where z ∈ V and t0 ∈ R, and let (t− , t+ ) := {t ∈ R | et ·z ∈ V }. Then, since V is
0
orbitally convex, equation (1) (alternatively, Proposition 2.4) implies that ϕ(et · z) = et+t · x
for all t ∈ (t− , t+ ). In particular, et · x ∈ ϕ(V ) for all t ∈ (t0 + t− , t0 + t+ ). Moreover, by
Lemma 2.6,
4
(1) either t+ = ∞, or there exists t ∈ (t0 + t− , t0 + t+ ) with Ψ(et · x) > δ; and
(2) either t− = −∞, or there exists t ∈ (t0 + t− , t0 + t+ ) with Ψ(et · x) < −δ.
Finally, by Lemma 1.1, the function t 7→ Ψ(et · x) is increasing. Therefore, ϕ(V ) is open and
orbitally convex. Thus, by Proposition 2.4, ϕ|V extends to a C× -equivariant biholomorphism
from C× · V to C× · ϕ(V ).
Since the function t 7→ Ψ(et ·x) is increasing, claims (1) and (2) above imply that Ψ(et ·x) 6∈
(−δ, δ) if t 6∈ (t0 + t− , t0 + t+ ). Since also et · x ∈ ϕ(V ) for all t ∈ (t0 + t− , t0 + t+ ), the above
proof has the following corollary.
Corollary 2.7. Let (M, J) be a complex manifold with a holomorphic C× -action, a sym1
1
plectic form ω ∈ Ω2 (M )S satisfying ω(ξM , J(ξM )) > 0 on M r M S , and a moment map
1
Ψ : M → R. Given p ∈ M S ∩ Ψ−1 (0) and an S1 -invariant neighbourhood U of p, there exists
δ > 0 and an S1 -invariant open neighbourhood V ⊆ U of p such that C× ·V ∩Ψ−1 (−δ, δ) ⊂ V .
3. Tame Reduction
It is well known that the reduced spaces of Kähler manifolds naturally inherit Kähler
structures [6, 9, 8]. The goal of this section is to prove a generalisation of this fact: the
reduced spaces of complex manifolds with tamed symplectic forms naturally inherit complex
structures that tame the reduced symplectic form.
Proposition 3.1. Let (M, J) be a complex manifold with a holomorphic C× -action, a sym1
1
plectic form ω ∈ Ω2 (M )S satisfying ω(ξM , J(ξM )) > 0 on M r M S , and a moment map
Ψ : M → R. Given a regular value a of Ψ, let Ua := C× · Ψ−1 (a). Then the following hold.
(1) The quotient Ua /C× is naturally a complex orbifold.
(2) There is a complex structure Ja on the reduced space M //a S1 so that the inclusion
Ψ−1 (a) ,→ Ua induces a biholomorphism M //a S1 → Ua /C× .
(3) For all x ∈ Ψ−1 (a), the natural map
{X ∈ Tx M | ω(ξM , X) = ω(ξM , J(X)) = 0} → T[x] (M //a S1 )
is an isomorphism of complex and symplectic vector spaces. Consequently, if J tames
ω at x then Ja tames the reduced symplectic form at [x].
Remark 3.2. In the situation of Proposition 3.1, if (M, J, ω) is Kähler then the reduced
space M //a S1 is Kähler by Claim (3). Thus, the standard theorem for Kähler manifolds is
a special case. It is easy to see that analogous claims hold for cutting (Proposition 5.1) and
blow-ups (Proposition 6.1 and 7.7), though in the last case we rely on the final claims in
Lemmas 7.5 and 7.6.
To prove this, we will need the following important technical lemma, which only depends
on Lemma 1.1 and the fact that Ψ is equivariant.
Lemma 3.3. Let (M, J) be a complex manifold with a holomorphic C× -action, a symplectic
1
1
form ω ∈ Ω2 (M )S satisfying ω(ξM , J(ξM )) > 0 on M rM S , and a moment map Ψ : M → R.
Define
1
U := {(x, s) ∈ M r M S × R | s ∈ Ψ(C× · x)}.
5
Then for all (x, s) ∈ U, there exists a unique f (x, s) ∈ R such that
Ψ(ef (x,s) · x) = s;
moreover, U is open and f : U → R is a smooth S1 -invariant function.
Proof. Given (x, s) ∈ U, let γx (t) := Ψ(et · x) for all t ∈ R. The moment map is S1 invariant; hence, by definition, there exists f (x, s) ∈ R such that γx (f (x, s)) = s. Moreover,
by Lemma 1.1, γx is strictly increasing, and so f (x, s) is unique. By the implicit function
theorem, there exist open neighbourhoods V of (x, s) ∈ U and W of f (x, s) ∈ R, and a
smooth function fe: V → W such that
(y, u), fe(y, u) (y, u) ∈ V = {(y, u, t) ∈ V × W | Ψ(et · y) = u}.
Therefore U is open and f is smooth. Finally, since Ψ is S1 -invariant, the function f is as
well. (Here, S1 acts trivially on R.)
Proof of Proposition 3.1. By Lemma 3.3, the C× -invariant set
1
Ua := C× · Ψ−1 (a) = {x ∈ M r M S | a ∈ Ψ(C× · x)}
is open and there exists a smooth function f : Ua → R such that
Ψ(ef (x) · x) = a
for all x ∈ Ua .
Define Θ : C× × M → M × M by Θ(eu+iv , x) = (eu+iv · x, x) for all u, v ∈ R and x ∈ M .
Given a compact set L ⊆ M ×M , there exists a closed interval [s, t] ⊂ R so that f (x1 ) ∈ [s, t]
and f (x2 ) ∈ [s, t] for all (x1 , x2 ) ∈ L. Since Ψ is S1 -invariant,
f (eu+iv · x) = f (x) − u
for all u, v ∈ R and x ∈ Ua . Thus, for all (eu+iv , x) ∈ Θ−1 (L), we have f (x) − u ∈ [s, t] and
f (x) ∈ [s, t], and so u ∈ [s − t, t − s]. Thus Θ−1 (L) is contained in a compact set. Since Θ
is continuous, this implies that Θ−1 (L) is compact, that is, the C× -action on Ua is proper.
Thus, by [4, Corollary B.32], Ua /C× is a complex orbifold.
The natural inclusion map Ψ−1 (a) ,→ Ua descends to a well-defined smooth map i : M //a S1 →
Ua /C× . Similarly, the map Ua → Ψ−1 (a) defined by x 7→ ef (x) · x descends to a smooth map
g : Ua /C× → M //a S1 . Moreover, these induced maps are inverses of each other. Under the
resulting identification, the symplectic quotient M //a S1 inherits a natural complex structure
from Ua /C× .
Fix x ∈ Ψ−1 (a). By the preceding paragraph, the natural map T[x] (M //a S1 ) → T[x] (Ua /C× )
is an isomorphism of complex vector spaces. Since ω(ξM , J(ξM )) > 0 on Ψ−1 (a), we
can represent every vector in T[x] (M //a S1 ) by a unique vector in the J-invariant subpace
{X ∈ Tx M | ω(ξM , X) = ω(ξM , J(X)) = 0} of Tx M . The third claim follows.
6
4. Tame Birational Equivalence
In this section, we study how the reduced spaces that we defined in the previous section
change as we vary the value at which we reduce. Our main goal is to prove the proposition
below, which shows that the birational equivalence theorem of Guillemin and Sternberg also
holds for tamed symplectic forms [7] (if the fixed points are isolated); see also [2]. However,
most of the tools that we develop here work for arbitrary weights. For example, we use them
in §7 to prove Proposition 7.8, which is the tame analog of (a special case of) a theorem of
Godinho [5].
Proposition 4.1. Let (M, J) be a complex manifold with a holomorphic C× -action, a sym1
1
plectic form ω ∈ Ω2 (M )S satisfying ω(ξM , J(ξM )) > 0 on M r M S , and a proper moment
map Ψ : M → R. Assume that dimC (M ) > 2, and that Ψ−1 (a− , a+ ) contains exactly k fixed
points1; each lies in Ψ−1 (0) and has weights {−1, 1, . . . , 1}. Then there exists a complex
orbifold (X, I) and κ, η ∈ H 2 (X) so that
• for all s ∈ (a− , 0), the reduced space M //s S1 is biholomorphically symplectomorphic
to (X, I, σs ), where σs ∈ Ω2 (X) satisfies [σs ] = κ + 2πs η; and
• for all s ∈ (0, a+ ), the reduced space M //s S1 is biholomorphically symplectomorphic
b I,
bσ
b satisfies
to (X,
bs ), where σ
bs ∈ Ω2 (X)
[b
σs ] = q ∗ κ + 2πs q ∗ η − 2πs
k
X
Ej .
j=1
b is the blow-up of X at smooth points x1 , . . . , xk ∈ X, the map q : X
b → X is the
Here, X
−1
blow-down map, and Ej is the Poincaré dual of the exceptional divisor q (xj ) for all j.
