# Kirwan surjectivity and Lefschetz-Sommese theorems for a generalized
hyperkähler reduction

Jonathan Fisher<EMAIL_ADDRESS>,  Lisa Jeffrey  Department of
Mathematics, University of Toronto, Canada<EMAIL_ADDRESS>,
Alessandro Malusà  Department of Mathematics, University of Toronto, Canada
<EMAIL_ADDRESS>and  Steven Rayan  Centre for Quantum Topology and
Its Applications (quanTA) and Department of Mathematics and Statistics,
University of Saskatchewan, Canada<EMAIL_ADDRESS>
###### Abstract.

Let $G$ be a compact Lie group. We study a class of Hamiltonian $(G\times
S^{1})$-manifolds _decorated_ with a function $s$ with certain equivariance
properties, under conditions on the $G$-action which we call of _(semi-)linear
type_. In this context, a close analogue of hyperkähler reduction is defined,
and our main result establishes surjectivity of an appropriate analogue of
Kirwan’s map. As a particular case, our setting includes a class of
hyperkähler manifolds with trihamiltonian torus actions, to which our
surjectivity result applies.

##  1\. Introduction

In light of the remarkable success of the Atiyah-Bott-Kirwan theory for
symplectic quotients [1, 11], it seems natural to consider Morse theory on a
hyperkähler manifold $M$ with function $\left|\mu_{\mathrm{G,s}}\right|^{2}$,
the norm squared of the trimoment map of a group action, in order to study the
cohomology ring of the corresponding hyperkähler quotient. However, one
immediately bumps into trouble due to the non-compactness of $M$. Under the
assumption that the gradient flow of $\left|\mu_{\mathbf{C}}\right|^{2}$
converges, the requisite Morse theory was developed in [10]. Partial results
on the convergence of the gradient flow were obtained by the first author [4],
but it appears that in general this convergence is difficult to establish.

In their study of hyperpolygon spaces, which arise as hyperkähler quotients by
a compact group $G$, two of the authors [5] established Kirwan surjectivity
using circle compactness with respect to an additional $S^{1}$-action. The
construction does not, in fact, use the whole hyperkähler structure, but only
one Kähler form for which the $S^{1}$-action is Hamiltonian. Correspondingly,
the quotient is viewed as the Kähler reduction of the subvariety cut by the
complex moment map for $G$, regarded simply as an equivariant holomorphic
function without using its relation to the group action.

In this work, we aim to extend that technique to a more general setting and
obtain similar results in cohomology, which is always understood to take
rational coefficients unless otherwise specified. Namely, we consider a Kähler
manifold $M$, acted on by $G\times S^{1}$ with moment map
$\mu=(\mu_{G},\mu_{S^{1}})$, and equipped with a $G$-invariant holomoprhic
function $s$. Assuming that $s$ is homogeneous of some positive degree with
respect to the $S^{1}$-action, we call this a _decorated $G$-manifold_. Under
appropriate regularity conditions we define the _hyper-reduction_ of $M$ to be

(1) $M/\\!/\\!/G=\bigl{(}\mu^{-1}(0)\cap s^{-1}(0)\bigr{)}/G\,,$

which can be regarded as a symplectic quotient of the complex analytic
subvariety $s^{-1}(0)$ and is in general a non-compact complex orbifold,
smooth if $G$ acts freely on the appropriate locus. The embeddings

(2) $\mu^{-1}(0)\cap s^{-1}(0)\hookrightarrow\mu^{-1}(0)\hookrightarrow M$

induce natural maps

(3) $\begin{gathered}\kappa\colon H^{*}_{G}(M)\to
H^{*}_{G}\bigl{(}\mu^{-1}(0)\bigr{)}\cong H^{*}(M/\\!/G)\,,\\\
\kappa_{G,s}\colon H^{*}_{G}(M)\to H^{*}_{G}\bigl{(}\mu^{-1}(0)\cap
s^{-1}(0)\bigr{)}\cong H^{*}(M/\\!/\\!/G)\end{gathered}$

in rational cohomology, which we call the _Kirwan map_ and _hyper-Kirwan map_
, respectively. Since $s$ is $G$-invariant, it also descends to a function on
$M/\\!/G$, and $M/\\!/\\!/G$ can equivalently be described as the zero locus
of the function on the quotient. The inclusion $\iota\colon
M/\\!/\\!/G\hookrightarrow M/\\!/G$ gives then a factorisation of the hyper-
Kirwan map as

(4)
$H^{*}_{G}(M)\stackrel{{\scriptstyle\kappa}}{{\longrightarrow}}H^{*}(M/\\!/G)\stackrel{{\scriptstyle\iota^{*}}}{{\longrightarrow}}H^{*}(M/\\!/\\!/G)\,.$

We approach surjectivity of $\kappa_{G,s}$ by studying the two maps $\kappa$
and $\iota^{*}$ separately. As it turns out, our results actually hold under
weaker conditions on $M$, which we may only required to be _almost_ Kähler or,
for some parts, just symplectic. Our main results are as follows.

######  Theorem 1 (Cf. Theorem 2.4).

Suppose $(M,\omega)$ is a Hamiltonian $G$-manifold, equipped with an
additional Hamiltonian $S^{1}$-action commuting with $G$. If
$0\in\mathfrak{g}^{\ast}$ is regular for $\mu_{G}$, $M$ and $M/\\!/G$ are
circle compact (see Definition 2.1), and $\overline{M/\\!/G}$ is smooth, then
the Kirwan map $\kappa\colon H^{*}_{G}(M)\to H^{*}(M/\\!/G)$ is surjective.

######  Theorem 2 (Cf. Theorem 3.9).

Suppose $M$ is an almost Kähler manifold equipped with a Hamiltonian $(G\times
S^{1})$-action and a holomorphic function $s$ valued in some complex vector
space $V$ such that

  1. (1)

$s$ is $G$-invariant and $d$-homogeneous of some positive degree for the
$S^{1}$-action;

  2. (2)

$0$ is a regular value of both $s$ and $\mu_{G}$, $\mu_{G}^{-1}(0)\pitchfork
s^{-1}(0)$, and $G$ acts freely on $\mu_{G}^{-1}(0)$;

  3. (3)

$M$ and $M/\\!/G$ are circle compact (see Definition 2.1) for their respective
$S^{1}$-actions.

Then the restriction map

(5) $H^{*}(M/\\!/G;R)\longrightarrow H^{*}(M/\\!/\\!/G;R)$

is an isomorphism for every coefficient ring $R$.

Combining these, we obtain the main result of this paper.

######  Theorem 3 (Cf. Theorem 3.10).

When the assumptions of both Theorems 1 and 2 are satisfied, the hyper-Kirwan
map is surjective.

Our theorem applies in particular to hypertoric varieties [2, 12] and
hyperpolygon spaces [13, 5].

######  Remark 4.

Independently, McGerty and Nevins have obtained a surjectivity result for a
large class of algebraic symplectic quotients, based on their results on the
derived categories of quotient stacks [17].

Our results are also related to those of [21].

