Lemma 3.13 and a result of Lagarias and Ziegler [33] imply the following
result.

######  Corollary 3.14.

Up to the action of $\mathrm{GL}(\ell^{*})$, there are only finitely many
reflexive polytopes in $\mathfrak{t}^{*}$.

For instance, there are sixteen two-dimensional reflexive polytopes (see [47,
Figure 2]), five of which are also Delzant (see Figure 3.1).

Figure 3.1. The five reflexive Delzant polygons.

For a rational polytope $\Delta\subset\mathfrak{t}^{*}$, given a vertex $v$ of
$\Delta$ one can choose the vectors $\alpha_{i}$’s in (2.14), which support
the edges coming out of $v$, to be primitive elements of $\ell^{*}$; these
vectors are uniquely determined and referred to as the weights of the vertex
$v$.

Reflexive Delzant polytopes are in particular rational and they can be
characterized in terms of the weights of their vertices. More precisely, the
following result, proved in various contexts by various authors, holds (see,
in particular, [13, Prop 1.8], [40, Sect. 3] and [16, Prop. 2.12]).

######  Proposition 3.15.

Let $\Delta\subset\mathfrak{t}^{*}$ be a $d$-dimensional Delzant polytope. The
following conditions are equivalent:

  1. (i)

$\Delta$ is a reflexive polytope.

  2. (ii)

$\Delta$ satisfies the weight sum formula, i.e., for each vertex $v\in\Delta$,

(3.2) $v=-\sum_{j=1}^{d}\alpha_{j}\,,$

where $\alpha_{1},\ldots,\alpha_{d}$ are the weights of $v$.

###### Remark 3.16

In [16, Prop. 2.12] it is assumed that the origin is an interior point of
$\Delta$ to prove that (ii) implies (i). However this follows by (3.2).
Indeed, consider the multiset $\mathcal{W}$ of all the primitive vectors
appearing as weights of vertices of $\Delta$. Note that, if
$\alpha\in\mathcal{W}$ has multiplicity $r$, then so does $-\alpha$.

Hence the sum of all the weights in $\mathcal{W}$ is $0\in\mathfrak{t}^{*}$.
Therefore, if $\Delta$ satisfies (3.2), then

(3.3) $\sum_{v\in\mathcal{V}}v=0\,,$

where $\mathcal{V}$ is the set of vertices of $\Delta$. Since $\Delta$ is the
convex hull of its vertices, the interior points of $\Delta$ are precisely
those that can be written as follows:

(3.4) $\sum_{v\in\mathcal{V}}\lambda_{v}v\quad\text{with
}\lambda_{v}>0\;\;\text{for all
}v\in\mathcal{V}\;\;\text{and}\;\;\sum_{v\in\mathcal{V}}\lambda_{v}=1\,.$

Let $\lambda_{v}=\frac{1}{|\mathcal{V}|}$ for all $v\in\mathcal{V}$. Then by
(3.3), we have $0=\sum_{v\in\mathcal{V}}\lambda_{v}v$. Hence, (3.4) yields
that $0$ belongs to the interior of $\Delta$.

The following technical lemma regarding reflexive Delzant polytopes is used
extensively in Sections 4 and 5 below.

######  Lemma 3.17.

Let $\Delta$ be a reflexive Delzant polytope in $\mathfrak{t}^{*}$, let
$\mathcal{F}$ be a facet of $\Delta$ supported on the affine hyperplane
$\\{w\in\mathfrak{t}^{*}\mid\langle w,\nu\rangle=-1\\}$, let $v$ be a vertex
of $\Delta$ in $\mathcal{F}$, and let
$\alpha_{1},\ldots,\alpha_{d}\in\ell^{*}$ be the weights of $v$ ordered so
that

(3.5) $\mathcal{F}\subset v+\mathbb{R}_{\geq
0}\langle\alpha_{1},\ldots,\alpha_{d-1}\rangle.$

Then $\langle\alpha_{d},\nu\rangle=1$. Moreover, if $e$ is the edge that is
incident to $v$ and comes out of $\mathcal{F}$, then there exists
$t_{\max}\in\mathbb{Z}_{>0}$ such that $e=\\{v+t\alpha_{d}\mid 0\leq t\leq
t_{\max}\\}$.

###### Proof.

By (3.5), $\langle\alpha_{i},\nu\rangle=0$ for all $i=1,\ldots,d-1$. By
Proposition 3.15, $\Delta$ satisfies the weight sum formula at $v$. Since
$v\in\mathcal{F}$, $\langle\alpha_{d},\nu\rangle=1$.

If $e$ is an edge of $\Delta$ as in the statement, then there exists
$t_{\max}>0$ such that $e=\\{v+t\alpha_{d}\mid 0\leq t\leq t_{\max}\\}$. It
remains to show that $t_{\max}$ is a positive integer. To this end, we observe
that $v^{\prime}:=v+t_{\max}\alpha_{d}$ is a vertex of $\Delta$. Since
$\Delta$ is reflexive, $v^{\prime}\in\ell^{*}$. By definition of weight,
$\alpha_{d}$ is primitive in $\ell^{*}$. Hence, $t_{\max}\in\mathbb{Z}_{>0}$.
∎

To conclude this section, we look at the relation between normalized monotone
complexity preserving $T$-spaces and reflexive Delzant polytopes in
$\mathfrak{t}^{*}$. We start with the following strengthening of Proposition
2.73 under the additional assumption of monotonicity.

######  Proposition 3.18.

The moment map image of a normalized monotone complexity preserving $T$-space
is a reflexive Delzant polytope in $\mathfrak{t}^{*}$. Conversely, for every
reflexive Delzant polytope $\Delta$ in $\mathfrak{t}^{*}$ and for every
integer $k\geq 0$, there exists a normalized monotone complexity preserving
Hamiltonian $T$-space $\left(M,\omega,\Phi\right)$ of complexity $k$ such that
$\Phi(M)=\Delta$.

Before proving Proposition 3.18, we recall the following result, stated below
without proof (see [16, Proposition 3.10]).

######  Proposition 3.19.

Let $\left(M,\omega,\Phi\right)$ be a compact symplectic toric manifold. If
$\Phi(M)$ is a reflexive Delzant polytope, then $\left(M,\omega,\Phi\right)$
is normalized monotone.

###### Proof of Proposition 3.18.

Let $\left(M,\omega,\Phi\right)$ be a normalized monotone complexity
preserving $T$-space. By Proposition 2.73, $\Phi(M)$ is a Delzant polytope in
$\mathfrak{t}^{*}$. It remains to show that $\Phi(M)$ is reflexive. Since
$\left(M,\omega,\Phi\right)$ is complexity preserving, by part (i) of
Corollary 2.34, the weights of $\Phi(M)$ at a vertex $v$ are equal to the non-
zero weights of any $p\in\Phi^{-1}(v)$. Since $\left(M,\omega,\Phi\right)$ is
normalized monotone, $\Phi$ satisfies the weight sum formula (3.1). Hence,
$\Phi(M)$ also satisfies the weight sum formula (3.2) and so, by Proposition
3.15, it is reflexive.

Conversely, let $\Delta$ be a reflexive Delzant polytope in $\mathfrak{t}^{*}$
and let $k\geq 0$ be an integer. We adapt the second half of the proof of
Proposition 2.73 (and fix the notation therein), to show that we can make
appropriate choices so that the resulting complexity preserving $T$-space of
complexity $k$ is normalized monotone. Since $\Delta$ is reflexive, by
Proposition 3.19, $(M^{\prime},\omega^{\prime})$ is normalized monotone.
Choose $M^{\prime\prime}=\mathbb{C}P^{k}$ and $\omega^{\prime\prime}$ to be a
monotone symplectic form on $\mathbb{C}P^{k}$ such that
$c_{1}(\mathbb{C}P^{k})=[\omega^{\prime\prime}]$. The complexity preserving
$T$-space $(M,\omega,\Phi)$ constructed in the second half of the proof of
Proposition 2.73 has complexity $k$ and is normalized monotone, as desired. ∎

We finish with the following simple, useful result.

######  Lemma 3.20.

Let $\left(M,\omega,\Phi\right)$ be a compact monotone complexity preserving
$T$-space. If $\Phi(M)$ is reflexive Delzant, then
$\left(M,\omega,\Phi\right)$ is normalized monotone.

###### Proof.

If $\dim M=0$, there is nothing to prove, so we may assume $\dim M>0$. Let $k$
be the complexity of $\left(M,\omega,\Phi\right)$. Since $(M,\omega)$ is
monotone, by the proof of Proposition 3.5, there exists a unique constant
$w\in\mathfrak{t}^{*}$ such that

$c^{T}_{1}+w=\lambda[\omega-\Phi].$

We fix a vertex $v\in\Phi(M)$ and a fixed point $p\in\Phi^{-1}(v)$. Evaluating
both sides of the above displayed equality at $p$,

(3.6) $\sum\limits_{i=1}^{n-k}\alpha_{i}+c=-\lambda v,$

where $\alpha_{1},\ldots,\alpha_{n-k}$ are the non-zero weights of $p$. Since
$\left(M,\omega,\Phi\right)$ is complexity preserving, the non-zero isotropy
weights of $p$ in $M$ are precisely the weights of $v$ in $\Phi(M)$. Hence, by
Proposition 3.15, (3.6) gives that $-v+w=-\lambda v$. Since this equality
holds for any vertex $v\in\Phi(M)$, we have $w=0$ and $\lambda=1$, as desired.
∎

##  4\. Complete invariants of compact monotone tall complexity one $T$-spaces

###  4.1. The genus and a minimal facet

In this section we explore the first consequences of the combination of
tallness and monotonicity of compact complexity one $T$-spaces, recovering and
extending some of the results in [49]. To this end, let
$\left(M,\omega,\Phi\right)$ be a monotone tall complexity one $T$-space of
dimension $2n$ and let $v\in\Phi(M)$ be a vertex. Let
$N=L_{1}\oplus\ldots\oplus L_{n-1}$ be the normal bundle to
$\Sigma:=\Phi^{-1}(v)$ in $M$ together with its $T$-equivariant splitting into
$T$-invariant complex line bundles as in (2.31).

######  Lemma 4.1.

Let $\left(M,\omega,\Phi\right)$ be a compact monotone tall complexity one
$T$-space of dimension $2n$ and let $v\in\Phi(M)$ be a vertex that attains the
minimum of $DH\left(M,\omega,\Phi\right)$. If $c_{1}(\Sigma),c_{1}(L_{i})$
denote the first Chern class of $\Sigma$ and of the complex line bundle
$L_{i}$ for any $i=1,\ldots,n-1$ respectively, then

(4.1) $c_{1}(\Sigma)[\Sigma]>-\sum\limits_{i=1}^{n-1}c_{1}(L_{i})[\Sigma].$

Moreover, the genus of $\left(M,\omega,\Phi\right)$ in the sense of Definition
2.76 equals zero.

###### Proof.

Since $(M,\omega)$ is monotone, by Proposition 3.3, it is positive monotone.
Hence, since $\Sigma$ is a symplectic submanifold of $(M,\omega)$,

(4.2)
$0<c_{1}[\Sigma]=c_{1}(\Sigma)[\Sigma]+c_{1}(N)[\Sigma]=c_{1}(\Sigma)[\Sigma]+\sum\limits_{i=1}^{n-1}c_{1}(L_{i})[\Sigma],$

whence (4.1) holds. By Lemma 2.63, the right hand side of (4.1) is non-
negative, which implies that $\Sigma$ is diffeomorphic to a sphere, as
desired. ∎

Lemma 4.1 is a stepping stone towards the following important result.

######  Proposition 4.2.

Let $\left(M,\omega,\Phi\right)$ be a compact monotone tall complexity one
$T$-space of dimension $2n$. There exists a facet $\mathcal{F}$ of $\Phi(M)$
such that $DH(M_{\mathcal{F}},\omega_{\mathcal{F}},\Phi_{\mathcal{F}})$ is
constant and equal to the minimum of $DH\left(M,\omega,\Phi\right)$, where
$(M_{\mathcal{F}},\omega_{\mathcal{F}},\Phi_{\mathcal{F}})$ is defined as in
(2.19). Moreover, for any vertex $v\in\mathcal{F}$, there exist $n-2$ non-zero
isotropy weights $\alpha_{1},\ldots,\alpha_{n-2}$ of $\Phi^{-1}(v)$ such that

  * •

$\mathcal{F}\subset v+\mathbb{R}_{\geq
0}\langle\alpha_{1},\ldots,\alpha_{n-2}\rangle$, and

  * •

the self-intersection of $\Phi^{-1}(v)$ in $\Phi^{-1}(e_{j})$ equals zero for
all $j=1,\ldots,n-2$, where $e_{j}\subset\mathcal{F}$ is the edge incident to
$v$ with tangent vector given by $\alpha_{j}$.

###### Proof.

By Proposition 2.75, $\left(M,\omega,\Phi\right)$ is complexity preserving.
Hence, by Corollary 2.71, for any face $\tilde{\mathcal{F}}$ of $\Phi(M)$

$DH(M_{\tilde{\mathcal{F}}},\omega_{\tilde{\mathcal{F}}},\Phi_{\tilde{\mathcal{F}}})=DH\left(M,\omega,\Phi\right)|_{\tilde{\mathcal{F}}}.$

Therefore, to prove the first statement, it suffices to prove that there
exists a facet $\mathcal{F}$ of $\Phi(M)$ such that
$DH\left(M,\omega,\Phi\right)|_{\mathcal{F}}$ is constant and equal to the
minimum of $DH\left(M,\omega,\Phi\right)$ – see Figure 4.1.