Remark 4.2. The statement of Proposition 4.1 is particularly simple if Ψ−1 (a− , a+ ) contains
no fixed points: There exists a complex orbifold (X, I) and κ, η ∈ H 2 (X) so that, for all
s ∈ (a− , a+ ), the reduced space M //s S1 is biholomorphically symplectomorphic to (X, I, σs ),
where σs ∈ Ω2 (X) satisfies [σs ] = κ + 2πs η.
Remark 4.3. In fact, we will show that the class η discussed in Proposition 4.1 is the Euler
class of a principal S1 -bundle P → X; hence η is integral. Moreover, for s ∈ (a− , 0), the
biholomorphic symplectomorphism from M //s S1 to (X, I, σs ) is induced
P by an equivariant
diffeomorphism from Ψ−1 (s) to P . A similar remark holds for q ∗ η − Ej and s ∈ (0, a+ ).
Remark 4.4. Let (M, J) be a complex manifold with a holomorphic C× -action, a symplectic
1
form ω ∈ Ω2 (M )S , and a moment map Ψ : M → R. Then the C× action on M given by
(λ, x) 7→ λ−1 · x for all λ ∈ C× and x ∈ M is also holomorphic. The vector field induced by
the corresponding S1 -action is −ξM , and the weights at each fixed point are the negatives
of the original weights. Hence, Ψ0 := −Ψ is a moment map for this action. The reduced
space (Ψ0 )−1 (s)/S1 is naturally biholomorphically symplectomorphic to the reduced space
Ψ−1 (−s)/S1 ; however, the Euler class of the S1 -bundle (Ψ0 )−1 (s) → (Ψ0 )−1 (s)/S1 is the
negative of the Euler class of Ψ−1 (−s) → Ψ−1 (−s)/S1 .
1
Here, and throughout this section, we allow the case that Ψ−1 (a− , a+ ) contains no fixed points.
7
Therefore, Proposition 4.1 implies that the same claim holds if we make the following
modifications: Assume that the weights are {1, −1, . . . , −1} at each fixed point in Ψ−1 (0),
instead of {−1, 1, . . . , 1}. In this case, Claim (1) holds for all t ∈ (0, a+ ), not all t ∈ (a− , 0).
Claim (2) holds for all t ∈ (a− , 0), not all t ∈ (0, a+ ); moreover, the displayed equality is
replaced by
k
X
[b
σs ] = q ∗ κ + 2πs q ∗ η + 2πs
Ei .
j=1
Analogous statements apply to Proposition 5.1, Proposition 7.1, and Proposition 7.8.
By Proposition 3.1, the reduced space M //s S1 is isomorphic to C× · Ψ−1 (s) /C× for all
regular s ∈ R. Since x ∈ C× · Ψ−1 (s) exactly if s ∈ Ψ(C× · x), our first goal is to understand
Ψ(C× · x) for x ∈ M ; we accomplish this in the next two lemmas.
Lemma 4.5. Let (M, J) be a complex manifold with a holomorphic C× -action, a symplectic
1
1
form ω ∈ Ω2 (M )S satisfying ω(ξM , J(ξM )) > 0 on M r M S , and a proper moment map
Ψ : M → R. Given a ∈ R, the following hold for all x ∈ M :
(1) If Ψ(x) < a then a ∈ Ψ(C× · x) exactly if
1
{et · x | t ≥ 0} ∩ Ψ−1 [Ψ(x), a] ∩ M S = ∅.
(2) If Ψ(x) > a then a ∈ Ψ(C× · x) exactly if
1
{et · x | t ≤ 0} ∩ Ψ−1 [a, Ψ(x)] ∩ M S = ∅.
1
Proof. By symmetry, it suffices to consider x ∈ M with Ψ(x) < a. If x ∈ M S , then – since
the action is holomorphic – x is also fixed by C× . Hence, a 6∈ Ψ(C× · x) and the equality
1
displayed in Claim (1) does not hold. Thus, we may assume that x 6∈ M S .
Assume first that
1
p ∈ {et · x | t ≥ 0} ∩ Ψ−1 [Ψ(x), a] ∩ M S 6= ∅.
1
Since p ∈ M S , et · x 6= p for any t ∈ R. Hence, since p ∈ {et · x | t ≥ 0}, there exists a
sequence of ti ∈ R with
lim ti = ∞ and lim eti · x = p.
i→∞
i→∞
t
Moreover, by Lemma 1.1, the map t 7→ Ψ(e · x) is strictly increasing, and so Ψ(et · x) <
Ψ(p) ≤ a for all t ∈ R. Therefore, since Ψ is S1 -invariant, a 6∈ Ψ(C× · x).
So assume instead that
1
{et · x | t ≥ 0} ∩ Ψ−1 [Ψ(x), a] ∩ M S = ∅.
If a 6∈ Ψ(C× · x), then since the map t 7→ Ψ(et · x) is increasing, Ψ(et · x) ∈ [Ψ(x), a) for all
t ≥ 0. Hence, since Ψ is proper, the set K := {et · x | t ≥ 0} is a compact set
that contains
d
no fixed points. Thus, Lemma 1.1 implies that there exists ε > 0 so that dt Ψ(et · y) > ε
t=0
for all y ∈ K. Since es · (et · y) = est · y, this implies that dtd Ψ(et · x) > ε for all t ≥ 0. Thus,
Ψ(et · x) > Ψ(x) + tε for all t > 0. Since this gives a contradiction, a ∈ Ψ(C× · x).
8
Lemma 4.6. Let (M, J) be a complex manifold with a holomorphic C× -action, a symplectic
1
1
form ω ∈ Ω2 (M )S satisfying ω(ξM , J(ξM )) > 0 on M r M S , and a proper moment map
1
Ψ : M → R. Assume that that M S ∩ Ψ−1 (a− , a+ ) = {p1 , . . . , pk } ⊂ Ψ−1 (0). Then there
exists a C× -invariant neighbourhood W of {p1 , . . . , pk } in M which is C× -equivariantly bi`
`
−
+
holomorphic to a neighbourhood of kj=1 {0} in kj=1 Cnj × Cnj , where C× acts linearly on
−
+
each Cnj (respectively, Cnj ) with negative (respectively, positive) weights; we identify these
neighbourhoods. Then, for any x ∈ C× · Ψ−1 (a− , a+ ),

{0}
if x ∈ {p1 , . . . , pk },



`
−
(a , 0)
if x ∈ kj=1 Cnj r {0} × {0} ⊂ W,
−
×
Ψ(C · x) ∩ (a− , a+ ) =
`k
n+
j r {0}

⊂ W,
{0}
×
C
(0,
a
)
if
x
∈
+

j=1


(a− , a+ ) otherwise.
Proof. The first claim is an immediate consequence of Proposition 2.1.
For each j, the connected component Wj of W containing pj is open and C× -invariant.
Hence, if pj ∈ {et · x | t ∈ R} then x ∈ Wj . This implies that, for any x ∈ M ,
−
pj ∈ {et · x | t ≥ 0} exactly if x ∈ Cnj × {0} ⊂ Wj .
Moreover, p1 , . . . , pk are the only fixed points in Ψ−1 (a− , a+ ). Thus, for any x ∈ M ,
{et
−1
· x | t ≥ 0} ∩ Ψ (a− , a+ ) ∩ M
S1
6= ∅ exactly if x ∈
k
a
−
Cnj × {0} ⊂ W.
j=1
By a similar argument, for any x ∈ M ,
{et
−1
· x | t ≤ 0} ∩ Ψ (a− , a+ ) ∩ M
S1
6= ∅ exactly if x ∈
k
a
+
{0} × Cnj
⊂ W.
j=1
1
Therefore, since M S ∩ Ψ−1 (a− , a+ ) = {p1 , . . . , pk } ⊂ Ψ−1 (0), the lemma follows from
Lemma 4.5.
By Proposition 3.1, Lemma 4.6 implies the following, which describes the reduced spaces
up to biholomorphism.
Corollary 4.7. In the situation of Lemma 4.6, define
k
k
a
a
−
n+
×
−1
×
−1
j
Cnj ×{0} .