###  1.1. Hypertoric case

The setup above includes the case of hypertoric manifolds, i.e. hyperkähler
manifolds with a trihamiltonian action of a compact abelian group $T$, in the
following way.

Recall that a hyperkähler manifold is a Riemannian manifold $(M,g)$ equipped
with three orthogonal parallel complex structures
$(\mathbf{i},\mathbf{j},\mathbf{k})$ satisfying the quaternion relations. We
denote by $(\omega_{1},\omega_{2},\omega_{3})$ the respective symplectic
(Kähler) forms. Focusing on the particular complex structure $\mathbf{i}$, it
is common to denote by $\omega_{\mathbf{R}}\coloneqq\omega_{1}$ and
$\omega_{\mathbf{C}}\coloneqq\omega_{2}+\sqrt{-1}\omega_{3}$ the _real_ and
_complex_ (or _holomorphic_) symplectic forms, respectively.

If a Lie group $G$ acts on $M$, we say the action is _trihamiltonian_ if there
exists a _hyperkähler moment map_ , i.e. a function

(6) $\mu_{G}=(\mu_{1},\mu_{2},\mu_{3})\colon
M\to\mathfrak{g}^{\ast}\otimes\mathbf{R}^{3}\,,$

where $\mathfrak{g}$ denotes the Lie algebra of $G$, such that each $\mu_{i}$
is an ordinary moment map with respect to $\omega_{i}$. Again, it is customary
to call $\mu_{\mathbf{R}}\coloneqq\mu_{1}$ and
$\mu_{\mathbf{C}}\coloneqq\mu_{2}+\sqrt{-1}\mu_{3}$ the _real_ and _complex_
(or _holomorphic_) moment maps. It is easy to see that $\mu_{\mathbf{C}}\colon
M\to\mathfrak{g}^{\ast}_{\mathbf{C}}$ is indeed holomorphic with respect to
the complex structure $\mathbf{i}$, and so its level sets are complex analytic
subvarieties.

Given such data, the hyperkähler quotient construction [9] is a quaternionic
(or holomorphic symplectic) analogue of the Marsden-Weinstein reduction.
Assuming that $0$ is a regular value for the real moment map, one can note
that $(0,0,0)$ is regular for the hyperkähler moment map. The Kähler and
hyperkähler quotients of $M$ are then defined, respectively, to be

(7) $\displaystyle M/\\!/G$
$\displaystyle\coloneqq\mu_{\mathbf{R}}^{-1}(0)/G\,,$ $\displaystyle
M/\\!/\\!/G$
$\displaystyle\coloneqq\mu_{G}^{-1}(0,0,0)/G=\bigl{(}\mu_{\mathbf{R}}^{-1}(0)\cap\mu_{\mathbf{C}}^{-1}(0)\bigr{)}/G\,.$

Although $M/\\!/\\!/G$ is naturally hyperkähler, if $G=T$ is abelian and one
only considers its orbifold or Kähler structure then taking
$s=\mu_{\mathbf{C}}$ this above is a particular case of hyper-reduction as
discussed above. Our theory may then be applied once an additional
$S^{1}$-action with the required properties is given—as in the setting of [5],
the latter will be assumed to act by hyperkähler rotations rather than fixing
the entire structure, as long as it is Hamiltonian for the Kähler form it
preserves.

### Plan of the exposition

Our paper is structured as follows. In Section 2, we will discuss the notion
of circle compactness and the relevant associated concepts, and prove Theorem
1. In Section 3, after introducing more definitions, we will show several
results relating the cohomology of manifolds with appropriate structure to
that of subspaces obtained as vanishing loci of functions or sections,
including Theorems 2 and 3. In the final Section 4 we focus on the case of a
hypertoric manifolds and show an extended version of Theorem 3 in that case.

### Acknowledgements

All authors wish to acknowledge the contributions of Young-Hoon Kiem, Frances
Kirwan, and Jon Woolf, who were co-authors of an earlier draft of this
manuscript. Many of the ideas in the present version originated in that
earlier draft. We would also like to acknowledge Tamás Hausel for carrying out
a detailed analysis which provided a counterexample to an argument stated in
that earlier version of this manuscript, which was written by JF and LJ
together with other authors. JF and LJ would like to thank Peter Crooks for
his reading of said prior version. AM and SR are grateful to JF and LJ for the
invitation to join the revised manuscript and for the ensuing collaborative
discussions. We also thank Jon Woolf for helpful comments at one point during
the preparation of the revised manuscript. LJ is partially supported by an
NSERC Discovery Grant. AM acknowledges generous funding from LJ’s Discovery
Grant as well as that of Marco Gualtieri. SR is partially supported by an
NSERC Discovery Grant, a Pacific Institute for the Mathematical Sciences
(PIMS) Collaborative Research Group (CRG) grant, and a Tri-Agency New
Frontiers in Research (NFRF) Exploration Stream grant.

##  2\. Kirwan surjectivity for circle compact manifolds

Throughout this section, $M$ will denote a symplectic manifold with a fixed
Hamiltonian $S^{1}$-action. We emphasise again that, wherever unspecified and
unless otherwise noted, all cohomology groups are taken with respect to
rational coefficients.

######  Definition 2.1.

$M$ is said to be _circle compact_ 111Also called _semiprojective_ [8] in the
Kähler case. if the $S^{1}$-action satisfies the following conditions:

  1. (1)

The fixed point set $M^{S^{1}}$ is compact;

  2. (2)

The $S^{1}$-moment map $\mu_{S^{1}}\colon M\to\mathbf{R}$ is proper and
bounded below.

Given a circle compact manifold $M$, we may construct a compactification
$\overline{M}$ as follows [16, 7]. Consider on $M\times\mathbf{C}$ the moment
map $\widetilde{\mu}_{S^{1}}$ of the diagonal $S^{1}$-action, and fix
$c\in\mathbf{R}$ larger than all the critical values of $\mu_{S^{1}}$. One
sees that

(8) $\widetilde{\mu}_{S^{1}}^{-1}(c)=\bigl{(}M^{<c}\times
S^{1}\bigr{)}\cup\mu_{S^{1}}^{-1}(c)\,,$

where $M^{<c}\coloneqq\mu_{S^{1}}^{-1}\bigl{(}(-\infty,c)\bigr{)}$ is, by
elementary Morse theory with the moment map, diffeomorphic to $M$ through an
equivariant map identifying $\mu_{S^{1}}$ with its restriction. Since $c$ is
regular for both $\mu_{S^{1}}$ and $\widetilde{\mu}_{S^{1}}$, one can write

(9)
$(M\times\mathbf{C}){/\\!/\\!}_{c}S^{1}=M^{<c}\cup\bigl{(}M{/\\!/\\!}_{c}S^{1}\bigr{)}\,,$

and it follows from the assumptions on $\mu_{S^{1}}$ that the above is
compact, and again by Morse theory different choices of $c$ produces
diffeomorphic orbifolds. Although $c$ does affect the symplectomorphism
classes of $M^{<c}$ and $\overline{M}$, as one may note by considering their
volumes, we shall neglect this dependence as it is of no consequence for our
arguments, and henceforth often identify $M^{<c}\subseteq\overline{M}$ with
$M$. There is also an additional Hamiltonian $S^{1}$-action on $\overline{M}$
descending from those on the first factor of $M\times\mathbf{C}$, and both
restrict the existing structure on $M$.