By Corollary 2.60, the minimum of $DH\left(M,\omega,\Phi\right)$ is attained
at a vertex of $\Phi(M)$, say $v_{0}$. Let $\alpha_{1},\ldots,\alpha_{n-1}$ be
the non-zero isotropy weights of the fixed surface $\Sigma:=\Phi^{-1}(v_{0})$.
By Lemma 2.63,

$DH\left(M,\omega,\Phi\right)\left(v_{0}+\sum\limits_{i=1}^{n-1}t_{i}\alpha_{i}\right)=\int\limits_{\Sigma}\omega-\sum\limits_{i=1}^{n-1}t_{i}c_{1}(L_{i})[\Sigma]$

for all $t_{1},\ldots,t_{n-1}\geq 0$ sufficiently small. By Lemma 4.1,
$2>-\sum\limits_{i=1}^{n-1}c_{1}(L_{i})[\Sigma]$; moreover, by Lemma 2.63,
$c_{1}(L_{i})[\Sigma]\leq 0$ for all $i=1,\ldots,n-1$. Thus at least $n-2$ of
$c_{1}(L_{1})[\Sigma],\ldots,c_{1}(L_{n-1})[\Sigma]$ vanish; there is no loss
of generality in assuming that $c_{1}(L_{i})[\Sigma]=0$ for all
$i=1,\ldots,n-2$. Hence,

$DH\left(M,\omega,\Phi\right)\left(v_{0}+\sum\limits_{i=1}^{n-1}t_{i}\alpha_{i}\right)=\int\limits_{\Sigma}\omega-
t_{n-1}c_{1}(L_{n-1})[\Sigma]$

for all $t_{1},\ldots,t_{n-1}\geq 0$ sufficiently small. In particular, the
restriction of $DH\left(M,\omega,\Phi\right)$ to a sufficiently small
neighborhood of $v_{0}$ in

$(v_{0}+\mathbb{R}_{\geq
0}\langle\alpha_{1},\ldots,\alpha_{n-2}\rangle)\cap\Phi(M)$

is constant and equal to $DH\left(M,\omega,\Phi\right)(v_{0})$, which, by
assumption, is the minimum of $DH\left(M,\omega,\Phi\right)$. Let
$\mathcal{F}$ be the facet of $\Phi(M)$ that is contained in
$v_{0}+\mathbb{R}_{\geq 0}\langle\alpha_{1},\ldots,\alpha_{n-2}\rangle$. Since
$DH\left(M,\omega,\Phi\right)$ is concave (see Proposition 2.59), since
$DH\left(M,\omega,\Phi\right)$ attains its minimum at $v_{0}$, and since
$v_{0}\in\mathcal{F}$, $DH\left(M,\omega,\Phi\right)|_{\mathcal{F}}$ is
constant and equal to the minimum of $DH\left(M,\omega,\Phi\right)$. Moreover,
since $c_{1}(L_{i})[\Sigma]=0$, the self-intersection of $\Sigma$ in
$\Phi^{-1}(e_{i})$ is zero for each $i=1,\ldots,n-2$.

It remains to show that the bullet points in the statement hold for all
vertices of $\mathcal{F}$. However, since
$DH\left(M,\omega,\Phi\right)|_{\mathcal{F}}$ is constant and equal to the
minimum of $DH\left(M,\omega,\Phi\right)$, it follows that any vertex of
$\mathcal{F}$ is a minimum of $DH\left(M,\omega,\Phi\right)$. Hence, the above
argument gives the desired result. ∎

Figure 4.1.

A facet as in Proposition 4.2 plays an important role throughout Section 4.

######  Definition 4.3.

Let $\left(M,\omega,\Phi\right)$ be a compact monotone tall complexity one
$T$-space. A facet of $\Phi(M)$ satisfying the conclusions of Proposition 4.2
is called a minimal facet of $\Phi(M)$ and denoted by
$\mathcal{F}_{\mathrm{min}}$. Given a minimal facet $\mathcal{F}_{\min}$, the
sheet corresponding to $\mathcal{F}_{\min}$ is denoted by
$(M_{\min},\omega_{\min},\Phi_{\min})$.

We observe that, in spite of the notation,
$(M_{\min},\omega_{\min},\Phi_{\min})$ clearly depends on
$\mathcal{F}_{\min}$. However, we trust that the notation does not cause
confusion.

######  Corollary 4.4.

Let $\left(M,\omega,\Phi\right)$ be a compact monotone tall complexity one
$T$-space. If $\mathcal{F}_{\min}$ is a minimal facet of $\Phi(M)$, then
$M_{\min}$ contains no isolated fixed point of $\left(M,\omega,\Phi\right)$.

###### Proof.

By Remark 2.70 and Proposition 2.75,
$(M_{\mathrm{min}},\omega_{\mathrm{min}},\Phi_{\mathrm{min}})$ is a compact
tall complexity one $T/H_{\mathcal{F}_{\min}}$-space. By Corollary 2.71 and
Proposition 4.2,
$DH(M_{\mathrm{min}},\omega_{\mathrm{min}},\Phi_{\mathrm{min}})$ is constant.
Hence, by Lemma 2.62, there are no singular values in the (relative) interior
of $\Phi_{\mathrm{min}}(M_{\min})$. Thus, by Lemma 2.72,
$(M_{\mathrm{min}},\omega_{\mathrm{min}},\Phi_{\mathrm{min}})$ has no isolated
fixed points for the $T/H_{\mathcal{F}_{\min}}$-action and, hence, for the
$T$-action. ∎

To conclude this section, we prove that certain self-intersections of the pre-
image of vertices in a minimal facet are independent of the vertices. To this
end, we say that an edge $e$ of a polytope $\Delta$ comes out of a facet
$\mathcal{F}$ if it is not contained in $\mathcal{F}$ but it is incident to a
vertex of $\Delta$ contained in $\mathcal{F}$.

######  Lemma 4.5.

Let $\left(M,\omega,\Phi\right)$ be a compact monotone tall complexity one
$T$-space and let $\mathcal{F}_{\min}$ be a minimal facet of $\Phi(M)$. There
exists $s\in\mathbb{Z}$ such that, for any vertex $v\in\mathcal{F}_{\min}$,
the self-intersection of $\Phi^{-1}(v)$ in $\Phi^{-1}(e)$ equals $s$, where
$e$ is the edge of $\Phi(M)$ that comes out of $\mathcal{F}_{\min}$ and is
incident to $v$.

###### Proof.

Let $v_{1},v_{2}\in\mathcal{F}_{\min}$ be vertices and let $e_{1},e_{2}$ be
the edges that come out of $\mathcal{F}_{\min}$ that are incident to $v_{1}$
and to $v_{2}$ respectively. By Proposition 4.2,
$DH\left(M,\omega,\Phi\right)(v_{1})=DH\left(M,\omega,\Phi\right)(v_{2})$. Let
$\Sigma_{i}:=\Phi^{-1}(v_{i})$ for $i=1,2$. Thus, by Lemma 2.63,
$[\omega](\Sigma_{1})=[\omega](\Sigma_{2})$. Since $(M,\omega)$ is monotone,
it is positive monotone by Proposition 3.3. Hence,
$c_{1}(\Sigma_{1})=c_{1}(\Sigma_{2})$. Moreover, by Proposition 4.2, the only
self-intersection of $\Sigma_{i}$ that can possibly be different from zero is
that in $\Phi^{-1}(e_{i})$, for $i=1,2$. Hence, since
$\Sigma_{1}\simeq\Sigma_{2}$, the result follows. ∎

###  4.2. A characterization of isolated fixed points

Henceforth, we assume that $\left(M,\omega,\Phi\right)$ is a normalized
monotone tall complexity one $T$-space unless otherwise stated. Moreover, we
fix a minimal facet $\mathcal{F}_{\min}$ of $\Phi(M)$ (which exists by
Proposition 4.2). By Proposition 3.18, $\Phi(M)$ is a reflexive Delzant
polytope. Hence, by Definition 3.12, there exists a unique primitive
$\nu_{\min}\in\ell$ such that

(4.3)
$\begin{split}\mathcal{F}_{\min}&\subset\\{w\in\mathfrak{t}^{*}\mid\langle
w,\nu_{\min}\rangle=-1\\},\\\
\Phi(M)&\subset\\{w\in\mathfrak{t}^{*}\mid\langle
w,\nu_{\min}\rangle\geq-1\\}.\end{split}$

Finally, if $v\in\mathcal{F}_{\min}$ is any vertex and
$\alpha_{1},\ldots,\alpha_{n-1}$ are the non-zero isotropy weights of
$\Phi^{-1}(v)$, then, by Proposition 4.2, we may assume that the weights are
ordered so that

(4.4) $\mathcal{F}_{\min}\subset v+\mathbb{R}_{\geq
0}\langle\alpha_{1},\ldots,\alpha_{n-2}\rangle.$

In particular, by Lemma 3.17,

(4.5) $\langle\alpha_{n-1},\nu_{\min}\rangle=1.$

######  Proposition 4.6.

Let $\left(M,\omega,\Phi\right)$ be a normalized monotone tall complexity one
$T$-space of dimension $2n$. If $p\in M^{T}$ is isolated, then there exists an
edge $e$ of $\Phi(M)$ that comes out of $\mathcal{F}_{\min}$ such that
$\Phi(p)$ is (the only element) in the intersection of $e$ with the linear
hyperplane $\\{w\in\mathfrak{t}^{*}\mid\langle w,\nu_{\min}\rangle=0\\}$.

Moreover, if $v\in\mathcal{F}_{\min}$ is the vertex that is incident to $e$
and if $\alpha_{1},\ldots,\alpha_{n-1}$ are the non-zero isotropy weights of
$\Phi^{-1}(v)$ ordered so that (4.4) holds, then the isotropy weights of $p$
are

(4.6) $\alpha_{1},\ldots,\alpha_{n-2},\alpha_{n-1},-\alpha_{n-1}.$

Figure 4.2. A schematic illustration of Proposition 4.6. The non-zero isotropy
weights of $\Phi^{-1}(v)$ are precisely $\alpha_{1}$ and $\alpha_{2}$.

###### Proof.

Since $\left(M,\omega,\Phi\right)$ is normalized monotone, by Corollary 3.8,
$\Phi(p)\in\ell^{*}$. Hence, since $\nu_{\min}\in\ell$,
$\langle\Phi(p),\nu_{\min}\rangle\in\mathbb{Z}$. Moreover, by Corollary 4.4,
$\Phi(p)\notin\mathcal{F}_{\min}$; hence, by (4.3),
$\langle\Phi(p),\nu_{\min}\rangle$ is a non-negative integer. Let
$\beta_{1},\ldots,\beta_{n}\in\ell^{*}$ be the isotropy weights of $p$. By
Lemma 2.39, there exists an isotropy weight $\beta$ of $p$ such that
$\langle\beta,\nu_{\min}\rangle<0$. Without loss of generality, we may assume
that $\beta_{n}=\beta$. Let $(N,\omega_{N},\Phi_{N})$ be the sheet along
$\beta_{n}$ given by Lemma 2.37, let

$H=\exp\left(\\{\xi\in\mathfrak{t}\mid\langle\beta_{n},\xi\rangle\in\mathbb{Z}\\}\right)$

be its stabilizer, and let $q\in M^{T}\cap N$ be a fixed point satisfying the
conclusions of Corollary 2.41, i.e.,

  * •

$\Phi(q)=\Phi_{N}(q)$ is a global extremum of $\Phi_{N}$,

  * •

$-\beta_{n}$ is an isotropy weight of $q$, and $\alpha_{1}$ $\alpha_{2}$
$-\alpha_{2}$ $\mathcal{F}_{\min}$

(4.7) $\langle\Phi(q),\nu_{\min}\rangle<\langle\Phi(p),\nu_{\min}\rangle.$

We split the proof in two cases: first, we assume that
$\langle\Phi(p),\nu_{\min}\rangle$ is minimal among isolated fixed points and,
second, we deduce the general case from this special one.

Case 1: We suppose that $\langle\Phi(p),\nu_{\min}\rangle$ is minimal among
isolated fixed points. By (4.7), the fixed point $q$ is not isolated. Hence,
it lies on a fixed surface. By Proposition 2.67, $v$ is a vertex of $\Phi(M)$.
Let $\alpha_{1},\ldots,\alpha_{n-1}$ be the non-zero isotropy weights of
$\Phi^{-1}(\Phi(q))$ ordered so that $\alpha_{n-1}=-\beta_{n}$. By Proposition
2.73, $\alpha_{1},\ldots,\alpha_{n-1}$ are a basis of $\ell^{*}$. Hence,
$\beta_{n}$ is a primitive element in $\ell^{*}$. Moreover, since $H$ must be
one of the stabilizers of dimension $n-2$ for points sufficiently close to $q$
in $M$, it follows that $\dim N=4$ and that
$\Phi(p)+\mathbb{R}\langle\beta_{n}\rangle$ contains an edge of $\Phi(M)$.
Hence, since $\left(M,\omega,\Phi\right)$ is tall, $\Phi(p)$ lies in the
(relative) interior of this edge.

Hence, by Corollary 2.34, there exists precisely one $i=1,\ldots,n-1$ such
that $\beta_{i}$ is a multiple of $\beta_{n}$; without loss of generality, we
assume that $i=n-1$. By Remark 2.10, the $\mathbb{Z}$-span of
$\beta_{1},\ldots,\beta_{n-2},\beta_{n}$ equals $\ell^{*}$. Hence, by a
dimension count, $\beta_{1},\ldots,\beta_{n-2},\beta_{n}$ are linearly
independent. We claim that $\langle\beta_{j},\nu_{\min}\rangle\geq 0$ for all
$j=1,\ldots,n-2$. Suppose, on the contrary, that there exists an index
$j=1,\ldots,n-2$ such that $\langle\beta_{j},\nu_{\min}\rangle<0$. By
Corollary 2.41 applied to $\beta_{j}$ and by the above argument, $\Phi(p)$
must lie in the (relative) interior of an edge of $\Phi(M)$ that is contained
in $\Phi(p)+\mathbb{R}\langle\beta_{j}\rangle$. (Observe that this uses the
fact that $\langle\Phi(p),\nu_{\min}\rangle$ is minimal among all isolated
fixed points.) Since any point in a convex polytope is contained in the
(relative) interior of at most one edge, it follows that
$\mathbb{R}\langle\beta_{j}\rangle=\mathbb{R}\langle\beta_{n}\rangle$, which
contradicts the linear independence of
$\beta_{1},\ldots,\beta_{n-2},\beta_{n}$.