U− := C ·Ψ (a− , a+ )r
{0}×C
and U+ := C ·Ψ (a− , a+ )r
j=1
j=1
The quotients U± /C× are naturally complex orbifolds; moreover, the inclusion Ψ−1 (s± ) ,→
U± induces a biholomorphism M //s± S1 → U± /C× for each s− ∈ (a− , 0) and s+ ∈ (0, a+ ).
If we remove the claims about the cohomology classes [σs ] and [b
σs ] from Proposition 4.1,
then the revised statement follows easily from Corollary 4.7. (For details, see the proof at
the end of this section.) Moreover, the statements in Proposition 4.1 about the cohomology
9
classes of the reduced symplectic forms are formally identical to those in [7, Theorem 13.2].
However, the blow-down map in our paper is not identical to the one that appears in their
paper. Therefore, we include a proof for completeness.
Lemma 4.8. Let G be a closed subgroup of U(n). Let U be a G-invariant neighbourhood of
0 ∈ Cn , and let β ∈ Ω2 (U )G be closed. Then there exists ν ∈ Ω1 (U )G with compact support
such that βe := β − dν vanishes in a neighbourhood of 0. Finally, if β ∈ Ω1,1 (U )G , then we
may also choose ν so that βe ∈ Ω1,1 (U )G .
Proof. Pick a G-invariant smooth function ρ : U → [0, 1] with compact support that is 1 on
a neighbourhood of 0. By the Poincaré Lemma (and averaging), after possibly shrinking U
there exists µ ∈ Ω1 (U )G such that dµ = β. Let ν = ρµ ∈ Ω1 (U )G . If β ∈ Ω1,1 (U ), then
by the Poincaré Lemma for d and ∂, there exists a smooth
G-invariant potential function
h : U → R such that β = 2i ∂∂h. Let ν := 4i ∂(ρh) − ∂(ρh) .
Lemma 4.9. Let S1 act on a symplectic manifold (M, ω) with moment map
Ψ : M → R.
1
Assume that M S = {p1 , . . . , pk } ⊂ Ψ−1 (0). Then there exists κ ∈ H 2 M/S1 so that
[ωs ] = i∗s (κ) + 2πs ηs
for every regular value s ∈ R, where ωs ∈ Ω2 (M //s S1 ) is the reduced symplectic form,
is : M //s S1 → M/S1 is the natural inclusion, and ηs ∈ H 2 (M //s S1 ) is the Euler class of the
circle bundle Ψ−1 (s) → M //s S1 .
1
1
1
1
Proof. Let θ ∈ Ω1 (M r M S )S be a connection one-form. Since ω − d(Ψθ) ∈ Ω2 (M r M S )S
1
1
is basic, it is the pull-back of a closed form ω on (M rM S )/S1 . Let Ω ∈ Ω2 (M rM S )/S1 )
be the curvature form associated to θ. Then i∗s ([ω]) = [ωs ] − s[ Ω|M //s S1 ] = [ωs ] − 2πs ηs for
all regular s ∈ R.
By Lemma 4.8 and the Bochner linearisation theorem (cf. Lemma 2.2), there exists
1
ν ∈ Ω1 (M )S such that ω 0 := ω − dν vanishes on a neighbourhood of {p1 , . . . , pk }; let
Ψ0 = Ψ − ξM y ν. Since ξM y ω 0 = −dΨ0 and Ψ0 (pj ) = Ψ(pj ) = 0 for all j, the function
1
Ψ0 also vanishes on a neighbourhood of {p1 , . . . , pk }. Therefore, ω 0 − d(Ψ0 θ) ∈ Ω2 (M )S is
well-defined; it is the pull-back of a closed form ω 0 on M/S1 . Let κ = [ω 0 ] ∈ H 2 (M/S1 ).
1
1
Since ω 0 − d(Ψ0 θ) and ω − d(Ψθ) differ by the basic form ν − (ξM y ν)θ ∈ Ω1 (M r M S )S ,
i∗s (κ) = i∗s ([ω]) for all regular s ∈ R.
Remark 4.10. Alternately, we could replace the second paragraph of the proof above with
the following more sophisticated argument:
In the de Rham model of equivariant cohomology, ω + Ψ represents an equivariant coho1
mology class on M [1]. Moreover, under the natural isomorphism HS∗1 (M rM S ) → H ∗ (M r
1
1
1
M S )/S1 , [ω + Ψ] maps to [ω]. Since Ψ(M S ) = 0, the restriction [ω + Ψ]M S1 ∈ HS21 (M S ) is
1
1
in the image of the natural inclusion H ∗ (M S ) ,→ HS∗1 (M S ). Hence, by the Leray spectral
sequence for the natural map M ×S1 ES1 → M/S1 , the class [ω + Ψ] lies in the image of the
pull-back H ∗ (M/S1 ) → HS∗1 (M ).
Lemma 4.11. Let (M, J) be a complex manifold with a holomorphic C× -action, a symplectic
1
1
form ω ∈ Ω2 (M )S satisfying ω(ξM , J(ξM )) > 0 on M r M S , and a proper moment map
10
1
Ψ : M → R. Assume that 0 ∈ (a− , a+ ) and that M S ∩ Ψ−1 (a− , a+ ) = {p1 , . . . , pk } ⊂ Ψ−1 (0).
Then for all x ∈ C× · Ψ−1 (a− , a+ ) and τ ∈ [0, 1], there exists a unique G(x, τ ) ∈ M such that
G(x, τ ) ∈ {et · x | t ∈ R} and Ψ(G(x, τ )) = τ Ψ(x);
moreover, G : C× · Ψ−1 (a− , a+ ) × [0, 1] → M is a continuous S1 -equivariant map.
Proof. Define
1
U := {(x, s) ∈ M r M S × R | s ∈ Ψ(C× · x)}.
By Lemma 3.3, for all (x, s) ∈ U, there exists a unique f (x, s) ∈ R such that
Ψ(ef (x,s) · x) = s;
moreover, U is open, and f : U → R is a smooth S1 -invariant function. By Lemma 4.6, given
x ∈ C× · Ψ−1 (a− , a+ ) r {p1 , . . . , pk } and s ∈ (a− , a+ ),
(x, s) 6∈ U ⇔ x ∈
k
a
C
n−
j
k
a
n+
j
× {0} and s ≥ 0 or x ∈
{0} × C
and s ≤ 0 .
j=1
j=1
Therefore, for all x ∈ C× · Ψ−1 (a− , a+ ) and τ ∈ [0, 1], there exists a unique G(x, τ ) ∈ M
such that G(x, τ ) ∈ {et · x | t ∈ R} and Ψ(G(x, τ )) = τ Ψ(x):


if x = pj for some j,
p j
−
+ G(x, τ ) = pj
if x ∈ Cnj × {0} ∪ {0} × Cnj
for some j and τ = 0,

 f (x,τ Ψ(x))
e
· x otherwise.
Clearly, G : C× · Ψ−1 (a− , a+ ) × [0, 1] → M is an S1 -equivariant map that is continuous on the
complement of G−1 ({p1 , . . . , pk }). Let U ⊆ Ψ−1 (a− , a+ ) be an S1 -invariant neighbourhood
of pj . By Corollary 2.7, there exists δ > 0 and an S1 -invariant open neighbourhood V ⊆ U
of pj such that C× · V ∩ Ψ−1 (−δ, δ) ⊂ V . Then
(x, τ ) ∈ (C× · V ) × [0, 1] |τ Ψ(x)| < δ
is an open subset of G−1 (U ) that contains G−1 (pj ). Therefore, G is continuous.
Corollary 4.12. Assume that we are in the situation of Lemma 4.11. There exists κ ∈
H 2 (Ψ−1 (a− , a+ )/S1 ) with the following property: Given a regular s ∈ (a− , a+ ), let Us :=
C× · Ψ−1 (s). There exists a continuous map
hs : Us /C× → Ψ−1 (a− , a+ )/S1 such that hs ([x])
is the unique S1 -orbit in C× · x ∩ Ψ−1 (0) /S1 . Moreover,
[ωs ] = js∗ h∗s (κ) + 2πs ηs ,
where ωs ∈ Ω2 (M //s S1 ) is the reduced symplectic form, js : M //s S1 → Us /S1 is induced by the
inclusion map, and ηs ∈ H 2 (M //s S1 ; Z) is the Euler class of the S1 -bundle Ψ−1 (s) → M //s S1 .
Proof. Applying Lemma 4.9 to Ψ−1 (a− , a+ ), there exists κ ∈ H 2 (Ψ−1 (a− , a+ )/S1 ) so that
[ωs ] = i∗s (κ) + 2πs ηs
for every regular value s ∈ (a− , a+ ), where is : M //s S1 ,→ Ψ−1 (a− , a+ )/S1 is the natural
inclusion.