######  Definition 2.2.

The _cut compactification_ of a circle compact symplectic manifold $M$ is the
orbifold

(10) $\overline{M}\coloneqq(M\times\mathbf{C}){/\\!/\\!}_{c}S^{1}\,.$

The _boundary divisor_ of $M$ is

(11) $D_{M}\coloneqq\overline{M}\setminus M\cong M{/\\!/\\!}_{c}S^{1}\,.$

The term “boundary divisor”, while a slight abuse of notation in this context,
is inspired by the case where $M$ is Kähler.

######  Lemma 2.3.

If $M$ is circle compact, and $S^{1}$ acts freely on $\mu_{S^{1}}^{-1}(c)$,
then the natural restriction $H^{\ast}(\overline{M})\to H^{\ast}(M)$ is
surjective.

###### Proof.

Under our assumptions, both $\overline{M}$ and $D_{M}$ are smooth, the latter
an embedded submanifold. The boundary divisor is, by construction, the
absolute maximum locus of the moment map on $\overline{M}$. Therefore, it is a
critical component of index equal to its codimension, i.e. $2$, while all
other components are contained in $M$.

According to Proposition 5.8 of [11] (and 5.9 therein), the moment map of an
$S^{1}$-action on a compact symplectic manifold is perfect as a Morse-Bott
function. In her proof, Kirwan establishes the converse of the Morse
inequalities by inspecting the Serre spectral sequence of a homotopy quotient,
an argument that does not rely on compactness. The conclusion then holds
whenever Morse inequalities can be established, which is the case for $M$
since $\mu_{S^{1}}$ is proper and bounded below. We may then conclude that

(12) $P_{t}(\overline{M})=P_{t}(M)+t^{2}P_{t}(D_{M})\,,$

where $P_{t}$ denotes Poincaré polynomial in the variable $t$. Therefore, by
induction, the Thom-Gysin sequence of $D_{M}\subseteq\overline{M}$ splits into
short exact portions

(13) $0\longrightarrow H^{\ast-2}(D_{M})\longrightarrow
H^{\ast}(\overline{M})\longrightarrow H^{\ast}(M)\longrightarrow 0\,,$

whence the result. ∎

We now turn our attention to the case where $M$ is equipped with an additional
action of a compact Lie group $G$. Suppose the overall $G\times S^{1}$-action
is Hamiltonian, so in particular the moment map $\mu=(\mu_{G},\mu_{S^{1}})$ is
equivariant, and that $0$ is a regular value for $\mu_{G}$. In that case, the
$S^{1}$-action and its Hamiltonian function descend to the reduction
$M/\\!/G$, and it makes sense to talk about circle compactness of the latter
space.

######  Theorem 2.4.

Suppose $G$ is a compact Lie group, $(M,\omega)$ a Hamiltonian $G\times
S^{1}$-space with $0\in\mathfrak{g}^{\ast}$ as a regular value of the
$G$-moment map. If both $M$ and $M/\\!/G$ are circle compact, the latter with
smooth compactification, then the Kirwan map $\kappa\colon H^{*}_{G}(M)\to
H^{*}(M/\\!/G)$ is surjective.

###### Proof.

Let $\overline{M/\\!/G}$ denote the cut compactification of $M/\\!/G$ as in
Definition 2.2. By construction, the inclusion map
$M/\\!/G\hookrightarrow\overline{M/\\!/G}$ can be regarded as

(14) $\bigl{(}M\times\mathbf{C}^{*}\bigr{)}/\\!/\bigl{(}G\times
S^{1}\bigr{)}\hookrightarrow\bigl{(}M\times\mathbf{C}\bigr{)}/\\!/\bigl{(}G\times
S^{1}\bigr{)}\,,$

from which we obtain a commutative diagram

${H^{\ast}_{G\times
S^{1}}(M\times\mathbf{C})}$${H^{\ast}(\overline{M/\\!/G})}$${H^{\ast}_{G\times
S^{1}}(M\times\mathbf{C}^{\ast})}$${H^{\ast}(M/\\!/G)}$

Since $M$ is circle compact, in particular the Hamiltonian of the
$S^{1}$-action is proper, and therefore so is the full $G\times S^{1}$-moment
map on $M\times\mathbf{C}$. Therefore, the top horizontal map in the diagram
is surjective by usual Atiyah-Bott-Kirwan theory. Since the right vertical
arrow is surjective by Lemma 2.3, we may conclude that the bottom horizontal
map is also onto. On the other hand, the $S^{1}$-action on
$M\times\mathbf{C}^{*}$ is free, so $H^{*}_{G\times
S^{1}}(M\times\mathbf{C}^{*})\cong H^{*}_{G}(M)$. Thus $H^{*}_{G}(M)\to
H^{*}(M/\\!/G)$ is surjective. ∎

That concludes our results for symplectic manifolds with no additional
structure; starting with the next section, we will be requiring additional
structure on $M$.

##  3\. Lefschetz-Sommese theorems for actions of semi-linear type

The next step in our program is to study the effect induced on cohomology,
both ordinary and equivariant, by the operation of restricting to vanishing
loci of appropriate functions or sections. This is reminiscent of the Sommese
Theorem and the Lefschetz Hyperplane Theorem [20, 15, 3, 14]. For later
reference, let us recall their precise statements.

######  Theorem 3.1 (Sommese [20, 14]).

Suppose $X$ is a non-singular projective variety of dimension $n$, $E\to X$ an
ample vector bundle of rank $e$. If $\sigma$ is a section of $E$ and
$Z=\sigma^{-1}(0)$, then

(15) $H^{i}(X,Z;\mathbf{Z})=0\qquad\text{for $i\leq n-e$.}$

In particular, the restriction map $H^{i}(X;\mathbf{Z})\to
H^{i}(Z;\mathbf{Z})$ is an isomorphism for $i<n-e$ and an embedding for
$i=n-e$.

######  Theorem 3.2 (Lefschetz Hyperplane [3]).

Suppose $X$ is a compact complex manifold of dimension $n$, $E\to X$ a
positive holomorphic line bundle. If $\sigma$ is a section of $E$ intersecting
$0$ transversely and $Z=\sigma^{-1}(0)$, then

(16) $H^{i}(X,Z;R)=0\qquad\text{for $i\leq n-1$}$

for every coefficient ring $R$, and the restriction map $H^{i}(X;R)\to
H^{i}(Z;R)$ is an isomorphism for $i<n-1$ and an embedding for $i=n-1$.

Although the compactness condition is generally broken in our setting, we
shall overcome this difficulty by using cut compactifications, working
equivariantly, or both. Furthermore, it turns out that almost complex
structures are enough for some of our results, without requiring
integrability.