Since $\beta_{n-1}$ is a multiple of $\beta_{n}$, since $\beta_{n}$ is
primitive, and since $\Phi(p)$ lies in the (relative) interior of an edge of
$\Phi(M)$, there exists a positive integer $\lambda$ such that
$\beta_{n-1}=-\lambda\beta_{n}$. Since $\left(M,\omega,\Phi\right)$ is
normalized, the weight sum formula at $p$

(4.8) $\Phi(p)=-\sum\limits_{j=1}^{n}\beta_{j}$

implies that

(4.9)
$0\leq\langle\Phi(p),\nu_{\min}\rangle=\underbrace{-\sum\limits_{j=1}^{n-2}\langle\beta_{j},\nu_{\min}\rangle}_{\leq
0}+\underbrace{(\lambda-1)}_{\geq
0}\underbrace{\langle\beta_{n},\nu_{\min}\rangle}_{<0}\leq 0.$

Therefore $\langle\Phi(p),\nu_{\min}\rangle=0$, $\lambda=1$ and
$\langle\beta_{j},\nu_{\min}\rangle=0$ for all $j=1,\ldots,n-2$. Since
$\langle\Phi(q),\nu_{\min}\rangle<\langle\Phi(p),\nu_{\min}\rangle$ by (4.7),
since $\langle\Phi(q),\nu_{\min}\rangle\in\mathbb{Z}$ by Corollary 3.8, and by
(4.3), we have that $\langle\Phi(q),\nu_{\min}\rangle=-1$, i.e., $v:=\Phi(q)$
is a vertex of $\mathcal{F}_{\min}$. Moreover, since $\alpha_{n-1}=-\beta_{n}$
and since $\langle\beta_{n},\nu_{\min}\rangle<0$, the edge incident to $v$
contained in $\Phi(p)+\mathbb{R}\langle\beta_{n}\rangle$ comes out of
$\mathcal{F}_{\min}$. Since $\beta_{n-1}=-\lambda\beta_{n}$, then
$\beta_{n-1}=\alpha_{n-1}$. We observe that the set (!)
$\\{\beta_{1},\ldots,\beta_{n-2}\\}$ is precisely the multiset of isotropy
weights for the $T$-action on the normal bundle to the $T$-invariant
submanifold $N$ at the point $p$. Hence, this set equals
$\\{\alpha_{1},\ldots,\alpha_{n-2}\\}$ modulo
$\mathbb{Z}\langle\beta_{n}\rangle=\mathbb{Z}\langle\alpha_{n-1}\rangle$,
since the latter is the multiset of isotropy weights for the $T$-action on the
normal bundle to the $T$-invariant submanifold $N$ at another point. The
affine hyperplane $v+\mathbb{R}\langle\alpha_{1},\ldots,\alpha_{n-2}\rangle$
contains $\mathcal{F}_{\min}$ by (4.4). Hence,
$\langle\alpha_{j},\nu_{\min}\rangle=0$ for all $j=1,\ldots,n-2$. Since
$\langle\alpha_{n-1},\nu_{\min}\rangle>0$ and
$\langle\beta_{j},\nu_{\min}\rangle=0$ for all $j=1,\ldots,n-2$, then
$\\{\beta_{1},\ldots,\beta_{n-2}\\}=\\{\alpha_{1},\ldots,\alpha_{n-2}\\}$.
This completes the proof of the result under the hypothesis that
$\langle\Phi(p),\nu_{\min}\rangle$ is minimal among all isolated fixed points.

Case 2: To conclude the proof, it suffices to show that, if $p\in M^{T}$ is
isolated, then $\langle\Phi(p),\nu_{\min}\rangle$ is minimal. Suppose not;
then, by the above argument and since
$\langle\Phi(p),\nu_{\min}\rangle\in\mathbb{Z}_{\geq 0}$, there exists
$p^{\prime}\in M^{T}$ such that $\langle\Phi(p^{\prime}),\nu_{\min}\rangle>0$
and minimal among fixed points with positive pairing. Let
$\beta_{1}^{\prime},\ldots,\beta_{n}^{\prime}\in\ell^{*}$ be the isotropy
weights of $p^{\prime}$. By Lemma 2.39, we may assume that
$\langle\beta_{n}^{\prime},\nu_{\min}\rangle<0$. We consider the sheet
$(N^{\prime},\omega_{N^{\prime}},\Phi_{N^{\prime}})$ along
$\beta_{n}^{\prime}$ given by Lemma 2.37 and the point $q^{\prime}\in
M^{T}\cap N^{\prime}$ satisfying the conclusions of Corollary 2.41. In analogy
with (4.7),
$\langle\Phi(q^{\prime}),\nu_{\min}\rangle<\langle\Phi(p^{\prime}),\nu_{\min}\rangle$.
Hence, by minimality, $q^{\prime}$ is either isolated and satisfies
$\langle\Phi(q^{\prime}),\nu_{\min}\rangle=0$, or is not isolated. In either
case, by the arguments used in the special case above,
$\beta^{\prime}_{n}\in\ell^{*}$ is primitive, $\dim N^{\prime}=4$ and
$\Phi(p^{\prime})$ lies in the (relative) interior of an edge of $\Phi(M)$. In
particular, there is precisely one other isotropy weight of $p^{\prime}$ that
is collinear with $\beta^{\prime}_{n}$, say $\beta^{\prime}_{n-1}$. Moreover,
as above, $\langle\beta^{\prime}_{j},\nu_{\min}\rangle\geq 0$ for all
$j=1,\ldots,n-2$. Set $\beta^{\prime}_{n-1}=-\lambda^{\prime}\beta_{n}$ for
some positive integer $\lambda^{\prime}$; since $\beta^{\prime}_{n}$ is
primitive, $\lambda^{\prime}\geq 1$. We use again the weight sum formula (4.8)
and, in analogy with (4.9), we obtain the following absurd string of
inequalities

$0<\langle\Phi(p^{\prime}),\nu_{\min}\rangle=\underbrace{-\sum\limits_{j=1}^{n-2}\langle\beta^{\prime}_{j},\nu_{\min}\rangle}_{\leq
0}+\underbrace{(\lambda^{\prime}-1)}_{\geq
0}\underbrace{\langle\beta^{\prime}_{n},\nu_{\min}\rangle}_{<0}\leq 0.$

∎

###  4.3. The space of exceptional orbits and the painting

Motivated by Proposition 4.6, our first aim is to prove properties of an
isolated fixed point with isotropy weights
$\alpha_{1},\ldots,\alpha_{n-2},\alpha_{n-1},-\alpha_{n-1}$ in a complexity
one $T$-space (see Lemma 4.9 below). By Remark 2.10,
$\alpha_{1},\ldots,\alpha_{n-1}$ form a basis of $\ell^{*}$; therefore, by a
dimension count, $\alpha_{j}$ is primitive for all $j=1,\ldots,n-1$. We define
the following subgroups of $T$:

$H:=\exp\left(\\{\xi\in\mathfrak{t}\mid\langle\alpha_{i},\xi\rangle=0,\text{
for all }i=1,\ldots,n-2\\}\right),$

which is of dimension 1, and

$T^{\prime}:=\exp\left(\\{\xi\in\mathfrak{t}\mid\langle\alpha_{n-1},\xi\rangle=0\\}\right),$

which is of codimension 1. Observe that $T\simeq T^{\prime}\times H$;
moreover, we use the given inner product to identify the duals
$\mathfrak{h}^{*},(\mathfrak{t}^{\prime})^{*}$ of the Lie algebras of $H$ and
$T$ with $\mathbb{R}\langle\alpha_{n-1}\rangle$ and
$\mathbb{R}\langle\alpha_{1},\ldots,\alpha_{n-2}\rangle$ respectively.

We start by looking at the local model determined by the above isotropy
weights (see Section 2.1.4). We consider the following $T$-action on
$\mathbb{C}^{n}$

(4.10) $\begin{split}\exp(\xi)\cdot&(z_{1},\ldots,z_{n-2},z_{n-1},z_{n})=\\\
&(e^{2\pi i\langle\alpha_{1},\xi\rangle}z_{1},\ldots,e^{2\pi
i\langle\alpha_{n-2},\xi\rangle}z_{n-2},e^{2\pi
i\langle\alpha_{n-1},\xi\rangle}z_{n-1},e^{2\pi
i\langle-\alpha_{n-1},\xi\rangle}z_{n})\text{ for
}\xi\in\mathfrak{t},\end{split}$

with moment map $\Phi_{0}:\mathbb{C}^{n}\to\mathfrak{t}^{*}$ given by

(4.11)
$\Phi_{0}(z_{1},\ldots,z_{n})=\pi\left(\sum\limits_{j=1}^{n-2}\alpha_{j}|z_{j}|^{2}+\alpha_{n-1}(|z_{n-1}|^{2}-|z_{n}|^{2})\right).$

From (4.10) it is clear that $0\in\mathbb{C}^{n}$ is a fixed point, that the
circle $H$ acts trivially on $\mathbb{C}^{n-2}=\mathbb{C}\langle
z_{1},\ldots,z_{n-2}\rangle$, and that the $(n-2)$-dimensional torus
$T^{\prime}$ acts trivially on $\mathbb{C}^{2}=\mathbb{C}\langle
z_{n-1},z_{n}\rangle$. Therefore the linear $T$-action on $\mathbb{C}^{n}$ of
(4.10) splits as the product of a toric $T^{\prime}$-action on
$\mathbb{C}^{n-2}=\mathbb{C}\langle z_{1},\ldots,z_{n-2}\rangle$, and a
complexity one $H$-action on $\mathbb{C}^{2}=\mathbb{C}\langle
z_{n-1},z_{n}\rangle$. Moreover,

  1. (1)

the stabilizer in $T$ of a point $q:=(z_{1},\ldots,z_{n})\in\mathbb{C}^{n}$ is
the product of the stabilizer in $T^{\prime}$ of
$q_{1}:=(z_{1},\ldots,z_{n-2})\in\mathbb{C}^{n-2}$ and of the stabilizer in
$H$ of $q_{2}:=(z_{n-1},z_{n})\in\mathbb{C}^{2}$,

  2. (2)

the symplectic slice representation of $q\in\mathbb{C}^{n}$ for the action of
$T$ splits as the product of the symplectic slice representations of
$q_{1}\in\mathbb{C}^{n-2}$ for the action of $T^{\prime}$ and of
$q_{2}\in\mathbb{C}^{2}$ for the action of $H$, and

  3. (3)

a point $q\in\mathbb{C}^{n}$ is exceptional with respect to the action of $T$
if and only if at least one of $q_{1}\in\mathbb{C}^{n-2}$ and
$q_{2}\in\mathbb{C}^{2}$ is exceptional with respect to the corresponding
actions of $T^{\prime}$ and $H$.

We observe that property (3) follows from properties (1) and (2). Hence, in
order to understand properties of the product, we consider each factor
separately. This is the content of the following two results.

######  Lemma 4.7.

Consider $\mathbb{C}^{n-2}$ with the above linear toric $T^{\prime}$ action.
Then every point in $\mathbb{C}^{n-2}$ is regular for the action of
$T^{\prime}$. Moreover, for each
$q_{1}=(z_{1},\ldots,z_{n-2})\in\mathbb{C}^{n-2}$, the subset
$J:=\\{j\in\\{1,\ldots,n-2\\}\mid z_{j}\neq 0\\}$ is the unique subset such
that

  * •

the moment map image
$\pi\sum\limits_{j=1}^{n-2}\alpha_{j}|z_{j}|^{2}\in(\mathfrak{t}^{\prime})^{*}$
lies in $\mathbb{R}_{>0}\langle\\{\alpha_{j}\mid j\in J\\}\rangle$ (if
$J=\emptyset$, then $q_{1}=0$ and the moment map image equals zero),

  * •

the stabilizer of $q_{1}$ is

(4.12)
$K_{J}:=\exp\left(\\{\xi\in\mathfrak{t}^{\prime}\mid\langle\alpha_{j},\xi\rangle=0\text{
for all }j\in J\\}\right),$

and

  * •

the isotropy weights of $q_{1}$ are $\\{\alpha_{j}\mid j\notin
J\\}\subset(\mathfrak{t}^{\prime})^{*}$, where we identify
$\mathrm{Lie}(K_{J})^{*}$ with $\mathbb{R}\langle\\{\alpha_{j}\mid j\notin
J\cup\\{n-1\\}\\}\rangle\subseteq(\mathfrak{t}^{\prime})^{*}\subset\mathfrak{t}^{*}$.

Conversely, given any subset $J\subseteq\\{1,\ldots,n-2\\}$ and any
$w\in\mathbb{R}_{>0}\langle\\{\alpha_{j}\mid j\in J\\}\rangle$, there exists
$q_{1}=(z_{1},\ldots,z_{n-2})\in\mathbb{C}^{n-2}$ such that
$\pi\sum\limits_{j=1}^{n-2}\alpha_{j}|z_{j}|^{2}=w$, the stabilizer of $q_{1}$
is $K_{J}$, and the isotropy weights of $q_{1}$ are $\\{\alpha_{j}\mid j\notin
J\\}$.

Finally, the subset of points with trivial stabilizer is path-connected and
dense.

###### Proof.

By Lemma 2.19, every point in a complexity zero Hamiltonian space is regular.
The linear toric $T^{\prime}$-action on $\mathbb{C}^{n-2}$ is given explicitly
by

(4.13) $\exp(\xi^{\prime})\cdot(z_{1},\ldots,z_{n-2})=(e^{2\pi
i\langle\alpha_{1},\xi^{\prime}\rangle}z_{1},\ldots,e^{2\pi
i\langle\alpha_{n-2},\xi^{\prime}\rangle}z_{n-2})\quad\text{for}\;\;\xi^{\prime}\in\mathfrak{t}^{\prime}\,,$

(cf. (2.8)). By definition of $J$, the moment map image
$\pi\sum\limits_{j=1}^{n-2}\alpha_{j}|z_{j}|^{2}$ lies in
$\mathbb{R}_{>0}\langle\\{\alpha_{j}\mid j\in J\\}\rangle$. Since
$\alpha_{1},\ldots,\alpha_{n-2}$ are linearly independent, $J$ is the only
such subset of $\\{1,\ldots,n-2\\}$. This proves the first bullet point.

Next we prove the second bullet point. Let $K$ be the stabilizer of $q_{1}$.
By (4.13), if $\xi^{\prime}\in\mathfrak{t}^{\prime}$ then
$\exp(\xi^{\prime})\in K$ if and only if
$\langle\alpha_{j},\xi^{\prime}\rangle\in\mathbb{Z}$ for all $j\in J$.
However, since each $\alpha_{j}$ is primitive,

$K=\exp\left(\\{\xi^{\prime}\in\mathfrak{t}^{\prime}\mid\langle\alpha_{j},\xi^{\prime}\rangle\in\mathbb{Z}\text{
for all }j\in
J\\}\right)=\exp\left(\\{\xi^{\prime}\in\mathfrak{t}^{\prime}\mid\langle\alpha_{j},\xi^{\prime}\rangle=0\text{
for all }j\in J\\}\right),$

which, by definition, is exactly $K_{J}$.