11
Let G : C× · Ψ−1 (a− , a+ ) × [0, 1] → M be defined as in Lemma 4.11. Given a regular s ∈ (a− , a+ ), Lemma 4.11 implies that hs ([x]) := [G(x, 0)] is the unique S1 -orbit in
C× · x ∩ Ψ−1 (0) /S1 for any x ∈ Us /C× ; moreover, hs : Us /S1 → Ψ−1 (a− , a+ )/S1 is continuous. Additionally, since G(x, 1) is the identity map, the inclusion map is : M //s S1 ,→
Ψ−1 (a− , a+ )/S1 is isotopic to the composition hs ◦ js .
Proof of Proposition 4.1. We may assume without loss of generality that 0 ∈ (a− , a+ ). By
1
Proposition 4.6, there exists a C× -invariant neighbourhood W of Ψ−1 (0) ∩ M S which is
`
`
C× -equivariantly biholomorphic to a neighbourhood of kj=1 {0} in kj=1 Cn , where in each
case C× acts on Cn by
λ · (z1 , . . . , zn ) = (λ−1 z1 , λz2 , . . . , λzn )
(5)
for all λ ∈ C× and z ∈ Cn ; we identify these neighbourhoods. Define
U− := C× ·Ψ−1 (a− , a+ )r
k
a
{0}×Cn−1
and U+ := C× ·Ψ−1 (a− , a+ )r
j=1
k
a
C×{0} ,
j=1
and let η± ∈ H 2 (U± /C× ) be the Euler classes of the C× -bundles U± → U± /C× . By
Corollary 4.7, the quotients U± /C× are naturally complex orbifolds; moreover, the inclusion
Ψ−1 (s± ) ,→ U± induces a biholomorphism js± : M //s± S1 → U± /C× for each s− ∈ (a− , 0) and
s+ ∈ (0, a+ ). Additionally, js∗± (η± ) is the Euler class of the S1 -bundle Ψ−1 (s± ) → M //s± S1
for each s− ∈ (a− , 0) and s+ ∈ (0, a+ ). By Corollary 4.12, there exists κ ∈ H 2 (Ψ−1 (a− , a+ ))
with the following property: There exist continuous maps h± : U± /C× → Ψ−1 (a− , a+ )/S1
such that h± ([x]) ∈ C× · x ∩ Ψ−1 (0) /S1 for all [x] ∈ U± /C× . Moreover,
[ωs± ] = js∗± h∗± (κ) + s± η±
for each s− ∈ (a− , 0) and s+ ∈ (0, a+ ), where ωs± ∈ Ω2 (M //s± S1 ) is the reduced symplectic
form.
The blow-up of Cn−1 ∼
= (C r {0}) ×C× Cn−1 at 0 is C ×C× (Cn−1 r {0}); the blow-down
map sends [u; z1 , . . . , zn−1 ] to [1; uz1 , . . . , uzn−1 ]. Therefore, U+ /C× is the blow-up of U− /C×
at the smooth points
a
{x1 , . . . , xk } :=
(C r {0}) × {0} /C× ;
j
×
moreover, if q : U+ /C×→ U− /C
map,
is the blow-down
then h− ◦ q = h+ . (To see this, note
that if x ∈ C r {0} × {0} ∪ {0} × Cn−1 r {0} ⊂ Cn , then h± ([x]) = [0] ∈ Cn /S1 .)
Thus, h∗+ (κ) = q ∗ (h∗− (κ)).
1
Finally, let Wj be the component of the C× -invariant neighbourhood W of Ψ−1 (0) ∩ M S
that contains j. We may assume that W is chosen so that Wj+ := (Wj ∩ U+ )/C× is a tubular
neighbourhood of the exceptional divisor Ej := q −1 (xj ) for all j. This implies that Wj+
is homotopy equivalent to Ej ∼
= CPn−2 and Wj+ r Ej is homotopy equivalent to S2n−3 ; in
particular, since n > 2, H 1 (Wj+ r Ej ; R) = 0. Therefore, by the Mayer-Vietoris argument
`
`
for j Wj+ and U+ /C× r j Ej , a cohomology class in H 2 (U+ /C× ) is uniquely determined
12
by its restrictions to
`
j
Ej+ and U+ /C× r
`
Ej . Both sides of
X
η+ = q ∗ (η− ) −
Ej
j
(6)
j
2
restrict to`the positive generator of H (Ej ; Z) for each j. Since the restriction of (6) to
U+ /C× r j Ej = (U+ ∩ U− )/C× also holds, we can conclude that (6) is true.
5. Tame Cutting
Next, we show that symplectic cutting, developed by by Lerman in [10], also works in our
category.
Proposition 5.1. Let (M, J) be a complex manifold with a holomorphic C× -action, a sym1
1
plectic form ω ∈ Ω2 (M )S satisfying ω(ξM , J(ξM )) > 0 on M r M S , and a moment map
Ψ : M → R. Assume that 0 ∈ R is a regular value of Ψ. Then there exists a complex orbifold
(Mcut , Jcut ) with a holomorphic C× -action, an S1 -invariant symplectic form ωcut ∈ Ω2 (Mcut )
S1
satisfying ωcut (ξMcut , Jcut (ξMcut )) > 0 on Mcut r Mcut
, and a moment map Ψcut : Mcut → R
so that the following hold.
(1) Ψcut (Mcut ) ⊆ (−∞, 0].
1
(2) Ψ−1
cut (0) is a fixed component that is biholomorphically symplectomorphic to M //0 S .
−1
1
−1
(3) There exists an S -equivariant symplectomorphism from Ψ (−∞, 0) to Ψcut (−∞, 0)
that intertwines the moment maps and induces a biholomorphism between the reduced
spaces at all regular s < 0.
(4) Fix s ∈ R. If J tames ω near Ψ−1 (s), then Jcut tames ωcut near Ψ−1
cut (s).
(5) If Ψ is proper, then Ψcut is proper.
Proof. Let J 0 be the product complex structure on M 0 = M × C. The diagonal C× -action
on M 0 is holomorphic, where C× acts on C by multiplication. Consider the symplectic form
1
1
1
ω 0 = ω + idz ∧ dz/2 ∈ Ω2 (M 0 )S . If (x, z) ∈ M 0 r (M 0 )S , then either x 6∈ M S or z 6= 0. In
either case, the fact that ξM 0 = ξM + ξC implies that
ω 0 (ξM 0 , J 0 (ξM 0 ))
= ω(ξM , J(ξM )) + |z|2 > 0.
(x,z)
x
The function Ψ0 : M 0 → R sending (x, z) to Ψ(x) + |z|2 /2 is a moment map. Since 0 is a
regular value of Ψ, it is also a regular value of Ψ0 .
Define U00 := C× · (Ψ0 )−1 (0) ⊆ M 0 . By Proposition 3.1, the quotient U00 /C× is naturally a
complex orbifold. Moreover, the reduced space
Mcut := M 0 //0 S1 = (x, z) ∈ M × C Ψ(x) + |z|2 /2 = 0 /S1
(7)
inherits a complex structure Jcut so that the inclusion (Ψ0 )−1 (0) ,→ U00 induces a biholomorphism Mcut → U00 /C× . Finally, for all (x, z) ∈ (Ψ0 )−1 (0), the natural map
{X 0 ∈ T(x,z) M 0 | ω 0 (ξM 0 , X 0 ) = ω 0 (ξM 0 , J(X 0 )) = 0} → T[x,z] (Mcut )
is an isomorphism of complex and symplectic vector spaces.
13
(8)
Since the holomorphic C× -action on M 0 given by λ · (x, z) = (λx, z) commutes with the
diagonal action, it descends to a holomorphic C× -action on Mcut . Moreover, the fact that
1
ω 0 is invariant under the associated S1 -action on M 0 implies that ωcut ∈ Ω2 (Mcut )S . The
function Ψcut : Mcut → R defined by
Ψcut ([x, z]) = Ψ(x)
(9)
is a moment map for the S1 -action on Mcut . The vector field Ξ on M 0 given by
ω(ξM , J(ξM ))
|z|2 ξM − ω(ξM , J(ξM ))ξC
0
Ξ := ξM −
ξ
=
(10)
M
ω(ξM , J(ξM )) + |z|2
ω(ξM , J(ξM )) + |z|2
descends to ξMcut on Mcut and satisfies ω 0 (ξM 0 , Ξ) = ω 0 (ξM 0 , J 0 (Ξ)) = 0. Since (8) is an
isomorphism of symplectic and complex vector spaces, this implies that
|z|2 ω(ξM , J(ξM ))
ωcut (ξMcut , Jcut (ξMcut )) = ω 0 (Ξ, J 0 (Ξ)) =
.