Before we start, a few definitions are in order. Henceforth we will assume
$(M,\omega)$ to be endowed with a compatible almost complex structure $I$ and
a Hamiltonian isometric $S^{1}$-action. In this case, if the manifold is
circle compact then the cut compactification $\overline{M}$ is itself an
almost Kähler manifold, and if $I$ is integrable then $D_{M}$ becomes a
divisor in the usual sense.

We will also assume $M$ to be decorated in the following sense.

######  Definition 3.3.

We call $M$ a _decorated manifold_ if it is endowed with a holomorphic
function $s$ valued in a vector space $V$, homogeneous of some positive degree
$d$ under the $S^{1}$-action, which is to say that

(17) $s(tx)=t^{d}s(x)$

for all $t\in S^{1}$ and $x\in M$. If $G$ is a compact Lie group, we will say
that $M$ is a _decorated $G$-manifold_ if it is equipped with a Hamiltonian
$G\times S^{1}$-action preserving $s$ and the almost Kähler structure. We will
denote $\mu=(\mu_{G},\mu_{S^{1}})$ the corresponding moment map.

Note that, if $M$ is circle compact under the $S^{1}$-action, $s$ does not
descend to a function on the cut compactification, but rather to a section of
an appropriate vector bundle $E$. Upon further inspection, when the
terminology makes sense $E$ is isomorphic to
$\underline{V}\otimes\mathcal{O}_{\overline{M}}(D_{M})^{d}$, where
$\underline{V}$ is the trivial vector bundle with fibre $V$.

######  Definition 3.4.

We will say that a decorated $G$-manifold is

  * •

_$s$ -regular_ if $0\in V$ is a regular value for $s$;

  * •

_$G$ -regular_ if $0\in\mathfrak{g}^{\ast}$ is a regular value for $\mu_{G}$
and the $G$-action is free on $\mu_{G}^{-1}(0)$;

  * •

_hyper-regular_ if it is both $s$\- and $G$-regular, and moreover
$\mu^{-1}_{G}(0)\pitchfork s^{-1}(0)$.

If $M$ is $G$-regular, we will denote $\check{\mu}_{S^{1}}$ and $\check{s}$
the induced $S^{1}$ moment map and holomorphic function on the almost Kähler
manifold $M/\\!/G$. If it is hyper-regular, we will call

(18) $M/\\!/\\!/G\coloneqq\bigl{(}\mu_{G}^{-1}(0)\cap s^{-1}(0)\bigr{)}/G\,.$

the _hyper-reduction_ of $M$.

As a degenerate case, every decorated manifold is regular for the action of
the trivial group $G=\\{1\\}$, hyper-regular if and only if $s$-regular.

######  Definition 3.5.

We will say that the $G\times S^{1}$-action (or simply the $G$-action for
brevity) on a $G$-regular decorated $G$-manifold is

  * •

of _semi-linear type_ if both $M$ and $M/\\!/G$ are circle compact for their
respective $S^{1}$-actions;

  * •

of _linear type_ if moreover the almost complex structure $I$ is integrable,
the circle compactification $\overline{M}$ is smooth, and the line bundle
$\mathcal{O}_{\overline{M}}(D_{M})$ is ample.

In the degenerate case of $G=\\{1\\}$, the condition of semi-linear type
reduces to circle compactness of $M$. We also stress that, in the hyper-
regular semi-linear case, one can show that $M/\\!/\\!/G$ is circle compact by
using that it sits inside $M/\\!/G$ as a closed submanifold. Furthermore, if
$M$ is a circle compact Kähler manifold, $D_{M}$ smooth, and
$\mathcal{O}_{\overline{M}}(D_{M})$ ample, then $\overline{M}$ is a projective
variety, and the same goes for $\overline{M/\\!/G}$ and
$\overline{M/\\!/\\!/G}$ given the appropriate data.

For some of the following applications, one or more of the conditions in these
definitions may be relaxed. For instance, in some cases it is enough to assume
that $M$ is acted on by a central extension of $G$ by $S^{1}$, not necessarily
trivial, with due adjustments to the equivariance properties of $s$. In
others, the regularity conditions on $\mu_{G}$ and $s$ may be weakened. In
fact, one could alternatively require that $0\in\mathfrak{g}^{\ast}\oplus V$
be a regular value for $\mu_{G}\oplus s$ and that $G$ act freely on
$(\mu_{G}\oplus s)^{-1}(0)$. This alone would be enough to guarantee that the
hyper-reduction is a well defined smooth manifold carrying a natural almost
Kähler structure, but in our assumptions we also have smoothness of
$Z\coloneqq s^{-1}(0)$ and $M/\\!/G$. The transversality condition then
implies that $0$ is a regular value for both $\mu_{G}|_{Z}$ and $\check{s}$,
so $M/\\!/\\!/G$ can be realised equivalently as either $Z/\\!/G$ or
$\check{s}^{-1}(0)$. Regardless, we shall adhere to the definitions given
above, for the sake of simplicity.

The following result will serve as a justification for our definition.

######  Theorem 3.6.

Suppose $G$ is acting linearly on $\mathbf{C}^{n}$ and consider the naturally
induced tri-hamiltonian action on $M\coloneqq T^{*}\mathbf{C}^{n}$. Consider
$M$ as a Kähler manifold with the complex struture corresponding to the
identification with $\mathbf{C}^{2n}$, and for
$V\coloneqq\mathfrak{g}^{\ast}_{\mathbf{C}}$ let $s\colon M\to V$ be the
complex moment map of the $G$-action. Under the natural $S^{1}$-action on
$\mathbf{C}^{2n}$, $s$ is $2$-homogeneous, $M$ is circle compact, the
compactification $\overline{M}$ is smooth, and the line bundle
$\mathcal{O}_{\overline{M}}(D_{M})$ is ample. Furthermore, if
$\gamma\in\mathfrak{g}^{\ast}$ is a regular value for $\mu_{G}$ and
$M{/\\!/\\!}_{\gamma}G$ is smooth, then the latter is also circle compact.

In other words, $M$ is a decorated $G$-manifold, and up to the regularity
conditions the action is of linear type.

###### Proof.

Since the origin is the only fixed point for the $S^{1}$-action, and its
moment map is $\mu_{S^{1}}(x)=\frac{1}{2}\lVert x\rVert^{2}$, $M$ is clearly
circle compact. The cut compactification is $\overline{M}\cong\mathbf{P}^{2n}$
with the Fubini-Study Kähler structure (possibly up to a rescaling) and
boundary divisor $D\simeq\mathbf{P}^{2n-1}$, the projective space at infinity.
The line bundle $\mathcal{O}_{\overline{M}}(D_{M})$ is then isomorphic to
$\mathcal{O}_{\mathbf{P}^{2n}}(1)$, which is ample.

Using coordinates $(\mathbf{x},\mathbf{y})$ on
$M\cong\mathbf{C}^{n}\oplus\mathbf{C}^{n}$, the $S^{1}$-action reads
$t.(\mathbf{x},\mathbf{y})=(t\mathbf{x},t\mathbf{y})$. On the other hand, the
complex symplectic form is

(19)
$\omega_{\mathbf{C}}=\sum_{i=1}^{n}\operatorname{d}\\!x_{i}\wedge\operatorname{d}\\!y_{i}\,,$

so that clearly $t^{*}\omega_{\mathbf{C}}=t^{2}\omega_{\mathbf{C}}$. It
follows that $s$ is also $2$-homogeneous as claimed, being determined by
$\omega_{\mathbf{C}}$ and the $S^{1}$-invariant $\mathfrak{g}$-action.