We turn to the proof of the third bullet point. The symplectic slice
representation of $q_{1}$ is the following representation of $K_{J}$: We set

$\mathbb{C}^{J}:=\\{(w_{1},\ldots,w_{n-2})\in\mathbb{C}^{n-2}\mid
w_{j}=0\text{ for all }j\in J\\}.$

This is a $T^{\prime}$-invariant complex subspace of $\mathbb{C}^{n-2}$ that
can be identified symplectically with the symplectic normal to the
$T^{\prime}$-orbit of $q_{1}$, once the tangent space at $q_{1}$ is identified
with $\mathbb{C}^{n-2}$. Under this identification, since the
$T^{\prime}$-action on $\mathbb{C}^{n-2}$ is linear, the $K_{J}$-action on
$\mathbb{C}^{J}$ is given by the restriction of the $T^{\prime}$-action to
$K_{J}$. The isotropy weights of this $K_{J}$-action are given by the set
$\\{\alpha_{j}\mid j\notin J\\}\subset(\mathfrak{t}^{\prime})^{*}$.

Conversely, given a subset $J\subseteq\\{1,\ldots,n-2\\}$ and
$w\in\mathbb{R}_{>0}\langle\\{\alpha_{j}\mid j\in J\\}\rangle$, there exist
positive constants $\lambda_{j}$ for $j\in J$ such that $w=\sum\limits_{j\in
J}\lambda_{j}\alpha_{j}$. The point
$q_{1}=(z_{1},\ldots,z_{n-2})\in\mathbb{C}^{n-2}$ with coordinates given by

$z_{j}=\begin{cases}\pi^{-1}\sqrt{\lambda_{j}}&\text{if }j\in J\\\ 0&\text{if
}j\notin J\end{cases}$

is such that $\pi\sum\limits_{j=1}^{n-2}\alpha_{j}|z_{j}|^{2}=w$, its
stabilizer is $K_{J}$, and its isotropy weights are $\\{\alpha_{j}\mid j\notin
J\\}$.

Finally, $q_{1}=(z_{1},\ldots,z_{n-2})\in\mathbb{C}^{n-2}$ has trivial
stabilizer if and only if $z_{j}\neq 0$ for all $j=1,\ldots,n-2$. The subset

$\\{(z_{1},\ldots,z_{n-2})\in\mathbb{C}^{n-2}\mid z_{j}\neq 0\text{ for all
}j=1,\ldots,n-2\\}$

is clearly path-connected and dense. ∎

######  Lemma 4.8.

Consider $\mathbb{C}^{2}$ with the above linear complexity one $H$-action. A
point $q_{2}\in\mathbb{C}^{2}$ is exceptional if and only if $q_{2}=(0,0)$.
Moreover, $q_{2}=(0,0)$ is the only point stabilized by $H$. In this case, the
isotropy weights of $q_{2}$ are $\\{\pm\,\alpha_{n-1}\\}$.

###### Proof.

We fix an isomorphism between $H$ and $S^{1}$ so that $\alpha_{n-1}$
corresponds to $+1$. Under this isomorphism, the above linear $H$-action on
$\mathbb{C}^{2}$ can be identified with the linear $S^{1}$-action on
$\mathbb{C}^{2}$ with weights equal to $+1$ and $-1$. The result then follows
from a simple computation and from Lemma 2.19. ∎

Theorem 2.8 and Lemmas 4.7, 4.8 imply the following result that is central to
this section.

######  Lemma 4.9.

Let $\left(M,\omega,\Phi\right)$ be a complexity one $T$-space of dimension
$2n$ and let $p\in M^{T}$ be isolated with isotropy weights
$\alpha_{1},\ldots,\alpha_{n-2},\alpha_{n-1},-\alpha_{n-1}$. There exists an
open neighborhood $U$ of $p$ such that the following are equivalent:

  * •

$q\in U$ is exceptional, and

  * •

$\Phi(q)\in\Phi(p)+\mathbb{R}_{\geq 0}\langle\\{\alpha_{j}\mid
j=1,\ldots,n-2\\}\rangle$ and the stabilizer of $q$ contains $H$.

Moreover, given $q\in U$ exceptional, if $J\subseteq\\{1,\ldots,n-2\\}$ is
defined by

(4.14) $\Phi(q)\in\Phi(p)+\mathbb{R}_{>0}\langle\\{\alpha_{j}\mid j\in
J\\}\rangle,$

then

  1. (i)

the stabilizer of $q$ is $K_{J}\times H$, where $K_{J}$ is defined in (4.12),
and

  2. (ii)

the isotropy weights of $q$ are

$\\{\alpha_{j}\mid j\notin J\\}\cup\\{\pm\,\alpha_{n-1}\\},$

where we identify
$\text{Lie}(K_{J})^{*}\subseteq(\mathfrak{t}^{\prime})^{*}\subset\mathfrak{t}^{*}$
with $\mathbb{R}\langle\\{\alpha_{j}\mid j\notin J\cup\\{n-1\\}\\}\rangle$.

Conversely, given any subset $J\subseteq\\{1,\ldots,n-2\\}$ and any
$w\in\Phi(p)+\mathbb{R}_{>0}\langle\alpha_{j}\mid j\in J\\}\rangle$, there
exists an exceptional point $q\in U$ such that $\Phi(q)=w$, the stabilizer of
$q$ is $K_{J}\times H$, and the isotropy weights of $q$ are $\\{\alpha_{j}\mid
j\notin J\\}\cup\\{\pm\,\alpha_{n-1}\\}$.

Finally, the subset

$\\{q\in U\mid q\text{ is exceptional and has stabilizer }H\\}$

is path-connected and dense in $\\{q\in U\mid q\text{ is exceptional}\\}$.

###### Proof.

By Theorem 2.8, it suffices to consider the local model determined by the
isotropy weights, $p=0\in\mathbb{C}^{n}$ and $\Phi(p)=0$. By property (3), and
Lemmas 4.7 and 4.8, a point $q=(q_{1},q_{2})\in\mathbb{C}^{n}$ is exceptional
if and only if $q_{2}=(0,0)$, which is also equivalent to the stabilizer of
$q_{2}$ being $H$. Suppose that $q=(q_{1},q_{2})$ is exceptional and let
$J\subseteq\\{1,\ldots,n-2\\}$ be the subset given by Lemma 4.7. Since
$q_{2}=(0,0)$, by (4.11), $\Phi(q)$ lies in
$\Phi(p)+\mathbb{R}_{>0}\langle\\{\alpha_{j}\mid j\in J\\}\rangle$ if and only
if
$\pi\sum\limits_{j=1}^{n-2}\alpha_{j}|z_{j}|^{2}\in(\mathfrak{t}^{\prime})^{*}$
lies in $\mathbb{R}_{>0}\langle\\{\alpha_{j}\mid j\in J\\}\rangle$. Hence, $J$
is the unique subset of $\\{1,\ldots,n-2\\}$ such that (4.14) holds.
Properties (i) and (ii) in the statement follow immediately from (1) and (2)
in the discussion preceding Lemma 4.7, and from Lemmas 4.7 and 4.8.

Conversely, let $J\subseteq\\{1,\ldots,n-2\\}$ be a subset and
$w\in\mathbb{R}_{>0}\langle\\{\alpha_{j}\mid j\in J\\}\rangle$. Let
$q_{1}\in\mathbb{C}^{n-2}$ be the point given by Lemma 4.7. By property (3) in
the discussion preceding Lemma 4.7 and Lemma 4.8, the point $q=(q_{1},0,0)\in
V$ is exceptional. Moreover, by (4.11), $\Phi(q)=w$. By Lemmas 4.7 and 4.8,
and by properties (1) and (2), the stabilizer of $q$ is $K_{J}\times H$ and
the isotropy weights of $q$ are $\\{\alpha_{j}\mid j\notin
J\\}\cup\\{\pm\,\alpha_{n-1}\\}$, as desired.

Finally, by Lemmas 4.7 and 4.8,

(4.15) $\\{q=(q_{1},q_{2})\in\mathbb{C}^{n-2}\times\mathbb{C}^{2}\mid q\text{
is exceptional and has stabilizer }H\\}$

equals

(4.16) $\\{q=(q_{1},0,0)\in\mathbb{C}^{n-2}\times\mathbb{C}^{2}\mid
q_{1}\text{ has trivial stabilizer}\\}.$

By Lemma 4.7, $\\{q_{1}\in\mathbb{C}^{n-2}\mid q_{1}\text{ has trivial
stabilizer}\\}$ is path-connected and dense in $\mathbb{C}^{n-2}$, thus
completing the proof. ∎

The subset $J$ associated to an exceptional point near the isolated fixed
point of Lemma 4.9 has the following useful property.

######  Corollary 4.10.

Let $\left(M,\omega,\Phi\right)$ be a complexity one $T$-space of dimension
$2n$ and let $p\in M^{T}$ be isolated with isotropy weights
$\alpha_{1},\ldots,\alpha_{n-2},\alpha_{n-1},-\alpha_{n-1}$. Let $U$ be the
open neighborhood of $p$ given by Lemma 4.9. Given exceptional points
$q,q^{\prime}\in U$, let $J,J^{\prime}\subseteq\\{1,\ldots,n-2\\}$ be the
subsets corresponding to $q,q^{\prime}$ as in Lemma 4.9. The symplectic slice
representations of $q$ and $q^{\prime}$ are isomorphic if and only if
$J=J^{\prime}$.

###### Proof.

If $J=J^{\prime}$, then by parts (i) and (ii) of Lemma 4.9, the points $q$ and
$q^{\prime}$ have equal stabilizers and the same isotropy weights. Since their
common stabilizer is connected, it follows that they have isomorphic
symplectic slice representations. Conversely, suppose that $q$ and
$q^{\prime}$ have isomorphic symplectic slice representations. Hence, by Lemma
4.9, they have connected stabilizers, so that $K_{J}=K_{J^{\prime}}$. Since
the dual of the Lie algebra of $K_{J}$ can be identified with
$\mathbb{R}\langle\\{\alpha_{j}\mid j\notin J\cup\\{n-1\\}\\}\rangle$, and
since $\alpha_{1},\ldots,\alpha_{n-2}$ are linearly independent, it follows
that $J=J^{\prime}$. ∎

Throughout this section, we apply Lemma 4.9 and Corollary 4.10 to an isolated
fixed point in a normalized monotone tall complexity one $T$-space. In this
case, $H=H_{\mathcal{F}_{\min}}$, the stabilizer of
$(M_{\mathcal{F}_{\min}},\omega_{\mathcal{F}_{\min}},\Phi_{\mathcal{F}_{\min}})$,
see Definition 4.3. Intuitively speaking, the next result is the ‘global
version’ of Lemma 4.9 for normalized monotone tall complexity one $T$-spaces.

######  Lemma 4.11.

Let $\left(M,\omega,\Phi\right)$ be a normalized monotone tall complexity one
$T$-space of dimension $2n$. Let $q\in M$ be exceptional. There exist $p\in
M^{T}$ isolated and a unique subset $J\subseteq\\{1,\ldots,n-2\\}$ such that,
if $\alpha_{1},\ldots,\alpha_{n-2},\alpha_{n-1},-\alpha_{n-1}$ are the
isotropy weights of $p$ as in Proposition 4.6, then

  1. (i)

the moment map image $\Phi(q)$ lies in
$\Phi(p)+\mathbb{R}_{>0}\langle\\{\alpha_{j}\mid j\in J\\}\rangle$,

  2. (ii)

the stabilizer of $q$ is $K_{J}\times H_{\mathcal{F}_{\min}}$, where
$K_{J}\leq T^{\prime}$ is as in (4.12), and

  3. (iii)

the isotropy weights of $q$ are $\\{\alpha_{j}\mid j\notin
J\\}\cup\\{\pm\,\alpha_{n-1}\\}$ (see part (ii) of Lemma 4.9).

###### Proof.

By Lemma 2.30, the sheet $(N,\omega_{N},\Phi_{N})$ through $q$ is exceptional.
Since $N$ is compact, it contains a fixed point $p\in M^{T}$ that is
exceptional and therefore isolated by Lemma 2.20. Since $N$ is connected, by
the principal orbit theorem (see [12, Theorem 2.8.5]), there exists a
relatively open, dense and connected subset $N^{\prime}$ of $N$ such that, if
$q^{\prime}\in N^{\prime}$, then $q$ and $q^{\prime}$ have isomorphic
symplectic slice representations. In particular, if $U$ is the open
neighborhood of $p$ given by Lemma 4.9, then $U\cap N^{\prime}$ is not empty;
moreover, for all $q^{\prime}\in U\cap N^{\prime}$, the symplectic slice
representation of $q^{\prime}$ is isomorphic to that of $q$. By Lemma 4.9 and
Corollary 4.10, there exists a unique subset $J\subseteq\\{1,\ldots,n-2\\}$
such that, for all $q^{\prime}\in U\cap N^{\prime}$,

  * •

the moment map image $\Phi(q^{\prime})$ lies in
$\Phi(p)+\mathbb{R}_{>0}\langle\\{\alpha_{j}\mid j\in J\\}\rangle$,

  * •

the stabilizer of $q^{\prime}$ is $K_{J}\times H_{\mathcal{F}_{\min}}$, where
$K_{J}\leq T^{\prime}$ is as in (4.12), and

  * •

the isotropy weights of $q^{\prime}$ are $\\{\alpha_{j}\mid j\notin
J\\}\cup\\{\pm\,\alpha_{n-1}\\}$.