ω(ξM , J(ξM )) + |z|2
1
In particular, ωcut (ξMcut , Jcut (ξMcut ))[x,z] > 0 for any [x, z] ∈ Mcut unless z = 0 or x ∈ M S ;
1
S
in either case, [x, z] ∈ Mcut
.
Claims (1) and (5) are immediate consequences of (7) and (9).
Moreover, by (7) and (9),
1
Ψ−1
cut (0) = {(x, 0) ∈ M × C | Ψ(x) = 0}/S ,
and so the map from Ψ−1 (0) to M × C that takes x ∈ Ψ−1 (0) to (x, 0) induces a diffeomorphism from M //0 S1 to Ψ−1
cut (0). Additionally, by (8), under the natural isomorphism
0 ∼
T M = T M × T C, the natural map
{(X, 0) ∈ T(x,0) M 0 | ω(ξM , X) = ω(ξM , J(X)) = 0} → T[x,0] (Ψ−1
cut (0))
is an isomorphism of complex and symplectic vector spaces. Therefore, Claim (2) follows
from part (3) of Proposition 3.1.
It is
the map from Ψ−1 (−∞, 0) to Ψ−1
cut (−∞, 0) that sends
p
straightforward
to check that
−1
1
x to x, −2Ψ(x) for all x ∈ Ψ (−∞, 0) is an S -equivariant symplectomorphism that
intertwines the moment maps. Given a regular s < 0, it restricts to an S1 -equivariant
1
diffeomorphism from Ψ−1 (s) to Ψ−1
cut (s), and so induces a diffeomorphism from M //s S to
1
0 ∼
−1
Mcut //s S . Under the identification T M = T M × T C, it sends X ∈ Tx (Ψ (s)) to (X, 0) ∈
T[x,√−2s] (Ψ−1
cut (s)). Let Ξ be defined by (10); by Proposition 3.1 and (8), the natural map
from
{X 0 ∈ T(x,√−2s) M 0 | ω 0 (ξM 0 , X 0 ) = ω 0 (ξM 0 , J 0 (X 0 )) = ω 0 (Ξ, X 0 ) = ω 0 (Ξ, J 0 (X 0 )) = 0}
(11)
to T[x,√−2s] (Mcut //s S1 ) is an isomorphism of complex and symplectic vector spaces. Under
the identification T M 0 ∼
= T M × T C, the vector space in (11) can be rewritten as
{(X, 0) ∈ T(x,√−2s) M 0 | ω(ξM , X) = ω(ξM , J(X)) = 0}.
Therefore, Claim (3) follows from part (3) of Proposition 3.1.
Finally, fix s ∈ R and assume that J tames ω near Ψ−1 (s). If [x, z] ∈ Ψ−1
cut (s), then
x ∈ Ψ−1 (a) by (9). Hence, J tames ω near x, and so J 0 tames ω 0 near (x, z). Thus Jcut
tames ωcut near [x, z] by (8). This proves Claim (4).
14
6. Tame Blow-Up
It is well known that the blow-ups of Kähler manifolds admit Kähler forms. In this section,
we construct the blow-up of a complex manifold with a tamed symplectic form at one point.
Proposition 6.1. Let (M, J) be a complex manifold with a holomorphic C× -action, a sym1
plectic form ω ∈ Ω2 (M )S that is tamed on W ⊆ M and satisfies ω(ξM , J(ξM )) > 0 on
1
c, J)
b be the complex blow-up of M at
M r M S , and a moment map Ψ : M → R. Let (M
1
c)S1
p ∈ M S ∩ W . For sufficiently small t > 0, there exist a symplectic form ω
b ∈ Ω2 (M
b c)) > 0 on M
cr M
cS1 , and a moment map
that is tamed on q −1 (W ) and satisfies ω
b (ξM
c, J(ξM
b: M
c → R such that
Ψ
[b
ω ] = q ∗ [ω] − tE,
c → M is the blow-down map and E is the Poincaré dual of the exceptional divisor
where q : M
−1
b = q∗Ψ
q (p). Moreover, given a neighbourhood V of p, we may assume that ω
b = q ∗ ω and Ψ
c r q −1 (V ).
on M
Proof. We may assume that W is open. By Lemma 2.2, there exists an S1 -equivariant
biholomorphism from an S1 -invariant neighbourhood of 0 ∈ Cn to a neighbourhood U ⊆ W ∩
V of p, where S1 acts on Cn with weights α1 , ..., αn ∈ Z. We identify these neighbourhoods,
and also identify q −1 (U ) with a neighbourhood of the exceptional divisor E in
cn := (Cn r {0}) ×C× C,
C
where C× acts on (Cn r {0}) × C by
λ · (z1 , ..., zn ; u) = (λz1 , ..., λzn ; λ−1 u).
cn → Cn sends [z1 , ..., zn ; u] to (uz1 , ..., uzn ).
In these coordinates, the blow-down map q : C
cn → CPn−1 by π [z1 , ..., zn ; u] = [z1 , ..., zn ].
Define π : C
crE → M r{p} is equivariant and biholomorphic, the closed form
Since the restriction q : M
2 c S1
crE, tamed on q −1 (W )rE, and satisfies q ∗ ω ξ c, Jb ξ c >
q ω ∈ Ω (M ) is symplectic on M
M
M
cr M
cS1 ∪ E . Since q is holomorphic and ker(π∗ |m ) ∩ ker(q∗ |m ) = {0},
0 on M
∗
b
q ∗ ω(X, J(X))
≥0
c, with equality impossible if π∗ X = 0 and X 6= 0. Finally, since
for all m ∈ E and X ∈ Tm M
c → M is equivariant, ξ cy q ∗ ω = −dq ∗ Ψ.
q: M
M
Let ρ : R → R be a smooth function which is 1 on a neighbourhood of 0 and such that
1
z 7→ ρ(|z|2 ) has compact support in U . Define f : R+ → R by f (t) = 2π
ρ(t) ln t (c.f. [7,
Section 5]). As shown in [7, Section 4], on the complement of E,
i
π ∗ (Ω) = q ∗
∂∂ ln |z|2 ,
2π
15
where Ω is the Fubini-Study form on CPn−1 . Therefore, there exists a closed form η ∈
c)S1 with support in U equal to q ∗ (i∂∂(f (|z|2 )) on q −1 (U ) r E, and equal to π ∗ (Ω) near
Ω2 (M
c → R with support in U by
E. Define Φ : M
X
n
2 0
2
∗
αj |zj | f (|z| )
Φ(z) = q
j=1
on q −1 (U ) r E and by
Pn
π∗
2
j=1 αj |zj |
2π|z|2
!
(12)
near E. Then ξM
cy η = −dΦ by a straighforward calculation on U r {0}. The restriction of
2 c
[η] ∈ H (M ) to E is the positive generator of H 2 (E; Z) ∼
= Z, which is the negative of the
c
Euler class of the normal bundle to E in M . Moreover, since η is supported in a tubular
c r E vanishes. Hence, [η] = −E.
neighbourhood of E, the restriction of [η] to M
Therefore, for all t ∈ R,
∗
∗
[q ∗ ω + tη] = q ∗ [ω] − tE and ξM
cy (q ω + tη) = −d(q Ψ + tΦ).
1
c)S is symplectic, is tamed on q −1 (W ), and satisfies
It remains to show that q ∗ ω + tη ∈ Ω2 (M
b c)) > 0 on M
cr M
cS1 , for all sufficiently small t > 0. We will do this by
(q ∗ ω + tη)(ξM
c, J(ξM
looking at three regions.
(1) On a neighbourhood of the exceptional divisor, η = π ∗ (Ω). Since π is holomorphic and
b
c, with equality exactly
Ω is Kähler, this implies that η(X, J(X))
≥ 0 for all X ∈ Tm M
b
if π∗ (X) = 0. Moreover, by the second paragraph of this proof, q ∗ ω(X, J(X))
≥0
with equality impossible if π∗ X = 0 and X 6= 0. Therefore, for all t > 0,
b
q ∗ ω + tη)(X, J(X)
>0
c, that is, q ∗ ω + tη is tamed.
for all non-zero X ∈ Tm M
c, the form η vanishes; hence, by the sec(2) On the complement of q −1 (supp(ρ)) ⊆ M
∗
ond paragraph, q ω + tη is symplectic, is tamed on q −1 (W ), and satisfies (q ∗ ω +
b c) > 0 on M
cr M
cS1 , for all t.
tη) ξM
c, J(ξM
(3) The complement of the open sets considered in (1) and (2) is compact. Since q ∗ ω is
tamed on this set, q ∗ ω + tη is also tamed for all sufficiently small t.