The only remaining condition to check is that
$(M{/\\!/\\!}_{\gamma}G)^{S^{1}}$ is compact, since properness and boundedness
descend immediately from $M$. By properness, this is equivalent to showing
that the fixed point set has only finitely many connected components. However,
Morse theory with $\mu_{S^{1}}$ implies that if there were infinitely many
connected components then $H^{i}(M{/\\!/\\!}_{\gamma}G)$ would have infinite
rank for some $i$. On the other hand, ordinary Kirwan surjectivity implies
that $H^{*}(M{/\\!/\\!}_{\gamma}G)$ is a quotient of $H^{*}(BG)$, so it is
finite-dimensional in every degree. ∎

###  3.1. Equivariant Lefschetz-Sommese theorem

We are now ready to state our first result about restriction maps.

######  Theorem 3.7.

If $M$ is a circle compact $s$-regular decorated $G$-manifold, then the
inclusion $s^{-1}(0)\hookrightarrow M$ induces an isomorphism

(20) $H^{*}_{K}(M;R)\to H^{*}_{K}(s^{-1}(0);R)$

for every subgroup $K<G\times S^{1}$ and every coefficient ring $R$.

In particular, choosing $K=\\{1\\}$ or $K=S^{1}$ gives equivalences in
ordinary and $S^{1}$-equivariant cohomology. Note that, unlike the Sommese
Theorem and the Lefschetz Hyperplane Theorem, our result states that the
restriction map is an isomorphism in all degrees, something which in the
projective case would be obstructed by Poincaré duality. In our setting,
however, $M$ is necessarily non-compact, otherwise $s$ would be constant and
therefore vanish everywhere by $S^{1}$-homogeneity, which would violate the
$s$-regularity condition.

###### Proof.

Throughout this proof, we shall call $Z\coloneqq s^{-1}(0)$ for simplicity.

Notice first of all that $H^{*}_{K}(Z;R)$ makes sense since $Z$ is preserved
by the $G\times S^{1}$-action. The proof will use Morse-Bott theory with the
moment map $\mu_{S}^{1}$, and it will be structured as follows. First, we will
show that $M^{S^{1}}\subseteq Z$, i.e. the critical locus of $\mu_{S^{1}}$ is
contained in $Z$. Second, we will argue that, at every fixed point $x\in
M^{S^{1}}$, the restriction of the isotropy $S^{1}$-action to the normal space
$N_{x}Z$ has positive weights, and therefore, the indices of $\mu_{S^{1}}$ are
the same whether computed on $M$ or on $Z$. Next, we will discuss how, for
every critical component $\mathcal{C}$ of $\mu_{S^{1}}$, the immersions of the
associated disc bundle $D\mathcal{C}$ into $M$ and $Z$ commute with the
inclusion $Z\hookrightarrow M$. Moreover, we will see that if
$M_{\mathcal{C}}^{\pm}$ and $Z_{\mathcal{C}}^{\pm}$ denote the associated
super- and sub-level sets then the homotopy equivalences
$M_{\mathcal{C}}^{+}\cong M_{\mathcal{C}}^{-}\cup D\mathcal{C}$ and
$M_{\mathcal{C}}^{+}\cong M_{\mathcal{C}}^{-}\cup D\mathcal{C}$ are $(G\times
S^{1})$-equivariant. In the final part of the proof, we will tie the pieces
together and use induction to conclude that the inclusion maps induce
isomorphisms at all levels in the Thom-Gysin sequences arising from the Morse
stratification.

For the first step of our proof, note that for every $x\in M^{S^{1}}$ and
$t\in S^{1}$ we have that

(21) $s(x)=s(tx)=t^{d}s(x)$

since $s$ is $d$-homogeneous for $S^{1}$, and therefore $s(x)=0$, i.e. $x\in
Z$.

For the second part, suppose that $x\in M^{S^{1}}$. The induced isotropy
actions on $T_{x}M$ and $T_{x}Z$ are then linear with respect to $I$, so we
obtain a short exact sequence of complex $S^{1}$-representations

(22) $0\longrightarrow T_{x}Z\longrightarrow T_{x}M\longrightarrow
N_{x}Z\longrightarrow 0\,.$

Notice now that if $\xi\in T_{x}M$ is a vector of weight $w$ for the isotropy
action, then

(23)
$t^{w}\operatorname{d}_{x}\\!s(\xi)=\operatorname{d}_{x}\\!s(t^{w}\xi)=\operatorname{d}_{x}\\!s(t_{*}\xi)=\bigl{(}t^{*}\operatorname{d}_{x}\\!s\bigr{)}(\xi)=\bigl{(}\operatorname{d}_{x}\\!(t^{*}s)\bigr{)}(\xi)=t^{d}\operatorname{d}_{x}\\!s(\xi)\,,$

so $w=d$ unless $\operatorname{d}_{x}\\!s(\xi)=0$. Since $d$ is positive and
the weights of this action are the eigenvalues of the Hessian of
$\mu_{S^{1}}$, it follows that the latter is positive definite on $N_{x}Z$.
Thus, the negative subspace of $T_{x}M$ must sit inside $T_{x}Z$, and
therefore $\mu_{S^{1}}$ has the same index on $T_{x}Z$ as on $T_{x}M$.

We now turn our attention to our third claim, which is that the attachment
maps from the Morse stratifications of $M$ and $Z$ are compatible with the
inclusion—to that end we consider the gradient flows defined by $\mu_{S^{1}}$
on $M$ and $Z$. On the one hand, at every point $x\in Z$ the gradient vector
$\nabla_{x}^{Z}\mu_{S^{1}}$ is a priori the orthogonal projection of
$\nabla_{x}^{M}\mu_{S^{1}}$ onto $T_{x}Z$. On the other hand,
$\nabla^{M}\mu_{S^{1}}$ can be written as $I\xi_{S^{1}}$, where $\xi_{S^{1}}$
is the vector field generating the $S^{1}$-action. Since $Z$ is an almost
complex submanifold preserved by $S^{1}$, it follows that
$\nabla_{x}^{M}\mu_{S^{1}}$ is tangent to $Z$, and therefore
$\nabla_{x}^{Z}\mu_{S^{1}}=\nabla_{x}^{M}\mu_{S^{1}}$. Moreover, by the circle
compactness assumption $\mu_{S^{1}}$ is bounded below and proper on $M$, and
therefore the steepest descent flow is forward-complete. We then obtain Morse
stratifications on $M$ and $Z$, and for every critical component $\mathcal{C}$
the gradient flow lines emanating from $\mathcal{C}$ lie in $Z$ at all times.
This shows that the embeddings of the corresponding disc bundle $D\mathcal{C}$
into $M$ and $Z$ are compatible with the inclusion $Z\hookrightarrow M$, as
claimed. To verify our statement about equivariance of the homotopy
equivalences, notice that, for every element
$a\in\mathfrak{g}\oplus\mathbf{R}$ we have

(24)
$[a^{\\#},\nabla\mu_{S^{1}}]=[a^{\\#},I\xi_{S^{1}}]=\bigl{(}\mathcal{L}_{a^{\\#}}I\bigr{)}\xi_{S^{1}}+I[a^{\\#},\xi_{S^{1}}]=0\,,$

since $S^{1}$ is central and $G\times S^{1}$ preserves $I$. Thus the $(G\times
S^{1})$-action commutes with the gradient flow, which is what defines the
homotopy equivalences in question.