Since $U\cap N\neq\emptyset$, the second and third bullet points imply
properties (ii) and (iii). To see that property (i) holds, we observe that, by
the first bullet point, $\Phi(U\cap N^{\prime})$ is contained in
$\Phi(p)+\mathbb{R}_{>0}\langle\\{\alpha_{j}\mid j\in J\\}\rangle$. Since
$N_{\mathrm{reg}}$ is dense in $N$, we have that $\Phi(U\cap N)$ is contained
in $\Phi(p)+\mathbb{R}_{\geq 0}\langle\\{\alpha_{j}\mid j\in J\\}\rangle$. On
the other hand, since the stabilizer of $q$ is $K_{J}\times
H_{\mathcal{F}_{\min}}$, the sheet $(N,\omega_{N},\Phi_{N})$ is a compact
Hamiltonian $T^{\prime\prime}$-space, where $T^{\prime\prime}=T/(K_{J}\times
H_{\mathcal{F}_{\min}})\simeq T^{\prime}/K_{J}$. By construction, we may
identify the dual of the Lie algebra of $T^{\prime\prime}$ with
$\Phi(p)+\mathbb{R}\langle\\{\alpha_{j}\mid j\in J\\}\rangle$. Hence, by the
Convexity Package (Theorem 2.32), the moment map image $\Phi_{N}(N)=\Phi(N)$
is a convex polytope in $\Phi(p)+\mathbb{R}\langle\\{\alpha_{j}\mid j\in
J\\}\rangle$. Since $\Phi(p)+\mathbb{R}_{\geq 0}\langle\\{\alpha_{j}\mid j\in
J\\}\rangle$ is convex in $\Phi(p)+\mathbb{R}\langle\\{\alpha_{j}\mid j\in
J\\}\rangle$ and since $\Phi(U\cap N)$ is contained in
$\Phi(p)+\mathbb{R}_{\geq 0}\langle\\{\alpha_{j}\mid j\in J\\}\rangle$,
$\Phi(N)$ is contained in $\Phi(p)+\mathbb{R}_{\geq 0}\langle\\{\alpha_{j}\mid
j\in J\\}\rangle$. In particular, the interior of $\Phi(N)$ is contained in
$\Phi(p)+\mathbb{R}_{>0}\langle\\{\alpha_{j}\mid j\in J\\}\rangle$. Since $q$
is a regular point for the moment map $\Phi_{N}$, the moment map image
$\Phi_{N}(q)=\Phi(q)$ lies in the (relative) interior of $\Phi(N)$, as
desired. ∎

######  Remark 4.12.

By Lemma 4.11, if $q$ is exceptional, then the moment map $\Phi(q)$ can be
used to reconstruct the symplectic slice representation of $q$. To see this,
we observe that, by property (i) and Proposition 4.6, $\Phi(q)$ lies in the
affine hyperplane $\Phi(p)+\\{w\in\mathfrak{t}^{*}\mid\langle
w,\nu_{\min}\rangle=0\\}$. We recall that a basis for the linear subspace
$\\{w\in\mathfrak{t}^{*}\mid\langle w,\nu_{\min}\rangle=0\\}$ is given by
$\alpha_{1},\ldots,\alpha_{n-2}$ (cf. (4.4)). Hence, by Lemma 4.11, $\Phi(q)$
determines the subset $J$ uniquely, and $J$ determines the stabilizer and the
isotropy weights of $q$. Since the stabilizer of $q$ is connected, the claim
follows.

Our next aim is to prove Proposition 4.15, which plays an important role in
several key results below (e.g., Theorems 4.17 and 4.20. We start with the
following result.

######  Lemma 4.13.

Let $\left(M,\omega,\Phi\right)$ be a normalized monotone tall complexity one
$T$-space. The following are equivalent:

  1. (1)

there exists an isolated fixed point, and

  2. (2)

for each edge $e$ that comes out of $\mathcal{F}_{\min}$, there exists an
isolated fixed point $p\in M^{T}$ such that $\Phi(p)\in e$.

###### Proof.

Clearly (2) implies (1). Conversely, suppose that there exists an isolated
fixed point $p\in M^{T}$. If $\dim M=4$, there is nothing to prove, so we may
assume that $\dim M\geq 6$. By Proposition 4.6, there exists an edge $e$ that
comes out of $\mathcal{F}_{\min}$ such that $\Phi(p)\in e$. Let
$v\in\mathcal{F}_{\min}$ be the vertex to which $e$ is incident and let
$\alpha_{1},\ldots,\alpha_{n-2},\alpha_{n-1}$ be the non-zero isotropy weights
of $\Phi^{-1}(v)$ ordered so that (4.4) holds.

For $j=1,\ldots,n-2$, let $v_{j}\in\mathcal{F}_{\min}$ be the vertex that lies
on the edge supported on $v+\mathbb{R}_{\geq 0}\langle\alpha_{j}\rangle$ and
is not $v$. Let $e_{j}$ be the edge that comes out of $\mathcal{F}_{\min}$
that is incident to $v_{j}$. We claim that there exists an isolated fixed
point $p_{j}$ such that $\Phi(p_{j})\in e_{j}$ (see Figure 4.3). To this end,
by Lemma 4.9, there exists an exceptional point $q\in M$ arbitrarily close to
$p$ such that $\Phi(q)\in\Phi(p)+\mathbb{R}_{>0}\langle\alpha_{j}\rangle$;
moreover, the stabilizer of $q$ has codimension 1\. Let
$(N,\omega_{N},\Phi_{N})$ be the sheet through $q$. Since $q$ is exceptional,
so is $(N,\omega_{N},\Phi_{N})$; furthermore, $p\in N$ by construction. Hence,
$(N,\omega_{N},\Phi_{N})$ is a compact symplectic toric manifold with moment
map image contained in $\Phi(p)+\mathbb{R}_{\geq 0}\langle\alpha_{j}\rangle$.
Let $p_{j}\in M^{T}\cap N$ be the unique fixed point such that
$\Phi(p_{j})\in\Phi(p)+\mathbb{R}_{>0}\langle\alpha_{j}\rangle$. Since
$(N,\omega_{N},\Phi_{N})$ is exceptional, so is $p_{j}$; moreover, by Lemma
2.20, $p_{j}$ is isolated. Hence, by Proposition 4.6, the image $\Phi(p_{j})$
lies on an edge that comes out of $\mathcal{F}_{\min}$. This edge is
necessarily $e_{j}$: To see this, we observe that the moment map image
$\Phi(N)$ is contained in the affine two-dimensional plane
$v+\mathbb{R}\langle\alpha_{j},\alpha_{n-1}\rangle$. This plane supports a
two-dimensional face of $\Phi(M)$ that contains $e$ and the edge of
$\mathcal{F}_{\min}$ that is incident to both $v$ and $v_{j}$. Hence, there
exists only one other edge that is incident to $v_{j}$ that is contained in
this affine plane. Since this plane intersects $\mathcal{F}_{\min}$ precisely
in the edge that is incident to both $v$ and $v_{j}$, by (4.4) applied to the
non-zero isotropy weights of $\Phi^{-1}(v_{j})$, the other edge that is
incident to $v_{j}$ and contained in the above affine plane must come out of
$\mathcal{F}_{\min}$, i.e., it must be $e_{j}$.

Figure 4.3. A schematic illustration of the argument in the proof of Lemma
4.13.

By the last paragraph, (2) holds for each edge that comes out of
$\mathcal{F}_{\min}$ that is incident to a vertex of $\mathcal{F}_{\min}$ that
is adjacent to $v$ in $\mathcal{F}_{\min}$ (i.e., there exists an edge of
$\mathcal{F}_{\min}$ that is incident to both vertices). We define the
following relation on the set of vertices of $\mathcal{F}_{\min}$:

$v_{1}\sim v_{2}\Leftrightarrow\text{ either }v_{1}=v_{2}\text{ or
}v_{1}\text{ is adjacent to }v_{2}.$

Since the transitive closure of the above relation has one equivalence class
and since there is a one-to-one correspondence between edges that come out of
$\mathcal{F}_{\min}$ and vertices of $\mathcal{F}_{\min}$, (2) holds. ∎

As a consequence of Lemma 4.13, we obtain the following sufficient condition
for a normalized monotone tall complexity one $T$-space to be without isolated
fixed points.

######  Corollary 4.14.

Let $\left(M,\omega,\Phi\right)$ be a normalized monotone tall complexity one
$T$-space. If there is a vertex of $\Phi(M)$ on the linear hyperplane
$\\{w\in\mathfrak{t}^{*}\mid\langle w,\nu_{\min}\rangle=0\\}$, then there are
no isolated fixed points. Moreover, $M_{\mathrm{exc}}=\emptyset$.

###### Proof.

Let $v\in\Phi(M)$ be a vertex of $\Phi(M)$ such that $\langle
v,\nu_{\min}\rangle=0$. First we show that there is an edge $e$ that comes out
of $\mathcal{F}_{\min}$ that is incident to $v$. Let
$\alpha_{1},\ldots,\alpha_{n-1}$ be the non-zero isotropy weights of
$\Phi^{-1}(v)$. By Lemma 2.39, we may assume that
$\langle\alpha_{n-1},\nu_{\min}\rangle<0$. Let $e$ be the edge that is
contained in $v+\mathbb{R}_{\geq 0}\langle\alpha_{n-1}\rangle$ and let
$v^{\prime}\in\Phi(M)$ be the other vertex to which $e$ is incident. By
construction, $\langle v^{\prime},\nu_{\min}\rangle<0$. Moreover, since
$\left(M,\omega,\Phi\right)$ is normalized monotone, $\Phi(M)$ is integral.
Therefore, by (4.3), $\langle v^{\prime},\nu_{\min}\rangle=-1$, i.e.,
$v^{\prime}$ is a vertex of $\mathcal{F}_{\min}$. Since
$\langle\alpha_{n-1},\nu_{\min}\rangle<0$, $e$ is an edge of $\Phi(M)$ that
comes out of $\mathcal{F}_{\min}$. Hence, $v$ is the only element in the
intersection of $e$ and $\\{w\in\mathfrak{t}^{*}\mid\langle
w,\nu_{\min}\rangle=0\\}$. By Theorem 2.32 and Proposition 4.6, there is no
isolated fixed point that is mapped to $e$ under $\Phi$. Hence, by Lemmas
2.20, 2.36 and 4.13, the result follows. ∎

The next result plays a key role throughout the paper.

######  Proposition 4.15.

Let $\left(M,\omega,\Phi\right)$ be a normalized monotone tall complexity one
$T$-space of dimension $2n$. If $(N,\omega_{N},\Phi_{N})$ is an exceptional
sheet that is stabilized by a one-dimensional subgroup $H$, then
$H=H_{\mathcal{F}_{\min}}$ and

(4.17) $\Phi(N)=\Phi(M)\cap\\{w\in\mathfrak{t}^{*}\mid\langle
w,\nu_{\min}\rangle=0\\}.$

###### Proof.

Since $(N,\omega_{N},\Phi_{N})$ is exceptional, every point in $N$ is
exceptional. Moreover, since $(N,\omega_{N},\Phi_{N})$ is stabilized by $H$,
there exists a point $q\in N$ with stabilizer equal to $H$. Hence, by part
(ii) of Lemma 4.11, there exist $p\in M^{T}$ isolated and a unique subset
$J\subseteq\\{1,\ldots,n-2\\}$ such that, if
$\alpha_{1},\ldots,\alpha_{n-2},\alpha_{n-1},-\alpha_{n-1}$ are the isotropy
weights of $p$ as in Proposition 4.6, then the stabilizer of $q^{\prime}$ is
$K_{J}\times H_{\mathcal{F}_{\min}}$, where $K_{J}\leq T^{\prime}$ is as in
(4.12). By definition, $K_{J}$ is connected. Hence, if the dimension of the
stabilizer of $q$ is one, then it must be $H_{\mathcal{F}_{\min}}$, thus
proving the first statement.

By Proposition 3.18, $\Phi(M)$ is a reflexive Delzant polytope and therefore
the origin lies in the interior of $\Phi(M)$ (see Lemma 3.13). Hence, the
interior of $\Phi(M)\cap\\{w\in\mathfrak{t}^{*}\mid\langle
w,\nu_{\min}\rangle=0\\}$ in $\\{w\in\mathfrak{t}^{*}\mid\langle
w,\nu_{\min}\rangle=0\\}$ is non-empty. Since both $M$ and $N$ are compact,
and since $\\{w\in\mathfrak{t}^{*}\mid\langle w,\nu_{\min}\rangle=0\\}$ is a
linear hyperplane in $\mathfrak{t}^{*}$, by the Convexity Package (Theorem
2.32), both $\Phi(M)\cap\\{w\in\mathfrak{t}^{*}\mid\langle
w,\nu_{\min}\rangle=0\\}$ and $\Phi_{N}(N)=\Phi(N)$ are convex polytopes.
Therefore, in order to prove that (4.17) holds, it suffices to show that
$\Phi(M)\cap\\{w\in\mathfrak{t}^{*}\mid\langle w,\nu_{\min}\rangle=0\\}$ and
$\Phi(N)$ have the same vertices.

Since $M_{\mathrm{exc}}\neq\emptyset$, by Corollary 4.14, there is no vertex
of $\Phi(M)$ lying on $\\{w\in\mathfrak{t}^{*}\mid\langle
w,\nu_{\min}\rangle=0\\}$. Hence, a point
$\hat{v}\in\Phi(M)\cap\\{w\in\mathfrak{t}^{*}\mid\langle
w,\nu_{\min}\rangle=0\\}$ is a vertex if and only if there exists an edge $e$
of $\Phi(M)$ that comes out of $\mathcal{F}_{\min}$ such that $\hat{v}$ is the
intersection of $e$ with $\\{w\in\mathfrak{t}^{*}\mid\langle
w,\nu_{\min}\rangle=0\\}$. On the other hand, since $(N,\omega_{N},\Phi_{N})$
is exceptional and since the complexity of $\left(M,\omega,\Phi\right)$ is
one, by Proposition 2.26, the complexity of $(N,\omega_{N},\Phi_{N})$ is zero,
i.e., it is a compact symplectic toric manifold. Therefore,
$\hat{v}\in\Phi(N)$ is a vertex if and only if there exists an isolated fixed
point $p\in N$ such that $\Phi(p)=\hat{v}$.