Remark 6.2. The exact same argument works if (M, J) is a complex orbifold, as long as
we blow up at a smooth point p ∈ M .
b so that Ψ
b is
Remark 6.3. In Proposition 6.1, if Ψ is proper, then we may choose ω
b and Ψ
proper. To see this, simply choose V to be a neighbourhood with compact closure.
16
7. Application
We are now ready to build the specific machinery that the first author needs to construct
a non-Hamiltonian symplectic circle action with isolated fixed points on a closed, connected
symplectic manifold in [14]. While we also need to prove Proposition 7.8, our main goal is
to prove the following proposition.
Proposition 7.1. Let (M, J) be a complex manifold with a holomorphic C× -action, a sym1
plectic form ω ∈ Ω2 (M )S that is tamed near Ψ−1 (0) and satisfies ω(ξM , J(ξM )) > 0 on
1
M r M S , and a proper moment map Ψ : M → R. Assume that the S1 -action on Ψ−1 (0)
is free except for k orbits with stabilisers Z2 . Then for sufficiently small ε > 0 there exist
1
f, Je with a holomorphic C× -action, a symplectic form ω
f S
a complex manifold M
e ∈ Ω2 M
e f > 0 on M
frM
fS1 , and a proper moment map Ψ
e: M
f → R so that
satisfying ω
e ξM
f, J ξM
the following hold:
e −1 (−ε, 0] contains exactly k fixed points; each lies in Ψ
e −1 (0) and has weights {−2, 1, . . . , 1}.
(1) Ψ
e −1 (−∞, −ε/2) to Ψ−1 (−∞, −ε/2)
(2) There is an equivariant symplectomorphism from Ψ
f//s S1 to M //s S1 for all regular s ∈ (−∞, −ε/2).
that induces a biholomorphism from M
−1
e (−ε, ε).
(3) ω
e is tamed on Ψ
To prove this, we need to construct the blow-up of a complex orbifold with a tamed
symplectic form at an isolated Z2 -singularity. The first step is to define the blow-up of a
complex orbifold at an isolated Z2 -singularity.
Definition 7.2. Let Z2 act diagonally on Cn . The blow-up of Cn /Z2 at [0] is
n /Z := (Cn r {0}) × × C,
\
C
2
C
where C× acts on (Cn r {0}) × C by
λ · (z1 , ..., zn ; u) = (λz1 , ..., λzn ; λ−2 u).
n /Z → Cn /Z is given by
\
The blow-down map q : C
2
2
√
√
q([z1 , ..., zn ; u]) := [ uz1 , ..., uzn ].
Unfortunately, although the map q is continuous and the pullback q ∗ f is holomorphic
for every holomorphic function f : Cn /Z2 → C, the blow-down map q is not smooth. For
example, [w] 7→ |w|2 is a smooth function on Cn /Z2 , but its pullback [z; u] 7→ |u||z|2 is not a
√
n /Z . (Note that, when n = 1, C/Z
\
[2 ∼
smooth function on C
= C and q(u) = [ u ].) However,
2
the exceptional divisor E := q −1 ([0]) is biholomorphic to CPn−1 . Moreover, q restricts to a
n /Z r E to Cn /Z r {[0]}, and so there is a well-defined blow-up of
\
biholomorphism from C
2
2
a complex orbifold at any isolated Z2 -singularity.
Since the blow-down map is not smooth, we need to modify the symplectic form locally
before pulling it back to the blow-up. We will do this in two stages, and use the following
criterion for Kähler forms, which we adapted from [7, Lemma 5.3].
17
Lemma 7.3. Given n > 1 and a smooth function f : R → R, the form ω = 2i ∂∂f |z|2 ∈
Ω1,1 (Cn ) is Kähler at z0 ∈ Cn exactly if
f 0 |z0 |2 > 0 and f 0 |z0 |2 + |z0 |2 f 00 |z0 |2 > 0.
Proof. Define g : Cn → R by g(z) = f |z|2 . The form ω = 2i ∂∂g is Kähler exactly if the
eigenvalues of the Hermitian matrix
2 h
i
∂ g
= f 0 |z|2 δjk + f 00 |z|2 z j zk
∂zj ∂z k
2
are all positive. Since the matrix
eigenvalues 0 and |z| , the matrix
rank21 00and 2has
[z j z0k ] is 2of
2
0
above has eigenvalues f |z| and f |z| + |z| f |z| .
i
2
Remark 7.4. Similarly, given a smooth function
f
:
R
→
R,
the
form
ω
=
∈
|z|
∂∂f
2
1,1
0
2
2 00
2
Ω (C) is Kähler at z0 ∈ C exactly if f |z0 | + |z0 | f |z0 | > 0.
The next lemma translates [13, Theorem 1.10] to our setting.
Lemma 7.5. Let G be a closed subgroup of U(n). Let U be a G-invariant neighbourhood
of 0 ∈ Cn , and let ω ∈ Ω2 (U )G be a tamed symplectic form. Then there exists ν ∈ Ω1 (U )G
with compact support such that ω
e := ω − dν is a tamed symplectic form and is constant in a
neighbourhood of 0. Moreover, if ω ∈ Ω1,1 (U )G is Kähler, then we may also choose ν so that
ω
e is Kähler.
Proof. We may assume that U is a ball centred at 0. Pick a G-invariant smooth function
ρ : Cn → [0, 1] so that ρ has compact support K in U and is 1 on a neighbourhood of 0.
Given λ > 0, define ρλ : Cn → [0, 1] by ρλ (x) = ρ(λx); the support of ρλ is λ1 K.
Identify Cn with R2n . Let ω 0 ∈ Ω2 (U )G be the unique constant
ω 0 |0 = ω|0 .
P form satisfying
1
G
By the Poincaré Lemma (and averaging), there exists µ = i fi dxi ∈ Ω (U ) such that
∂fi
(0) = 0 for all i, j. Therefore,
dµ = ω − ω 0 . Since dµ|0 = 0, we may assume that fi (0) = ∂x
j
∂fi
by the Mean Value Theorem, there exists C > 0 so that ∂x
(x) ≤ C|x| and fi (x) ≤ C|x|2 for
j
all x ∈ K and all i, j. Since tameness is an open condition, a straightforward calculation in
coordinates shows that ω − d(ρλ µ) is tamed for sufficiently large λ. Let ν = ρλ µ ∈ Ω1 (U )G .
If ω ∈ Ω1,1 (U ), then by the Poincaré Lemma for d and ∂, there exists a smooth G-invariant
potential function h : U → R such that ω − ω 0 = 2i ∂∂(ρλ h). Since ω|0 = ω 0 |0 , we may assume
that h(0) = 0, ∂h|0 = 0, and ∂h|0 = 0. By a calculation similar to the one above, if we let
ν := 4i ∂(ρλ h) − ∂(ρλ h) ∈ Ω1 (U )G , then ω
e := ω − dν is Kähler for sufficiently large λ. Lemma 7.6. Let G be a closed subgroup of U(n). Let U be a G-invariant neighbourhood
of 0 ∈ Cn , and let ω ∈ Ω2 (U )G be a tamed symplectic form. Given a smooth function
f : R → R so that the function z 7→ f (|z|2 ) is strictly plurisubharmonic on Cn r {0}, there
exists ν ∈ Ω1 (U )G with compact support such that ω
e := ω − dν satisfies the following:
(1) ω
e is a tamed symplectic
form on U r {0}; and
(2) ω
e 1,1 = 2i ∂∂f |z|2 , and ω
e 2,0 and ω
e 0,2 are constant, on a neighbourhood of 0 ∈ U .
18
Moreover, if ω ∈ Ω1,1 (U )G is Kähler, then we may also choose ν so that ω
e is Kähler on
U r {0}.
Proof. By Lemma 7.5, we may assume that ω is constant. Write ω = ω 1,1 + ω 2,0 + ω 0,2 ,
j,k
where ω j,k ∈ Ω
(Cn ) for all j, k. By a linear change of variables, we may further assume
P
that ω 1,1 = 2i j dzj ∧ dz j = 2i ∂∂|z|2 on U . Finally, U contains a closed ball Br of radius
r > 0 centred at 0.