We are now ready to approach the final part of our proof. For each critical
component $\mathcal{C}$ we may consider the pairs
$(M^{+}_{\mathcal{C}},M^{-}_{\mathcal{C}})$ and
$(Z^{+}_{\mathcal{C}},Z^{-}_{\mathcal{C}})$, their respective long exact
sequences, and the mapping between them induced by the inclusion maps
$Z_{\mathcal{C}}^{\pm}\hookrightarrow M_{\mathcal{C}}^{\pm}$. If
$D\mathcal{C}$ denotes the unit disc bundle in the negative normal bundle of
$\mathcal{C}$, the cohomology of each pair can be replaced by that of
$(D\mathcal{C},\partial D\mathcal{C})$ via the usual excision argument. We
then obtain the following as a portion of the resulting commutative diagram:

${H_{K}^{*-1}(M^{-}_{\mathcal{C}})}$${H_{K}^{*-1}(D\mathcal{C},\partial
D\mathcal{C})}$${H_{K}^{*}(M^{+}_{\mathcal{C}})}$${H_{K}^{*}(M^{-}_{\mathcal{C}})}$${H_{K}^{*}(D\mathcal{C},\partial
D\mathcal{C})}$${H_{K}^{*-1}(Z^{-}_{\mathcal{C}})}$${H_{K}^{*-1}(D\mathcal{C},\partial
D\mathcal{C})}$${H_{K}^{*}(Z^{+}_{\mathcal{C}})}$${H_{K}^{*}(Z^{-}_{\mathcal{C}})}$${H_{K}^{*}(D\mathcal{C},\partial
D\mathcal{C})}$

$H^{*}_{K}$ is understood to be taken with coefficients in $R$. Since
$\mu_{S^{1}}$ is proper and bounded below, it has a critical component
$\mathcal{C}_{0}$, for which
$M^{-}_{\mathcal{C}_{0}}=Z^{-}_{\mathcal{C}_{0}}=\emptyset$. The diagram then
splits into a family of squares and we may conclude that $\iota^{*}\colon
H^{*}_{K}(M^{+}_{\mathcal{C}_{0}})\to H^{*}_{K}(Z^{+}_{\mathcal{C}_{0}})$ is
an isomorphism. On the other hand, if for some critical component
$\mathcal{C}$ the first and fourth vertical maps are isomorphisms, then so is
the third by the Five Lemma. Our conclusion then follows by induction on the
critical sets, which are finitely many since the fixed locus $M^{S^{1}}$ is
compact under the circle compactness assumption. ∎

######  Corollary 3.8.

If $M$ is a circle compact $s$-regular decorated $G$-manifold and the
$G$-action is free on $s^{-1}(0)$, then the natural map

(25) $H^{*}_{G\times S^{1}}(M;R)\to H^{*}_{S^{1}}(s^{-1}(0)/G;R)$

is an isomorphism for every coefficient ring $R$.

###### Proof.

By Theorem 3.7, the restriction map $H^{*}_{G\times S^{1}}(M,R)\to
H^{*}_{G\times S^{1}}(s^{-1}(0),R)$ is an isomorphism. On the other hand, if
the $G$-action on $s^{-1}(0)$ is locally free, then

(26) $H^{*}_{G\times S^{1}}(s^{-1}(0),R)\cong H^{*}_{S^{1}}(s^{-1}(0)/G,R)\,,$

showing our claim. ∎

######  Theorem 3.9.

Suppose $M$ is a hyper-regular decorated $G$-manifold and the action is of
semi-linear type. Then the maps

(27) $H^{*}(M/\\!/G;R)\to H^{*}(M/\\!/\\!/G;R)\,\qquad\text{and}\qquad
H^{*}_{S^{1}}(M/\\!/G;R)\to H^{*}_{S^{1}}(M/\\!/\\!/G;R)$

induced by the inclusion $M/\\!/\\!/G\hookrightarrow M/\\!/G$ are isomorphisms
for every coefficient ring $R$.

###### Proof.

For convenience, call $Y=M/\\!/G$ and $Z=\check{s}^{-1}(0)=M/\\!/\\!/G$. By
the regularity assumption, $Y$ is smooth, and for every $y=[x]\in Y$ the space
$T_{y}Y$ can be realised as the orthogonal complement of $T_{x}(Gx)$ in
$T_{x}\bigl{(}\mu_{G}^{-1}(0)\bigr{)}$ with respect to the Riemannian metric
$g$. On the other hand, $T_{x}\bigl{(}\mu_{G}^{-1}(0)\bigr{)}$ is the
orthogonal complement of $IT_{x}(Gx)$ in $T_{x}M$, so $T_{y}Y$ can
equivalently be expressed as the orthogonal in $T_{x}M$ of
$\mathbf{C}T_{x}(Gx)$.

Under the identification above, $\operatorname{d}_{y}\\!\check{s}$ is the
restriction of $\operatorname{d}_{x}\\!s$. Notice now that if $s(x)=0$ then
$\operatorname{d}_{x}\\!s\colon T_{x}M\to V$ is surjective, since $0\in V$ is
a regular value of $s$. On the other hand, $G$-invariance of $s$ implies that
$\operatorname{d}_{x}\\!s$ vanishes on $T_{x}(Gx)$, and therefore on
$\mathbf{C}T_{x}(Gx)$ by $\mathbf{C}$-linearity. Therefore,
$\operatorname{d}_{x}\\!s$ remains surjective when restricted to the
orthogonal complement of $\mathbf{C}T_{x}(Gx)$ in $T_{x}M$, which is to say
that $\operatorname{d}_{y}\\!\check{s}$ is surjective on $T_{y}Y$ if
$\check{s}(y)=0$. In summary, $0\in V$ is a regular value for $\check{s}$,
which is clearly homogeneous for the $S^{1}$-action with the same degree as
$s$. In summary, $Y$ itself satisfies the hypotheses of Theorem 3.7, and the
conclusion follows. ∎

The following summarises the main surjectivity result as presented in Theorem
3.

######  Theorem 3.10.

If $M$ is a hyper-regular decorated $G$-manifold with an action of semi-linear
type, then the induced maps

(28) $H^{*}_{G}(M)\to H^{*}(M/\\!/\\!/G)\qquad\text{and}\qquad
H^{*}_{G}(s^{-1}(0))\to H^{*}(s^{-1}(0)/\\!/G)$

in _rational_ cohomology are surjective.