Let $\hat{v}\in\Phi(N)$ be a vertex and let $p\in N$ be as above. By
Proposition 4.6, there exists $\hat{v}$ an edge $e$ of $\Phi(M)$ that comes
out of $\mathcal{F}_{\min}$ such that $\hat{v}$ is the intersection of $e$
with $\\{w\in\mathfrak{t}^{*}\mid\langle w,\nu_{\min}\rangle=0\\}$. Hence,
each vertex of $\Phi(N)$ is a vertex of
$\Phi(M)\cap\\{w\in\mathfrak{t}^{*}\mid\langle w,\nu_{\min}\rangle=0\\}$.
Moreover, by Lemma 4.9, there exists an open neighborhood $U$ of $p$ such that
$U\cap N$ is precisely the subset of exceptional points in $U$ and

(4.18) $\Phi(U\cap N)=\Phi(U)\cap\\{w\in\mathfrak{t}^{*}\mid\langle
w,\nu_{\min}\rangle=0\\}.$

Set $V:=\Phi(U)$. By the Convexity Package (Theorem 2.32), $V$ is an open
neighborhood of $\hat{v}$. Moreover, by (4.18),

(4.19) $V\cap\Phi(N)=V\cap\\{w\in\mathfrak{t}^{*}\mid\langle
w,\nu_{\min}\rangle=0\\},$

(see Figure 4.4).

Figure 4.4. The two possibilities in the proof of Proposition 4.15. The figure
on the left illustrates the case in which $\hat{v}$ is a vertex of $\Phi(N)$,
while that on the right shows the (absurd) case in which not all vertices of
$\Phi(M)\cap\\{w\in\mathfrak{t}^{*}\mid\langle w,\nu_{\min}\rangle=0\\}$ are
vertices of $\Phi(N)$.

Suppose that there exists a vertex of
$\Phi(M)\cap\\{w\in\mathfrak{t}^{*}\mid\langle w,\nu_{\min}\rangle=0\\}$ that
is not a vertex of $\Phi(N)$. Since both
$\Phi(M)\cap\\{w\in\mathfrak{t}^{*}\mid\langle w,\nu_{\min}\rangle=0\\}$ and
$\Phi(N)$ are convex polytopes of full dimension in
$\\{w\in\mathfrak{t}^{*}\mid\langle w,\nu_{\min}\rangle=0\\}$ and since the
vertices of the latter are a subset of those of the former, there exists a
vertex $\hat{v}$ of $\Phi(N)$ such that for any open neighborhood $V$ of
$\hat{v}$

$V\cap\Phi(N)\subsetneq V\cap\\{w\in\mathfrak{t}^{*}\mid\langle
w,\nu_{\min}\rangle=0\\},$

(see Figure 4.4). By (4.19), this is a contradiction. ∎

Our penultimate aim in this section is to prove Theorem 4.17 below. To this
end, first we prove the following result.

######  Lemma 4.16.

Let $\left(M,\omega,\Phi\right)$ be a normalized monotone tall complexity one
$T$-space of dimension $2n$. If $q\in M$ is exceptional, then there exists a
unique exceptional sheet $(N,\omega_{N},\Phi_{N})$ that is stabilized by
$H_{\mathcal{F}_{\min}}$ such that $q\in N$.

###### Proof.

First we prove uniqueness. Let $(N_{i},\omega_{i},\Phi_{i})$ be an exceptional
sheet that is stabilized by $H_{\mathcal{F}_{\min}}$ such that $q\in N_{i}$
for $i=1,2$. Hence, both $N_{1}$ and $N_{2}$ are connected components of
$M^{H_{\mathcal{F}_{\min}}}$. Since $q\in N_{1}\cap N_{2}$, it follows that
$N_{1}=N_{1}\cup N_{2}=N_{2}$, as desired.

Next we prove existence. Let $(N^{\prime},\omega^{\prime},\Phi^{\prime})$ be
the sheet through $q$. Since $q$ is exceptional, by Lemma 2.30,
$(N^{\prime},\omega^{\prime},\Phi^{\prime})$ is exceptional. Since
$N^{\prime}$ is compact, there exists $p\in M^{T}\cap N^{\prime}$ that, by
Lemma 2.20, is isolated. We claim that there exists an exceptional sheet
$(N,\omega_{N},\Phi_{N})$ stabilized by $H_{\mathcal{F}_{\min}}$ such that
$p\in N$. To this end, we use Proposition 4.6 and Lemma 4.9. Let $U$ be the
open neighborhood of $p$ given by Lemma 4.9 and let $U_{1}$ be the subset of
$U$ consisting of exceptional points stabilized by $H_{\mathcal{F}_{\min}}$,
which is path-connected and dense in the subset of $U$ consisting of
exceptional points. In particular, $U_{1}\neq\emptyset$. Given any point
$q^{\prime}\in U_{1}$, we consider the sheet $(N,\omega_{N},\Phi_{N})$ through
$q^{\prime}$. By Lemma 2.30, $(N,\omega_{N},\Phi_{N})$ is exceptional, while,
by definition, it is stabilized by $H_{\mathcal{F}_{\min}}$. Since $U_{1}$ is
dense in the subset of $U$ consisting of exceptional points and since $p\in U$
is exceptional by Lemma 2.20, it follows that $p\in N$, thus proving the
claim.

Hence, $N^{\prime}\cap N\neq\emptyset$. Moreover, since any point in
$N^{\prime}$ is exceptional, by Lemma 4.11 the stabilizer of any point in
$N^{\prime}$ contains $H_{\mathcal{F}_{\min}}$. Since both $N^{\prime}$ and
$N$ are connected and since $N$ is a connected component of
$M^{H_{\mathcal{F}_{\min}}}$, it follows that $N^{\prime}\cup N=N$, so that
$N^{\prime}$ is contained in $N$. Hence, $q\in N$, as desired. ∎

In fact, the proof of Lemma 4.16 yields a slightly stronger result, namely
that, under the hypotheses of the lemma, any exceptional sheet is contained in
one that stabilized by $H_{\mathcal{F}_{\min}}$.

######  Theorem 4.17.

Let $\left(M,\omega,\Phi\right)$ be a normalized monotone tall complexity one
$T$-space of dimension $2n$. Each connected component of $M_{\mathrm{exc}}$ is
mapped homeomorphically to $\Phi(M)\cap\\{w\in\mathfrak{t}^{*}\mid\langle
w,\nu_{\min}\rangle=0\\}$ by the orbital moment map. In particular, each
connected component of $M_{\mathrm{exc}}$ is contractible. Moreover,

  * •

if $m$ is the number of vertices of the minimal facet, then the number of
connected components of $M_{\mathrm{exc}}$ is precisely the number of isolated
fixed points divided by $m$, and

  * •

if $e$ is an edge of $\Phi(M)$ that comes out of $\mathcal{F}_{\mathrm{min}}$,
then the number of isolated fixed points lying on $\Phi^{-1}(e)$ equals the
number of connected components of $M_{\mathrm{exc}}$.

###### Proof.

First we show that the image of $\\{p\in M^{H_{\mathcal{F}_{\min}}}\mid
p\text{ is exceptional}\\}$ under the quotient map $M\to M/T$ equals
$M_{\mathrm{exc}}$ and that number of connected components of the latter
equals the number of exceptional sheets that are stabilized by
$H_{\mathcal{F}_{\min}}$. Clearly, the image of $\\{p\in
M^{H_{\mathcal{F}_{\min}}}\mid p\text{ is exceptional}\\}$ under the quotient
map $M\to M/T$ is contained in $M_{\mathrm{exc}}$. Conversely, given an
exceptional orbit $\mathcal{O}\in M_{\mathrm{exc}}$, every point in
$\mathcal{O}$ is exceptional by Remarks 2.6 and 2.16. Fix a point
$p\in\mathcal{O}$. By Proposition 4.15 and Lemma 4.16, $p\in
M^{H_{\mathcal{F}_{\min}}}$. Since $M^{H_{\mathcal{F}_{\min}}}$ is
$T$-invariant, it follows that $\mathcal{O}$ is contained in
$M^{H_{\mathcal{F}_{\min}}}$. Hence, $M_{\mathrm{exc}}$ is contained in the
image of $\\{p\in M^{H_{\mathcal{F}_{\min}}}\mid p\text{ is exceptional}\\}$
under the quotient map $M\to M/T$ and the first claim follows. To prove the
second claim, we observe that the connected components of $\\{p\in
M^{H_{\mathcal{F}_{\min}}}\mid p\text{ is exceptional}\\}$ are exactly the
exceptional sheets that are stabilized by $H_{\mathcal{F}_{\min}}$. Since
sheets are $T$-invariant and $T$ is connected, the restriction of the quotient
map $M\to M/T$ to $\\{p\in M^{H_{\mathcal{F}_{\min}}}\mid p\text{ is
exceptional}\\}$ induces a bijection between the connected components of
$\\{p\in M^{H_{\mathcal{F}_{\min}}}\mid p\text{ is exceptional}\\}$ and those
of $M_{\mathrm{exc}}$.

By Proposition 4.15, the image of an exceptional sheet that is stabilized by
$H_{\mathcal{F}_{\min}}$ is precisely
$\Phi(M)\cap\\{w\in\mathfrak{t}^{*}\mid\langle w,\nu_{\min}\rangle=0\\}$.
Moreover, any such sheet is a compact symplectic toric manifold, so that the
corresponding orbital moment map image is a homeomorphism onto its image.
Hence, the first claim follows.

By the above argument, in order to prove the bulleted statements, it suffices
to show that the number of exceptional sheets that are stabilized by
$H_{\mathcal{F}_{\min}}$ is precisely the number of isolated fixed points
divided by $m$. We begin by observing that $M_{\mathrm{exc}}=\emptyset$ if and
only if there are no isolated fixed points. This is a consequence of Lemma
2.36 and the complexity of $\left(M,\omega,\Phi\right)$ being one. So we may
assume that $M_{\mathrm{exc}}\neq\emptyset$. Let $(N,\omega_{N},\Phi_{N})$ be
such an exceptional sheet that is stabilized by $H_{\mathcal{F}_{\min}}$.
Since $(N,\omega_{N},\Phi_{N})$ is a compact symplectic toric manifold, the
number of vertices in $\Phi_{N}(N)=\Phi(N)$ is precisely the number of fixed
points. By Proposition 4.15, $\Phi(N)$ equals
$\Phi(M)\cap\\{w\in\mathfrak{t}^{*}\mid\langle w,\nu_{\min}\rangle=0\\}$;
moreover, there is a bijection between the vertices of $\Phi(N)$ and those of
$\mathcal{F}_{\min}$. Hence, the cardinality of $M^{T}\cap N$ equals $m$.

Moreover, if $(N^{\prime},\omega^{\prime},\Phi^{\prime})$ is another sheet
stabilized by $H_{\mathcal{F}_{\min}}$, then either $N=N^{\prime}$ or $N\cap
N^{\prime}=\emptyset$. In particular, if $N\neq N^{\prime}$, the subsets
$M^{T}\cap N$ and $M^{T}\cap N^{\prime}$ are disjoint. Finally, since the set
of isolated fixed points is a subset of the set of exceptional points by Lemma
2.20, by Lemma 4.16, $M^{T}_{\mathrm{isolated}}$ equals the disjoint union
over all exceptional sheets $(N,\omega_{N},\Phi_{N})$ stabilized by
$H_{\mathcal{F}_{\min}}$ of the intersections $M_{\mathrm{isolated}}^{T}\cap
N$. Since any such sheet is exceptional, by Lemma 2.20, $M^{T}\cap N$ equals
the intersection of $N$ with the set of isolated fixed points for any
exceptional sheet $(N,\omega_{N},\Phi_{N})$ stabilized by
$H_{\mathcal{F}_{\min}}$. Putting the above facts together, the bulleted
statements follow. ∎

We conclude this section with the following important result.

######  Theorem 4.18.

The equivalence class of paintings of a compact normalized monotone tall
complexity one $T$-space $\left(M,\omega,\Phi\right)$ is trivial.

###### Proof.

By Lemma 4.1, the genus of $\left(M,\omega,\Phi\right)$ is zero. Let
$f:M_{\mathrm{exc}}\to S^{2}$ be a painting of $\left(M,\omega,\Phi\right)$.
We need to construct a painting $f^{\prime}:M_{\mathrm{exc}}\to S^{2}$ that is
constant on each connected component of $M_{\mathrm{exc}}$ and that is
equivalent to $f$. If $M_{\mathrm{exc}}=\emptyset$, there is nothing to prove,
so we may assume that $M_{\mathrm{exc}}\neq\emptyset$.

Since $\Phi(M)\cap\\{w\in\mathfrak{t}^{*}\mid\langle w,\nu_{\min}\rangle=0\\}$
is a convex polytope, it is contractible. We fix a point $w_{0}$ in
$\Phi(M)\cap\\{w\in\mathfrak{t}^{*}\mid\langle w,\nu_{\min}\rangle=0\\}$ (for
instance, the origin). Since $\Phi(M)\cap\\{w\in\mathfrak{t}^{*}\mid\langle
w,\nu_{\min}\rangle=0\\}$ is contractible, there exists a deformation
retraction of $\Phi(M)\cap\\{w\in\mathfrak{t}^{*}\mid\langle
w,\nu_{\min}\rangle=0\\}$ onto $w_{0}$ that we denote by $H_{t}$, where
$H_{0}=\mathrm{id}$ and $H_{1}(w)=w_{0}$ for all
$w\in\Phi(M)\cap\\{w\in\mathfrak{t}^{*}\mid\langle w,\nu_{\min}\rangle=0\\}$.

Let $k>0$ be the number of connected components of $M_{\mathrm{exc}}$. We have
that

$M_{\mathrm{exc}}=\coprod\limits_{i=1}^{k}M_{i},$

where $M_{i}$ is a connected component of $M_{\mathrm{exc}}$ for
$i=1,\ldots,k$. By Theorem 4.17, the restriction of $\overline{\Phi}\colon
M/T\to\mathfrak{t}^{*}$ to $M_{i}$ is a homeomorphism onto
$\Phi(M)\cap\\{w\in\mathfrak{t}^{*}\mid\langle w,\nu_{\min}\rangle=0\\}$; let
$\overline{\Phi}^{-1}_{i}$ be the inverse to this homeomorphism. We denote the
restriction of $f$ to $M_{i}$ by $f_{i}$ and, for any $0\leq t\leq 1$, we
define $f_{i,t}:M_{i}\to S^{2}$ as the composite
$f_{i}\circ\overline{\Phi}_{i}^{-1}\circ H_{t}\circ\overline{\Phi}$. Since
$M_{\mathrm{exc}}$ is the disjoint union of the $M_{i}$’s, for any $0\leq
t\leq 1$, there exists a unique continuous map $f_{t}:M_{\mathrm{exc}}\to
S^{2}$ such that the restriction of $f_{t}$ to $M_{i}$ equals $f_{i,t}$ for
all $i=1,\ldots,k$.