Assume first that n > 1. By assumption, the form 2i ∂∂f |z|2 is Kähler on Cn r {0}. By
Lemma 7.3, this implies that f 0 (t) and tf 00 (t) + f 0 (t) are positive for all t > 0. Hence, there
exists a smooth function h : R → R such that h(t) = tf 0 (t) in a neighbourhood of 0, h(t) = t
for all t > r, and h(t) and h0 (t) are positive for all t > 0. Let fe: R → R be the smooth
function with fe0 (t) = h(t)/t and fe(0) = f (0). Then fe(t) = f (t) near 0, fe0 (t) = 1 for all
t > r, and fe0 (t) and tfe00 (t) + fe0 (t) are positive for all t > 0. Hence, Lemma 7.3 implies that
the form 2i ∂∂ fe |z|2 is Kähler on Cn r {0}. By Remark 7.4, a similar argument applies if
n = 1.
Define g : Cn → R by g(z) := |z|2 − fe(|z|2 ), and let ν := 4i (∂g − ∂g) ∈ Ω1 (Cn )U(n) . Define
ω
e := ω − dν ∈ Ω2 (U )G . Since fe0 (t) = 1 for all t > r, the support of ν is contained in Br ⊂ U .
Since dν = 2i ∂∂g ∈ Ω1,1 (Cn )U(n) , we have
i
ω
e 1,1 = ∂∂ fe |z|2 ,
2
ω
e 0,2 = ω 0,2 ,
and ω
e 2,0 = ω 2,0 .
This shows that ω
e 1,1 is Kähler on U r{0}, and proves Claim (2). If J is the standard complex
structure on Cn , then
ω 2,0 (X, JX) = ω 0,2 (X, JX) = 0
for all vectors X ∈ T U . This proves Claim (1). The last claim follows immediately.
We are now ready to extend Proposition 6.1 to the blow-up of a complex orbifold at an
isolated Z2 -singularity.
Proposition 7.7. Let (M, J) be a complex orbifold with a holomorphic C× -action, a sym1
plectic form ω ∈ Ω2 (M )S that is tamed on W ⊆ M and satisfies ω(ξM , J(ξM )) > 0 on
1
c, Jb be the complex blow-up of M at an
M r M S , and a moment map Ψ : M → R. Let M
1
isolated Z2 -singularity p ∈ M S ∩ W . For sufficiently small t > 0, there exist a symplectic
c)S1 that is tamed on q −1 (W ) and satisfies ω
b c)) > 0 on M
crM
cS1 ,
form ω
b ∈ Ω2 (M
b (ξM
c, J(ξM
b: M
c → R such that
and a moment map Ψ
t
[b
ω ] = q ∗ [ω] − E,
2
c → M the blow-down map and E is the Poincaré dual of the exceptional divisor
where q : M
b = q∗Ψ
q −1 (p). Moreover, given a neighbourhood V of p, we may assume that ω
b = q ∗ ω and Ψ
c r q −1 (V ).
on M
19
Proof. We may assume that W is open. By Lemma 2.2, there exists an S1 -equivariant
biholomorphism from an S1 -invariant neighbourhood of [0] ∈ Cn /Z2 to a neighbourhood
U ⊆ W ∩ V of p, where Z2 acts diagonally on Cn and S1 acts with weights α1 , ..., αn ∈
Z. We identify these neighbourhoods, and also identify q −1 (U ) with a neighbourhood of
n /Z := (Cn r {0}) × × C. (See Definition 7.2.) Define
\
the exceptional divisor E in C
2
C
n−1
n
\
π : C /Z2 → CP
by π([z1 , ..., zn ; u]) = [z1 , ..., zn ].
The function z 7→ |z|4 is strictly plurisubharmonic on Cn r {0}. Hence, by Lemma 7.6
there exists a closed S1 -invariant form ω
e ∈ [ω] ∈ H 2 (M ) so that ω
e = ω on M r U ; moreover
ω
e satisfies the following:
(1) ω
e is tamed on U r {p}; and
(2) ω
e 1,1 = 2i ∂∂(|z|4 ), and ω
e 2,0 and ω
e 0,2 are constant, on a neighbourhood of p.
e : M → R be the smooth function satisfying ξM y ω
e so that Ψ
e = Ψ on M r U .
Let Ψ
e = −dΨ
c r E → M r {p} is equivariant and biholomorphic, the form
Since the restriction q : M
1
b c)) > 0
c r E)S is symplectic, tamed on q −1 (W ) r E, and satisfies q ∗ ω
q∗ω
e ∈ Ω2 (M
e (ξM
c, J(ξM
1
cS ∪ E . A straightforward calculation in local coordinates shows that (2) implies
cr M
on M
c that restricts to q ∗ ω
c r E; by a slight
that there exists a unique closed form on M
e on M
1
c)S . Moreover,
abuse of notation, we will call this form q ∗ ω
e ∈ Ω2 (M
i
(q ∗ ω
e )1,1 = ∂∂ |u|2 |z|4
2
on a neighbourhood of E. Hence, another straightforward calculation implies that for all
c,
m ∈ E and X ∈ Tm M
b
q∗ω
e (X, J(X))
≥ 0,
c r E → M r {p} is
with equality impossible if π∗ X = 0 and X 6= 0. Finally, since q : M
∗
e is continuous and M
cr E
e on M
c r E. Since q ∗ Ψ
smooth and equivariant, ξM
e = −dq ∗ Ψ
cy q ω
e on M
c.
c, this implies that ξ cy q ∗ ω
is dense in M
e = −dq ∗ Ψ
M
The remainder of the proof is nearly identical to the proof of Proposition 6.1. The main
c is twice the (negative)
distinction is that here, the Euler class to the normal bundle of E in M
2
generator of H (E; Z). Thus, if we construct η as in the proof of Proposition 6.1, then
[η] = − 12 E. Hence, [e
ω + tη] = q ∗ [ω] − 2t E.
Proof of Proposition 7.1. Let (Mcut , Jcut ) be the complex orbifold with holomorphic C× 1
action, symplectic form ωcut ∈ Ω2 (Mcut )S satisfying ωcut (ξMcut , Jcut (ξMcut )) > 0 on Mcut r
S1
Mcut
, and proper moment map Ψcut : Mcut → R described in Proposition 5.1. Then Ψcut (Mcut ) ⊆
1
(−∞, 0], and the fixed component F := Ψ−1
cut (0) is diffeomorphic to M //0 S . Moreover, there
exists an S1 -equivariant symplectomorphism from Ψ−1 (−∞, 0) to Ψ−1
cut (−∞, 0) that intertwines the moment maps and induces a biholomorphism between the reduced spaces at
all regular s < 0. Hence, the orbifold Mcut is smooth except at isolated Z2 -singularities
p1 , ..., pk ∈ F . Finally, Jcut tames ωcut on a neighbourhood W of F .
20
S1
f ⊆ (−∞, 0], we may
There exists ε > 0 so that Ψ−1
cut (−ε, 0) ⊆ W r M . Since Ψcut q M
also assume that (q ∗ Ψcut )−1 (−∞, −ε/2) 6= ∅.
By the equivariant Thom isomorphism theorem for oriented orbifolds, there exist β ∈
1
1
Ω (Mcut )S and χ ∈ Ω0 (Mcut )S such that dβ = 0, ξMcut y β = −dχ, χ(F ) 6= 0, and β and χ
2
are supported in Ψ−1
cut (−ε/2, 0]. By Lemma 4.8, we may assume that β vanishes (and so χ
is the constant χ(F )) on a neighbourhood of {p1 , . . . , pk }. Since Ψcut is proper, the support
of β is compact. Hence, there exists δ > 0 such that ωcut + t0 β is symplectic, tamed on W ,
S1
for all t0 ∈ (−δ, δ).
and satisfies (ωcut + t0 β)(ξMcut , Jcut (ξMcut )) > 0 on Mcut r Mcut
f, Je be the complex blow-up of Mcut at p1 , ..., pk , let q : M
f → Mcut be the blowLet M
down map, and let Ej be the Poincaré dual to the exceptional divisor Ej := q −1 (pj ) for
all j. There exist pairwise disjoint neighbourhoods Vj ⊂ Ψ−1
cut (−ε/2, 0] r supp(β) of pj for
all j ∈ {1, . . . , k}. By Proposition 7.7, for each j and for sufficiently small tj > 0 there
1
f r S El S that is tamed on q −1 (W ) and satisfies
exist a symplectic form ω
ej ∈ Ω2 M
l6=j
e f) > 0 on the complement of the fixed set, and a moment map Ψ
ej : M
fr
ω
e
ξ
,
J(ξ
j
f
M
S M
l6=j El → R so that
[ tj
f r El .