###### Proof.

The first map is the hyper-Kirwan map $\kappa_{G,s}$, and as noted in the
introduction it splits as the composition

(29)
$H^{*}_{G}(M)\stackrel{{\scriptstyle\kappa}}{{\rightarrow}}H^{*}(M/\\!/G)\stackrel{{\scriptstyle\iota^{*}}}{{\rightarrow}}H^{*}(M/\\!/\\!/G)\,.$

We have established that $\kappa$ is surjective in Theorem 2.4 and that
$\iota^{*}$ is an isomorphism in Theorem 3.9. We have also shown with Theorem
3.7 that $H^{*}_{G}(M)\cong H^{*}_{G}(s^{-1}(0))$, and since
$s^{-1}(0)/\\!/G=M/\\!/\\!/G$ the map $H^{*}_{G}(s^{-1}(0))\to
H^{*}(s^{-1}(0)/\\!/G)$ is essentially the same as the hyper-Kirwan map. ∎

###  3.2. Additional results on the cut compactifications

Throughout this section we have thus far only invoked the properties of the
$S^{1}$-actions under the circle compactness assumption without actually
involving the cut compactifications of $M$ and its reductions. The properties
of these spaces are the focus of the remaining theorems in this section, which
are direct consequences of the Sommese Theorem [20] and the Lefschetz
Hyperplane Theorem [3].

######  Theorem 3.11.

Suppose $M$ is a decorated manifold and the $S^{1}$-action is of linear type.
Call $\sigma\in\Gamma(\overline{M},E)$ the section induced by $s$ and
$k\coloneqq\dim_{\mathbf{C}}M-\dim_{\mathbf{C}}V$. Then the restriction map

(30) $H^{i}\bigl{(}\overline{M}\bigr{)}\longrightarrow
H^{i}\bigl{(}\sigma^{-1}(0)\bigr{)}$

is an isomorphism for $i<k$ and injective for $i=k$.

As in the Sommese theorem, no particular regularity assumptions on $s$ are
strictly required, and the locus $\sigma^{-1}(0)$ may be singular.

###### Proof.

Recall that, according to Definition 3.5, the action being of linear (and not
just semi-linear) type implies that the complex structure on $M$ is integrable
and the boundary divisor defines an ample line bundle, which makes
$\overline{M}$ a projective variety. Furthermore, $E$ is isomorphic to
$\underline{V}\otimes\mathcal{O}_{\overline{M}}(D_{M})^{\otimes d}$, which is
itself ample. Our conclusion follows then from the ordinary Sommese Theorem
(see Theorem 3.1). ∎

######  Theorem 3.12.

Suppose $M$ is a hyper-regular decorated $G$-manifold with an action of linear
type, and call $k\coloneqq\dim_{\mathbf{C}}M-\dim G-\dim_{\mathbf{C}}V$. If
the compactification $\overline{M/\\!/G}$ is smooth, then the restriction map

(31) $H^{i}\bigl{(}\overline{M/\\!/G}\bigr{)}\longrightarrow
H^{i}\bigl{(}\overline{M/\\!/\\!/G}\bigr{)}$

is an isomorphism for $i<k$ and injective for $i=k$.

In this case, the hyper-regularity assumption implies that
$k=\dim_{\mathbf{C}}\bigl{(}M/\\!/\\!/G\bigr{)}$.

###### Proof.

Call for convenience $X\coloneqq M/\\!/G$. By hyper-regularity, $X$ is a
smooth manifold and $\check{s}$ a holomorphic function with $0\in V$ as a
regular value. The action being of linear type, $X$ is also circle compact,
which makes it a decorated manifold, and its almost complex structure is
integrable. Since $\overline{M}$ is a compact Kähler manifold and
$\mathcal{O}_{\overline{M}}(D_{M})$ is ample, it is a projective variety, and
therefore so is $\overline{M}/\\!/G$. It is then clear from the identification
$\overline{X}\cong\overline{M}/\\!/G$ that $\mathcal{O}_{\overline{X}}(D_{X})$
is also ample.

We have shown that $X$ is a decorated manifold with $S^{1}$-action of linear
type, and since $M/\\!/\\!/G$ is the zero locus of the section on $X$
corresponding to $s$, the result follows from Theorem 3.11. ∎

######  Theorem 3.13.

Suppose $M$ is an $s$-regular decorated Kähler manifold, circle compact for
the $S^{1}$-action and with smooth compactification $\overline{M}$. Call
$\sigma\in\Gamma(\overline{M},E)$ the section induced by $s$ and
$k\coloneqq\dim_{\mathbf{C}}M-\dim_{\mathbf{C}}V$. Then for every coefficient
ring $R$ the restriction map

(32) $H^{i}\bigl{(}\overline{M};R\bigr{)}\longrightarrow
H^{i}\bigl{(}\sigma^{-1}(0);R\bigr{)}$

is an isomorphism for $i<k$ and injective for $i=k$.

As in the Lefschetz Hyperplane Theorem, $\overline{M}$ is not required to be
projective, but there is a regularity condition on $s$.

###### Proof.

We will first of all show that $\sigma$ intersects the $0$ section of $E$
transversely. This is clearly the case by $s$-regularity at every point
$p\in\sigma^{-1}(0)$ in the interior of $\overline{M}$, as $\sigma$ is
identified with $s$ locally around $p$. Suppose, instead, that
$p\in\sigma^{-1}(0)$ is a boundary point, and choose $\widetilde{p}\in M$
representing $p$. If $X$ denotes the vector field generating the
$S^{1}$-action on $M$, then $T_{p}D_{M}$ can be realised as the orthogonal
complement of $\\{X_{\widetilde{p}},IX_{\widetilde{p}}\\}$ in
$T_{\widetilde{p}}M$. On the one hand, we know by assumption that
$\operatorname{d}_{\widetilde{p}}\\!s$ is surjective on $T_{\widetilde{p}}M$.
On the other hand

(33) $X_{\widetilde{p}}[s]=id\cdot s(\widetilde{p})=0\,,$

and therefore $IX_{\widetilde{p}}[s]$ also vanishes since
$\operatorname{d}\\!s$ is $\mathbf{C}$-linear. Thus $\operatorname{d}\\!s$
remains surjective when restricted to the orthogonal complement of
$\\{X_{\widetilde{p}},IX_{\widetilde{p}}\\}$, which is enough to conclude that
$\sigma$ intersects the $0$ section transversely at $p$.