We claim that $f_{t}:M_{\mathrm{exc}}\to S^{2}$ is a painting of
$\left(M,\omega,\Phi\right)$ for all $0\leq t\leq 1$. To this end, we fix
$0\leq t\leq 1$, consider the map $(\overline{\Phi},f_{t})$ and two distinct
points $[q_{i}],[q_{j}]\in M_{\mathrm{exc}}$ belonging to $M_{i}$ and $M_{j}$
respectively. We aim to show that
$(\overline{\Phi}([q_{i}]),f_{t}([q_{i}]))\neq(\overline{\Phi}([q_{j}]),f_{t}([q_{j}]))$.
If $i=j$, since the restriction of $\overline{\Phi}$ to $M_{i}=M_{j}$ is a
homeomorphism and since $[q_{i}]\neq[q_{j}]$, it follows that
$\overline{\Phi}([q_{i}])\neq\overline{\Phi}([q_{j}])$, so that the result
follows. Suppose next that $i\neq j$. If
$\overline{\Phi}([q_{i}])\neq\overline{\Phi}([q_{j}])$, then the result
follows, so we may assume that
$\overline{\Phi}([q_{i}])=\overline{\Phi}([q_{j}])=:w$. If
$f_{t}([q_{i}])=f_{t}([q_{j}])$, then the points
$\overline{\Phi}^{-1}_{i}(H_{t}(w))\in M_{i}$ and
$\overline{\Phi}^{-1}_{j}(H_{t}(w))\in M_{j}$ are distinct (since $i\neq j$),
but are such that

$(\overline{\Phi}(\overline{\Phi}^{-1}_{i}(H_{t}(w))),f(\overline{\Phi}^{-1}_{i}(H_{t}(w))))=(\overline{\Phi}(\overline{\Phi}^{-1}_{j}(H_{t}(w))),f(\overline{\Phi}^{-1}_{j}(H_{t}(w)))).$

This implies that $f$ is not a painting, which is absurd. Hence,
$f_{t}([q_{i}])\neq f_{t}([q_{j}])$, which implies that
$(\overline{\Phi}([q_{i}]),f_{t}([q_{i}]))\neq(\overline{\Phi}([q_{j}]),f_{t}([q_{j}]))$,
as desired.

We set $f^{\prime}:=f_{1}$. Since $f_{t}:M_{\mathrm{exc}}\to S^{2}$ is a
painting of $\left(M,\omega,\Phi\right)$ for all $0\leq t\leq 1$, $f=f_{0}$
and $f^{\prime}$ are homotopic through paintings. Moreover, the restriction of
$f^{\prime}$ to $M_{i}$ is constant by construction for any $i=1,\ldots,k$,
thus completing the proof. ∎

###  4.4. The Duistermaat-Heckman function

In order to prove the main result of this section (see Theorem 4.20), we prove
the following intermediate result. To this end, we recall that (4.3) and (4.4)
hold, that for any vertex $v$ and any edge $e$ of $\Phi(M)$, the preimages
$\Phi^{-1}(v)$ and $\Phi^{-1}(e)$ are a two dimensional sphere and a four
dimensional manifold respectively. Moreover, by Lemma 4.5, there exists
$s\in\mathbb{Z}$ such that, if $v\in\mathcal{F}_{\min}$ and $e$ is the edge
that comes out of $\mathcal{F}_{\min}$ that is incident to $v$, then the self-
intersection of $\Phi^{-1}(v)$ in $\Phi^{-1}(e)$ equals $s$ (and does not
depend on $v$).

######  Proposition 4.19.

Let $\left(M,\omega,\Phi\right)$ be a compact normalized monotone tall
complexity one $T$-space of dimension $2n$. Let $v\in\mathcal{F}_{\min}$ be a
vertex and let $e$ be the edge incident to $v$ that comes out of
$\mathcal{F}_{\min}$. Let $s\in\mathbb{Z}$ be as in Lemma 4.5 and let $k\geq
0$ be the number of isolated fixed points contained in $\Phi^{-1}(e)$. Then
the restriction of the Duistermaat-Heckman function
$DH\left(M,\omega,\Phi\right)$ to $e$ is the function

(4.20) $\begin{split}e&\to\mathbb{R}\\\ w&\mapsto 2-s\langle
w,\nu_{\min}\rangle-k\rho(w),\end{split}$

where $\rho:\mathfrak{t}^{*}\to\mathbb{R}$ is the function given by

$w\mapsto\begin{cases}0&\text{if }\langle w,\nu_{\min}\rangle\leq 0\\\ \langle
w,\nu_{\min}\rangle&\text{if }\langle w,\nu_{\min}\rangle\geq 0.\end{cases}$

###### Proof.

Let $\alpha_{1},\ldots,\alpha_{n-1}$ be the non-zero isotropy weights of
$\Phi^{-1}(v)$ ordered so that (4.4) holds; in particular,
$\langle\alpha_{n-1},\nu_{\min}\rangle=1$ (see Lemma 3.17 and (4.5)). Since
$e$ is the edge that comes out of $\mathcal{F}_{\min}$ and is incident to $v$,
by Lemma 3.17, there exists $t_{\max}\in\mathbb{Z}_{>0}$ such that

$e=\\{v+t\alpha_{n-1}\mid 0\leq t\leq t_{\max}\\}.$

Let $(M_{e},\omega_{e},\Phi_{e})$ be the sheet corresponding to $e$ as in
(2.19). Since $\alpha_{n-1}\in\ell^{*}$ is primitive, the stabilizer $H_{e}$
of $(M_{e},\omega_{e},\Phi_{e})$ is precisely
$\exp\left(\mathrm{Ann}\left(\mathbb{R}\langle\alpha_{n-1}\rangle\right)\right)$.
Since $\left(M,\omega,\Phi\right)$ is tall and compact,
$(M_{e},\omega_{e},\Phi_{e})$ is a compact tall complexity one Hamiltonian
$T/H_{e}\simeq S^{1}$-space by property (ii) of Corollary 2.68 and by
Proposition 2.75. In what follows, $\Phi_{e}$ is chosen so that
$\Phi_{e}(M_{e})=[0,t_{\max}]$. Moreover, the isolated fixed points for the
$S^{1}$-action on $M_{e}$ are precisely the isolated fixed points for the
$T$-action contained in $M_{e}$. By Corollary 2.71, the restriction of the
Duistermaat-Heckman function $DH\left(M,\omega,\Phi\right)$ to $e$ equals
$DH(M_{e},\omega_{e},\Phi_{e})$. Since $\langle
v+t\alpha_{n-1},\nu_{\min}\rangle=t-1$, in order to prove that (4.20) holds,
it suffices to check that the Duistermaat-Heckman function of
$(M_{e},\omega_{e},\Phi_{e})$ is the function $[0,t_{\max}]\to\mathbb{R}$
given by

(4.21) $t\mapsto 2-s(t-1)-k\Theta(t-1),$

where $\Theta:\mathbb{R}\to\mathbb{R}$ is the function

$t\mapsto\begin{cases}0&\text{if }t\leq 0\\\ t&\text{if }t\geq 0.\end{cases}$

By Proposition 4.6 and the definition of $\Phi_{e}$, any isolated fixed point
$p\in M_{e}$ for the $S^{1}$-action satisfies $\Phi_{e}(p)=1$ and its isotropy
weights are $+1,-1$. In particular, if $k>1$, then $t_{\max}>1$. Since
$(M_{e},\omega_{e},\Phi_{e})$ is tall and since the domain of the Duistermaat-
Heckman function is $\Phi_{e}(M_{e})=[0,t_{\mathrm{max}}]$ (see Definition
2.51), by [27, Lemma 2.12], we have that

(4.22)
$DH(M_{e},\omega_{e},\Phi_{e})(t)=\int\limits_{\Phi_{e}^{-1}(v)}\omega_{e}-tc_{1}(L_{e})[\Phi_{e}^{-1}(v)]-k\Theta(t-1)\qquad\text{for
all }t\in[0,t_{\mathrm{max}}],$

where $c_{1}(L_{e})$ is the first Chern class of the normal bundle $L_{e}$ to
$\Phi_{e}^{-1}(v)$ in $M_{e}$. Since $\left(M,\omega,\Phi\right)$ is
normalized monotone, since $\Phi^{-1}_{e}(v)=\Phi^{-1}(v)$ is a sphere by
Lemma 4.1, and by Lemma 4.2, we have that

(4.23)
$\int\limits_{\Phi_{e}^{-1}(v)}\omega_{e}=c_{1}(M)[\Phi^{-1}(v)]=2+\sum\limits_{j=1}^{n-1}c_{1}(L_{j})[\Phi^{-1}(v)]=2+c_{1}(L_{n-1})[\Phi^{-1}(v)],$

where $L_{1}\oplus\ldots\oplus L_{n-1}$ is the $T$-equivariant splitting of
the normal bundle to $\Phi^{-1}(v)$ in $M$, and the last equality follows from
Proposition 4.2. Combining equations (4.22) and (4.23), and observing that
$L_{e}=L_{n-1}$, we have that

$DH(M_{e},\omega_{e},\Phi_{e})(t)=2-s(t-1)-k\Theta(t-1),$

as desired. ∎

By Theorem 4.17, the number of isolated fixed points that lies in the preimage
of an edge that comes ouf of $\mathcal{F}_{\min}$ equals the number of
connected components of $M_{\mathrm{exc}}$. Hence, we can use Proposition 4.19
to prove the following result.

######  Theorem 4.20.

Let $\left(M,\omega,\Phi\right)$ be a compact normalized monotone tall
complexity one $T$-space of dimension $2n$. Let $s\in\mathbb{Z}$ be as in
Lemma 4.5 and let $k\geq 0$ be the number of connected components of
$M_{\mathrm{exc}}$. The Duistermaat-Heckman function
$DH\left(M,\omega,\Phi\right):\Phi(M)\to\mathbb{R}$ is given by

(4.24) $DH\left(M,\omega,\Phi\right)(w)=2-s\langle
w,\nu_{\min}\rangle-k\rho(w),$

where $\rho:\mathfrak{t}^{*}\to\mathbb{R}$ is the function given by

(4.25) $w\mapsto\begin{cases}0&\text{if }\langle w,\nu_{\min}\rangle\leq 0\\\
\langle w,\nu_{\min}\rangle&\text{if }\langle w,\nu_{\min}\rangle\geq
0.\end{cases}$

###### Proof.

First, we show that the interior of the intersection

$\Phi(M)\cap\\{w\in\mathfrak{t}^{*}\mid\pm\langle w,\nu_{\min}\rangle>0\\}$

consists entirely of regular values of $\Phi$. To this end, suppose that
$w\in\mathfrak{t}^{*}\smallsetminus\partial\Phi(M)$ is a singular value of
$\Phi$. Hence, there exists $q\in\Phi^{-1}(w)$ that has a stabilizer has
positive dimension. By [28, Corollary 4.9], $q$ is exceptional. Thus, by
Proposition 4.15 and Lemma 4.16, $\Phi(q)\in\\{w\in\mathfrak{t}^{*}\mid\langle
w,\nu_{\min}\rangle=0\\}$, as desired. Hence, by Remark 2.47, the restriction
of $DH\left(M,\omega,\Phi\right)$ to
$\Phi(M)\cap\\{w\in\mathfrak{t}^{*}\mid\pm\langle w,\nu_{\min}\rangle>0\\}$ is
the restriction of an affine function $f^{\pm}:\mathfrak{t}^{*}\to\mathbb{R}$
of the form $f^{\pm}(w)=c^{\pm}+\langle w,\beta^{\pm}\rangle$ for some
$c^{\pm}\in\mathbb{R}$ and some $\beta^{\pm}\in\ell$.

We fix a vertex $v\in\mathcal{F}_{\min}$, we let $e$ be the edge that comes
out of $v$ and $\alpha_{1},\ldots,\alpha_{n-1}$ be the non-zero isotropy
weights of $\Phi^{-1}(v)$ ordered so that (4.3) and (4.4) hold. Since
$DH\left(M,\omega,\Phi\right)$ is continuous (by Theorem 2.49) and since the
restriction of $DH\left(M,\omega,\Phi\right)$ to $\mathcal{F}_{\min}$ is
constant by Proposition 4.2, the restriction of the affine function $f^{-}$ to
the affine hyperplane supporting $\mathcal{F}_{\min}$ is constant. Since
$f^{-}$ is an affine function and since $\mathcal{F}_{\min}$ is supported on
the hyperplane $v+\mathbb{R}\langle\alpha_{1},\ldots,\alpha_{n-2}\rangle$, the
restriction of the linear part of $f^{-}$ to
$\mathbb{R}\langle\alpha_{1},\ldots,\alpha_{n-2}\rangle$ is identically zero.
In other words,
$\beta^{-}\in\mathrm{Ann}\left(\mathbb{R}\langle\alpha_{1},\ldots,\alpha_{n-2}\rangle\right)$.
Hence, since
$\nu_{\min}\in\mathrm{Ann}\left(\mathbb{R}\langle\alpha_{1},\ldots,\alpha_{n-2}\rangle\right)$,
since $\beta^{-},\nu_{\min}\in\ell$, and since $\nu_{\min}$ is primitive,
there exists $\lambda^{-}\in\mathbb{Z}$ such that
$\beta^{-}=\lambda^{-}\nu_{\min}$.

By Lemma 3.17, there exists $t_{\max}\in\mathbb{Z}_{>0}$ such that

$e=\\{v+t\alpha_{n-1}\mid 0\leq t\leq t_{\max}\\}.$

By (4.5) and since $v\in\mathcal{F}_{\min}$, $\langle
v+t\alpha_{n-1},\nu_{\min}\rangle=-1+t$ for all $0\leq t\leq t_{\max}$.
Therefore, by Proposition 4.19, the restriction of
$DH\left(M,\omega,\Phi\right)$ to $e\cap\\{w\in\mathfrak{t}^{*}\mid\langle
w,\nu_{\min}\rangle<0\\}$ is given by the function that sends $t$ to $2+s-st$,
where $0\leq t<1$. Hence,

(4.26) $c^{-}+\lambda^{-}(-1+t)=2+s-st\text{ for all }0<t<1.$

Equation (4.26) readily implies that $\lambda^{-}=-s$ and $c^{-}=2$. Hence,
$f^{-}(w)=2-s\langle w,\nu_{\min}\rangle$.