(13)
[e
ωj ] = q ∗ [ωcut ] − Ej ∈ H 2 M
2
l6=j
2
e j = q ∗ Ψcut on the complement of q −1 (Vj ). Given t0 ∈ (−δ, δ), we
Moreover, ω
ej = q ∗ ωcut and Ψ
1
f S and moment map
can glue these forms together to construct a symplectic form ω
e ∈ Ω2 M
e :M
f → R that are equal to q ∗ (ωcut +t0 β) and q ∗ (Ψcut +t0 χ), respectively, on M
fr S q −1 (Vj )
Ψ
j
0
−1
e
e
and equal to ω
ej and Ψj + t χ(F ) on q (Vj ) for each j. By construction, ω
e ξM
f, J(ξM
f) > 0
frM
fS1 . Moreover, since Ψcut and q are proper, and since Ψ
e = q ∗ Ψcut on M
fr
on M
e is proper.
(q ∗ Ψcut )−1 (−ε/2, 0], the moment map Ψ
By Lemma 2.2, for each j ∈ {1, . . . , k} there exists an S1 -equivariant biholomorphism
from an S1 -invariant neighbourhood of [0] ∈ Cn /Z2 to a neighbourhood of pj ∈ M , where
Z2 acts diagonally on Cn and S1 acts with weights −1, 0, . . . , 0. The corresponding S1 -action
n /Z is given by
\
on C
2
λ · [z1 , . . . , zn ; u] = [λ−1 z1 , z2 , . . . , zn ; u] = [z1 , λz2 , . . . , λzn ; λ−2 u]
n /Z . In particular, [z , . . . , z ; u] ∈ C
n /Z is fixed
\
\
for all λ ∈ S1 and [z1 , . . . , zn ; u] ∈ C
2
1
n
2
1
exactly if either z2 = · · · = zn = u = 0 or z1 = 0. Therefore, q −1 (F ) ∩ M S consists of
k isolated fixed points {pej ∈ Ej }kj=1 , each with weights {−2, 1, . . . , 1}, and a component Fe
that is the blow-up of F at p1 , . . . , pk . By (13), the restriction of [e
ω ] to Ej is tj times the
positive generator [Ω] of H 2 (CPn−1 ; Z); hence,
e Fe = Ψ
e Ej ∩ Fe = Ψ(e
e pj ) + tj .
Ψ
2π
e Fe) = Ψcut (F ) + t0 χ(F ) = t0 χ(F ), we can fix t0 ∈ (−δ, δ)
(Compare with (12).) Since also Ψ(
e pj ) = 0 for all j and Ψ(
e Fe) > 0. Moreover, if F 0 ⊂ M
f
and tj > 0 sufficiently small so that Ψ(e
2
If Mcut is a manifold, this follows from [1] (see also [4]); otherwise, Mcut is an oriented orbifold, and the
proof is similar.
21
e 0 ) = q ∗ Ψcut (F 0 ) ≤ −ε. This
is any other fixed component, then q ∗ Ψcut (F 0 ) ≤ −ε and so Ψ(F
proves Claim (1).
By construction, q restricts to an equivariant biholomorphic symplectomorphism from
(q Ψcut )−1 (−∞, −ε/2) to Ψ−1
cut (−∞, −ε/2) that intertwines the moment maps. Moreover,
∗
e
the fact that Ψ = q Ψcut on (q ∗ Ψcut )−1 (−∞, −ε/2] implies that
e −1 (−∞, −ε/2) = (q ∗ Ψcut )−1 (−∞, −ε/2)q Ψ
e −1 (−∞, −ε/2)∩(q ∗ Ψcut )−1 (−ε/2, ∞) . (14)
Ψ
∗
e is a proper moment map, the preimage Ψ
e −1 (−∞, −ε/2) is connected. Additionally,
Since Ψ
we chose ε so that (q ∗ Ψcut )−1 (−∞, −ε/2) 6= ∅. Hence, (14) implies that
e −1 (−∞, −ε/2) = (q ∗ Ψcut )−1 (−∞, −ε/2).
Ψ
This proves Claim (2).
Finally, ω
e is tamed on q −1 (W ) by construction. Moreover, since ω
e = q ∗ ωcut on the
e −1 (−ε, ε) ⊂ (q ∗ Ψcut )−1 (−ε, 0] ⊂
complement of (q ∗ Ψcut )−1 (−ε/2, 0], we have the inclusion Ψ
−1
q (W ). This proves Claim (3).
We can now generalize Proposition 4.1.
Proposition 7.8. Let (M, J) be a complex manifold with a holomorphic C× -action, a sym1
1
plectic form ω ∈ Ω2 (M )S satisfying ω(ξM , J(ξM )) > 0 on M r M S , and a proper moment
map Ψ : M → R. Assume that dimC (M ) > 2, and that Ψ−1 (a− , a+ ) contains exactly k
fixed points; each lies in Ψ−1 (0) and has weights {−2, 1, . . . , 1}. Then there exists a complex
orbifold (X, I) and classes κ, η ∈ H 2 (X) so that
• for all s ∈ (a− , 0), the reduced space M //s S1 is biholomorphically symplectomorphic
to (X, I, σs ), where σs ∈ Ω2 (X) satisfies [σs ] = κ + 2πs η; and
• for all s ∈ (0, a+ ), the reduced space M //s S1 is biholomorphically symplectomorphic
b I,
bσ
b satisfies
to (X,
bs ), where σ
bs ∈ Ω2 (X)
[b
σs ] = q ∗ κ + 2πs q ∗ η − πs
k
X
Ej .
j=1
b is the blow-up of X at isolated Z2 -singularities x1 , . . . , xk ∈ X, the map q : X
b →X
Here, X
−1
is the blow-down map, and Ej is the Poincaré dual of the exceptional divisor q (xj ) for all
j.
Proof. The first paragraph of the proof of Proposition 4.1 applies without change, except
that (5) should be replaced by
λ · (z1 , . . . , zn ) = (λ−2 z1 , λz2 , . . . , λzn ).
By Definition 7.2, U+ /C× is the blow-up of U− /C× at the isolated Z2 -singularities
a
{x1 , . . . , xk } :=
(C r {0}) × {0} /C× ;
j
22
moreover, if q : U+ /C× → U− /C× is the blow-down map, then h− ◦ q = h+ . The rest of the
argument is identical to that of the proof of Proposition 4.1, except that Wj+ r Ej is now
homotopy equivalent to RP2n−3 (instead of S2n−3 ) for all j and (6) should be replaced by
1X
Ej .
η+ = q ∗ (η− ) −
2 j
References
[1] M. F. Atiyah and R. Bott, The moment map and equivariant cohomology, Topology 23 (1984), 1–28.
[2] I. V. Dolgachev and Y. Hu, Variation of geometric invariant theory quotients, Inst. Hautes Études Sci.
Publ. Math. 87 (1998), 5-51.
[3] J. J. Duistermaat and G. J. Heckman, On the variation in the cohomology of the symplectic form of the
reduced phase space, Invent. Math. 69 (1982), 259-268.
[4] V. Guillemin, V. Ginzburg, and Y. Karshon, Moment Maps, Cobordisms, and Hamiltonian Group Actions, Appendix J by Maxim Braverman, Mathematical Surveys and Monographs, 98, American Mathematical Society, Providence, RI, 2002.
[5] L. Godinho, Blowing up symplectic orbifolds, Ann. Global Anal. Geom. 20 (2001), 117-162.
[6] V. Guillemin and S. Sternberg, Geometric quantization and multiplicities of group representations, Invent.
Math. 67 (1982), 515-538.
[7] V. Guillemin and S. Sternberg, Birational equivalence in the symplectic category, Invent. Math. 97 (1989),
485–522.
[8] N. J. Hitchin, A. Karlhede, U. Linström, M. Roček, Hyper-Kähler metrics and supersymmetry, Comm.
Math. Phys. 108 (1987), 535-589.
[9] F. C. Kirwan, Cohomology of quotients in symplectic and algebraic geometry, Mathematical Notes, 31,
Princeton University Press, Princeton, NJ, 1984.
[10] E. Lerman, Symplectic cuts, Math. Res. Lett. 2 (1995), 247-258.
[11] E. Lerman and S. Tolman, Hamiltonian torus actions on symplectic orbifolds and toric varieties, Tran.
Amer. Math. Soc. 349 (1997), 4201-4230.
[12] D. McDuff and D. Salamon, Introduction to symplectic topology, Oxford University Press, Oxford, 1998.
[13] R. Sjamaar, Holomorphic slices, symplectic reduction and multiplicities of representations, Ann. of Math.
(2) 141 (1995), 87-129.
[14] S. Tolman, Non-Hamiltonian actions with isolated fixed points,
preprint, http://www.math.uiuc.edu/~stolman/preprints.
Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois, USA 61801
E-mail address: [email protected]
Department of Mathematics, University of Colorado Boulder, Boulder, Colorado, USA
80309
E-mail address: [email protected]
23