Now choose a basis of $V$, to which corresponds a splitting of $E$ as a direct
sum of copies of the line bundle
$L\coloneqq\mathcal{O}_{\overline{M}}(D_{M})^{\otimes d}$. Since $d>0$ and the
Euler class of $\mathcal{O}_{\overline{M}}(D_{M})$ is represented by the
Kähler form $\omega$, $L$ is positive. We will show the result by realising
$\sigma^{-1}(0)$ as the intersection of the zero loci of the components
$\sigma_{i}$ of $\sigma$. To this end, call $Z_{0}=\overline{M}$ and
$Z_{i}\coloneqq Z_{i-1}\cap\sigma_{i}^{-1}(0)$ for $i>0$. Since
$\sigma\pitchfork 0$ in $E$, one can see that $\sigma_{i}\pitchfork 0$ in
$L\rvert_{Z_{i-1}}$, so by induction $Z_{i}$ is a smooth compact Kähler
submanifold, on which $L$ remains positive. We are then in the setting of the
Lefschetz Hyperplane Theorem (see Theorem 3.2), and we can conclude that the
restriction map

(34) $H^{j}(Z_{i-1};R)\longrightarrow H^{j}(Z_{i};R)$

is an isomorphism for $j<\dim_{\mathbf{C}}M-i$ and an embedding for
$j=\dim_{\mathbf{C}}M-i$. The result follows by composition. ∎

######  Theorem 3.14.

Suppose $M$ is a hyper-regular decorated $G$-manifold with an action of semi-
linear type, with $I$ integrable and $\overline{M/\\!/G}$ smooth, and call
$k=\dim_{\mathbf{C}}M-\dim G-\dim_{\mathbf{C}}V$. Then for every coefficient
ring $R$ the restriction map

(35) $H^{i}(\overline{M/\\!/G};R)\longrightarrow
H^{i}(\overline{M/\\!/\\!/G};R)$

is an isomorphism for $i<k$ and injective for $i=k$.

Again, by hyper-regularity $k=\dim M/\\!/\\!/G$.

###### Proof.

Under our assumptions, $M/\\!/G$ is a decorated Kähler manifold, regular for
the induced function $\check{s}$, circle compact, and with smooth
compactification. The result follows from Theorem 3.13 after noticing that
$\overline{M/\\!/\\!/G}$ is the zero locus of the section $\check{\sigma}$
corresponding to $\check{s}$. ∎

##  4\. Hypertoric case

We now return to the setting that first inspired the present work: that of a
hyperkähler manifold $M$ with a tri-hamiltonian action of a compact Lie group
$G$. We will assume the latter is part of a $G\times S^{1}$-action which is
Hamiltonian for $\omega_{1}$, with $S^{1}$ acting by hyperkähler rotations. In
that case, the complex moment map $s\coloneqq\mu_{\mathbf{C}}\colon
M\to\mathfrak{g}^{\ast}_{\mathbf{C}}\eqqcolon V$ is $I$-holomorphic and
$S^{1}$-homogeneous of some degree $d$, which we shall assume positive. If $0$
is a regular value for the real moment map $\mu_{G}=\mu_{\mathbf{R}}$, then
the same is true for $s$, and $\mu_{G}^{-1}(0)\pitchfork s^{-1}(0)$.

While this makes $M$ (with the Kähler structure $I$) a decorated manifold, the
function $s$ is _$G$ -equivariant_ in general, rather than invariant, unless
$G$ is abelian. For that reason, in order to apply the results from the
previous sections we will need to replace $G$ with a torus $T$, and we will
say that $M$ is hypertoric. That being the case, if $M$ is regular (and
therefore hyper-regular) then the hyper-reduction $M/\\!/\\!/T$ is the usual
hyperkähler quotient.

######  Theorem 4.1.

Let $M$ be a hyperkähler manifold acted on isometrically by $T\times S^{1}$,
with the two factors tri-hamiltonian and $\omega_{1}$-Hamiltonian by positive
hyperkähler rotations, respectively. Suppose that the resulting decorated
$T$-manifold is hyper-regular and that the action is of semi-linear type. Then
the restriction map

(36) $H^{*}(M/\\!/T;R)\longrightarrow H^{*}(M/\\!/\\!/T;R)$

is an isomorphism for every coefficient ring $R$. Moreover,

(37) $H^{i}(M/\\!/\\!/T)=0\qquad\text{for
$i>\dim_{\mathbf{C}}(M/\\!/\\!/T)=\dim_{\mathbf{C}}M-2\dim T$,}$

and therefore the same is true for $H^{i}(M/\\!/T)$. Finally, the hyper-Kirwan
map

(38) $\kappa_{T,s}\colon H^{*}_{T}(M;\mathbf{Q})\longrightarrow
H^{*}(M/\\!/\\!/T;\mathbf{Q})$

in _rational_ cohomology is surjective.

The first part of this statement is now a consequence of Theorem 3.9. By
Theorem 2.4, the Kirwan map

(39) $\kappa\colon H^{*}_{T}(M;\mathbf{Q})\longrightarrow
H^{*}(M/\\!/T;\mathbf{Q})$

is surjective, and the same follows for the hyper-Kirwan map by the
isomorphism $H^{*}(M/\\!/T;\mathbf{Q})\cong H^{*}(M/\\!/\\!/T;\mathbf{Q})$.
The only remaining part is the vanishing statement, which is a consequence of
the following.

######  Theorem 4.2.

Suppose that $M$ is an almost Kähler manifold, circle compact for an
$S^{1}$-action preserving the complex structure. If $M$ admits an additional
complex symplectic form $\omega_{\mathbf{C}}$ which is homogeneous of positive
degree $d$ for the $S^{1}$-action, then $H^{i}(M)$ vanishes for
$k>\dim_{\mathbf{C}}M$.

###### Proof.

The argument is essentially the same as [19, Lemma 5.6], but we repeat it here
for completeness. By Morse theory with the $S^{1}$-moment map, the claim will
follow if we establish that, for every critical component $\mathcal{C}$ of
$M^{S^{1}}$, we have the inequality
$\lambda_{\mathcal{C}}+\dim_{\mathbf{R}}\mathcal{C}\leq\dim_{\mathbf{C}M}$,
where $\lambda_{\mathcal{C}}$ denotes the Morse index of $\mathcal{C}$.

To show the inequality, suppose $\mathcal{C}$ is such a component, and for
$x\in\mathcal{C}$ and $k\in\mathbf{Z}$ let $T_{x}M(k)\subseteq T_{x}M$ denote
the subspace of weight $k$ in the isotropy representation. Since these
subspaces coincide with the eigenspaces of the Hessian of $\mu_{S^{1}}$, one
obtains that $\lambda_{\mathcal{C}}=\sum_{k<0}\dim_{\mathbf{R}}T_{x}M(k)$ and
$T_{x}M(0)=T_{x}\mathcal{C}$, so

(40) $\lambda_{\mathcal{C}}+\dim_{\mathbf{R}}\mathcal{C}=\sum_{k\leq
0}\dim_{\mathbf{R}}T_{x}M(k).$

Given that $\omega_{\mathbf{C}}$ is homogeneous of degree $d>0$, it pairs the
weight spaces $T_{x}M(k)$ and $T_{x}M(d-k)$ non-degenerately. Since at least
one of $k$ or $d-k$ is positive, we find that every subspace of non-positive
weight is matched in dimension by a positive one. Therefore, the sum of the
dimensions of such subspaces is at most half the dimension of $T_{x}M$, i.e.

(41)
$\lambda_{\mathcal{C}}+\dim_{\mathbf{R}}\mathcal{C}\leq\frac{1}{2}\dim_{\mathbf{R}}M=\dim_{\mathbf{C}}M\,,$

which is the desired inequality. ∎

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