We split the remainder of the proof in two cases, depending on whether
$t_{\max}=1$ or $t_{\max}\geq 2$. In the former case, the other vertex
$v^{\prime}$ of $\Phi(M)$ that is incident to $e$ lies on the linear
hyperplane $\\{w\in\mathfrak{t}^{*}\mid\langle w,\nu_{\min}\rangle=0\\}$.
Hence, by Corollary 4.14, there are no isolated fixed points. Therefore, by
Lemma 2.61, the function $DH\left(M,\omega,\Phi\right)$ is the restriction of
an affine function. Since the interior of
$\Phi(M)\cap\\{w\in\mathfrak{t}^{*}\mid-\langle w,\nu_{\min}\rangle>0\\}$ is
not empty and since the restriction of $DH\left(M,\omega,\Phi\right)$ to this
subset equals the affine function $f^{-}(w)=2-s\langle w,\nu_{\min}\rangle$,
it follows that

$DH\left(M,\omega,\Phi\right)(w)=2-s\langle w,\nu_{\min}\rangle=2-s\langle
w,\nu_{\min}\rangle-k\rho(w)\quad\text{for all }w\in\Phi(M),$

where the last equality follows from the fact that $k=0$, since there are no
isolated fixed points (see Lemma 2.20 and Corollary 2.21).

It remains to consider the case $t_{\max}\geq 2$. Since
$DH\left(M,\omega,\Phi\right)$ is continuous, then the restriction of $f^{-}$
and $f^{+}$ to $\Phi(M)\cap\\{w\in\mathfrak{t}^{*}\mid\langle
w,\nu_{\min}\rangle=0\\}$ are equal. Hence, the restriction of the affine
function $f^{+}$ to $\Phi(M)\cap\\{w\in\mathfrak{t}^{*}\mid\langle
w,\nu_{\min}\rangle=0\\}$ equals 2.  By Proposition 3.18, $\Phi(M)$ is a
reflexive Delzant polytope; thus, by Lemma 3.13, the origin lies in the
interior of $\Phi(M)$. Therefore, the interior of
$\Phi(M)\cap\\{w\in\mathfrak{t}^{*}\mid\langle w,\nu_{\min}\rangle=0\\}$ in
$\\{w\in\mathfrak{t}^{*}\mid\langle w,\nu_{\min}\rangle=0\\}$ is non-empty.
Hence, the restriction of the linear part of $f^{+}$ to the hyperplane
$\\{w\in\mathfrak{t}^{*}\mid\langle w,\nu_{\min}\rangle=0\\}$ is identically
zero. Arguing as above, this implies that there exists
$\lambda^{+}\in\mathbb{Z}$ such that $\beta^{+}=\lambda^{+}\nu_{\min}$.

Since $t_{\max}\geq 2$, the intersection
$e\cap\\{w\in\mathfrak{t}^{*}\mid\langle w,\nu_{\min}\rangle>0\\}$ is not
empty. By Proposition 4.19, the restriction of $DH\left(M,\omega,\Phi\right)$
to $e\cap\\{w\in\mathfrak{t}^{*}\mid\langle w,\nu_{\min}\rangle>0\\}$ is the
function that sends $t$ to $2+s-st-k(t-1)$ for $1<t\leq t_{\max}$. Hence, we
have that

(4.27) $\lambda^{+}(-1+t)+2=2+s-st-k(t-1)\text{ for all }1<t<t_{\max}.$

Equation (4.27) readily implies that $\lambda^{+}=-s-k$. Hence,
$f^{+}(w)=2-s\langle w,\nu_{\min}\rangle-k\langle w,\nu_{\min}\rangle$.

Thus the restriction of $DH\left(M,\omega,\Phi\right)$ to
$\Phi(M)\smallsetminus\\{w\in\mathfrak{t}^{*}\mid\langle
w,\nu_{\min}\rangle=0\\}$ is the function given by

$w\mapsto\begin{cases}2-s\langle w,\nu_{\min}\rangle&\text{if }\langle
w,\nu_{\min}\rangle<0\\\ 2-s\langle w,\nu_{\min}\rangle-k\langle
w,\nu_{\min}\rangle&\text{if }\langle w,\nu_{\min}\rangle>0.\end{cases}$

This function equals the restriction of (4.24) on the dense subset
$\Phi(M)\smallsetminus\\{w\in\mathfrak{t}^{*}\mid\langle
w,\nu_{\min}\rangle=0\\}$ of $\Phi(M)$. Since $DH\left(M,\omega,\Phi\right)$
is continuous, equation (4.24) holds. ∎

###  4.5. The Duistermaat-Heckman function is a complete invariant

We can proceed with the proof of our first main result.

###### Proof of Theorem 1.4.

Let $\left(M,\omega,\Phi\right)$ and
$(M^{\prime},\omega^{\prime},\Phi^{\prime})$ be compact monotone tall
complexity one $T$-spaces of dimension $2n$. If they are isomorphic, then they
have equal Duistermaat-Heckman measures. By Theorem 2.49, they have equal
Duistermaat-Heckman functions.

Suppose conversely that they have equal Duistermaat-Heckman functions. First,
we show that it suffices to show the result under the additional assumption
that both spaces are normalized. To this end, assume this special case. Since
$\left(M,\omega,\Phi\right)$ and $(M^{\prime},\omega^{\prime},\Phi^{\prime})$
are monotone, by Corollary 3.9, there exist $\lambda,\lambda^{\prime}>0$ and
$v,v^{\prime}\in\mathfrak{t}^{*}$ such that $(M,\lambda\omega,\lambda\Phi+v)$
and
$(M^{\prime},\lambda^{\prime}\omega^{\prime},\lambda^{\prime}\Phi^{\prime}+v^{\prime})$
are normalized monotone. Moreover, since $\left(M,\omega,\Phi\right)$ and
$(M^{\prime},\omega^{\prime},\Phi^{\prime})$ are tall and have equal
Duistermaat-Heckman functions, there exists a Delzant polytope $\Delta$ such
that $\Phi(M)=\Delta=\Phi^{\prime}(M^{\prime})$. Since
$(M,\lambda\omega,\lambda\Phi+v)$ and
$(M^{\prime},\lambda^{\prime}\omega^{\prime},\lambda^{\prime}\Phi^{\prime}+v^{\prime})$
are normalized monotone, by Proposition 3.18, the polytopes $\lambda\Delta+v$
and $\lambda^{\prime}\Delta+v^{\prime}$ are reflexive Delzant. We observe that
there exists at most one reflexive Delzant polytope obtained from $\Delta$ by
rescaling and translating because the vertices of a reflexive Delzant polytope
are determined by the tangent cones to its vertices (see Proposition 3.15).
Hence, $\lambda=\lambda^{\prime}$ and $v=v^{\prime}$; in particular,
$\lambda\Delta+v=\lambda^{\prime}\Delta+v^{\prime}$. On the other hand, we
observe that, for any $w\in\lambda\Delta+v=\lambda^{\prime}\Delta+v^{\prime}$,

(4.28) $\begin{split}DH(M,\lambda\omega,\lambda\Phi+v)(w)&=\lambda
DH\left(M,\omega,\Phi\right)(\lambda w+v)\\\
DH(M^{\prime},\lambda^{\prime}\omega^{\prime},\lambda^{\prime}\Phi^{\prime}+v^{\prime})(w)&=\lambda^{\prime}DH(M^{\prime},\omega^{\prime},\Phi^{\prime})(\lambda^{\prime}w+v^{\prime}).\end{split}$

This is an immediate consequence of the definition of the Duistermaat-Heckman
function, of the fact that its restriction to the set of regular values is
given by (2.21), and of the complexity of both spaces being one. Hence,
$DH(M,\lambda\omega,\lambda\Phi+v)=DH(M^{\prime},\lambda^{\prime}\omega^{\prime},\lambda^{\prime}\Phi^{\prime}+v^{\prime})$.
By assumption, $(M,\lambda\omega,\lambda\Phi+v)$ and
$(M^{\prime},\lambda^{\prime}\omega^{\prime},\lambda^{\prime}\Phi^{\prime}+v^{\prime})$
are isomorphic. Therefore, by Lemma 3.10, it follows that
$\left(M,\omega,\Phi\right)$ and $(M^{\prime},\omega^{\prime},\Phi^{\prime})$
are isomorphic, as desired.

It remains to prove the result under the additional assumption that the spaces
are normalized. By Remark 2.48 and Theorem 2.49, the two spaces have equal
Duistermaat-Heckman measures. Moreover, both spaces have genus zero by Lemma
4.1, and equal moment map images, both of which are equal to a reflexive
Delzant polytope $\Delta$ by Proposition 3.18. Since the Duistermaat-Heckman
functions are equal , there is a facet $\mathcal{F}_{\min}$ of $\Delta$ that
is a minimal facet for both spaces. Let $s,s^{\prime}\in\mathbb{Z}$ be the
integers as in Lemma 4.5 for $\left(M,\omega,\Phi\right)$ and
$(M^{\prime},\omega^{\prime},\Phi^{\prime})$ respectively. Since the
Duistermaat-Heckman functions of $\left(M,\omega,\Phi\right)$ and
$(M^{\prime},\omega^{\prime},\Phi^{\prime})$ are equal, by Theorem 4.20,

(4.29) $2-s\langle w,\nu_{\min}\rangle-k\rho(w)=2-s^{\prime}\langle
w,\nu_{\min}\rangle-k^{\prime}\rho(w)\text{ for all }w\in\Delta,$

where $\rho:\mathfrak{t}^{*}\to\mathbb{R}$ is the function of equation (4.25).
Since the interior of $\Delta$ is not empty and by (4.3), there exists a point
$w\in\Delta$ such that $-1<\langle w,\nu_{\min}\rangle<0$. Evaluating both
sides of (4.29) at this point, we obtain that $s=s^{\prime}$. By the same
argument and since $\Delta$ is reflexive so that the origin is an interior
point of $\Delta$, there exists $w^{\prime}\in\Delta$ such that $\langle
w^{\prime},\nu_{\min}\rangle>0$. Evaluating both sides of (4.29) at
$w^{\prime}$, we obtain that $k=k^{\prime}$. Hence, either both
$M_{\mathrm{exc}}$ and $M^{\prime}_{\mathrm{exc}}$ are empty or neither is.

We suppose first that neither $M_{\mathrm{exc}}$ nor
$M^{\prime}_{\mathrm{exc}}$ is empty. As in the proof of Theorem 4.18, we
write

$M_{\mathrm{exc}}=\coprod\limits_{j=1}^{k}M_{j}\qquad\text{ and }\qquad
M^{\prime}_{\mathrm{exc}}=\coprod\limits_{j=1}^{k}M^{\prime}_{j},$

where $M_{j}$ is a connected component of $M_{\mathrm{exc}}$ for all
$j=1,\ldots,k$, and analogously for $M^{\prime}_{j}$ and
$M_{\mathrm{exc}}^{\prime}$. By Theorem 4.18, there exist trivial paintings
$f:M_{\mathrm{exc}}\to S^{2}$ and $f^{\prime}:M^{\prime}_{\mathrm{exc}}\to
S^{2}$ of $\left(M,\omega,\Phi\right)$ and
$(M^{\prime},\omega^{\prime},\Phi^{\prime})$ respectively (see Definition
2.79). Since $f,f^{\prime}$ are paintings, if $i\neq j$, $f(M_{i})\neq
f(M_{j})$ and $f^{\prime}(M^{\prime}_{i})\neq f^{\prime}(M^{\prime}_{j})$. In
particular, the images of both $f$ and $f^{\prime}$ consist of $k$ distinct
points in $S^{2}$. We claim that we may assume that
$f(M_{j})=f^{\prime}(M^{\prime}_{j})$ for all $j=1,\ldots,k$. To see this, we
can use the argument of [10] to construct an orientation-preserving
diffeomorphism $\xi:S^{2}\to S^{2}$ such that
$\xi(f(M_{j}))=f^{\prime}(M^{\prime}_{j})$ for all $j=1,\ldots,k$. The
composite $\xi\circ f$ is a trivial painting of $\left(M,\omega,\Phi\right)$
that is equivalent to $f$.

By Theorem 4.17, for all $j=1,\ldots,n$, the orbital moment maps
$\overline{\Phi}$ (respectively $\overline{\Phi}^{\prime}$) maps each
connected component of $M_{j}$ (respectively $M^{\prime}_{j}$)
homeomorphically onto

$\Delta\cap\\{w\in\mathfrak{t}^{*}\mid\langle w,\nu_{\min}\rangle=0\\},$

where we use the fact that $\Phi(M)=\Delta=\Phi^{\prime}(M^{\prime})$. If
$\overline{\Phi}_{j}$ (respectively $\overline{\Phi}^{\prime}_{j}$) denotes
the restriction of the orbital moment map to $M_{j}$ (respectively
$M_{j}^{\prime}$), then the composite
$i_{j}:=(\overline{\Phi}^{\prime}_{j})^{-1}\circ\overline{\Phi}_{j}:M_{j}\to
M^{\prime}_{j}$ is a homeomorphism that satisfies
$\overline{\Phi}=\overline{\Phi}^{\prime}\circ i_{j}$. Hence, since
$M_{\mathrm{exc}}$ and $M^{\prime}_{\mathrm{exc}}$ have the same number of
connected components, there is a unique map $i:M_{\mathrm{exc}}\to
M^{\prime}_{\mathrm{exc}}$ that, when restricted to $M_{j}$, equals $i_{j}$
for all $j=1,\ldots,n$. Thus $i$ is a homeomorphism such that
$\overline{\Phi}=\overline{\Phi}^{\prime}\circ i$. Moreover, since the moment
map images of $\left(M,\omega,\Phi\right)$ and
$(M^{\prime},\omega^{\prime},\Phi^{\prime})$ are equal, by Remark 4.12, the
homeomorphism $i$ maps each orbit to an orbit with the same symplectic slice
representation. Hence, $i:M_{\mathrm{exc}}\to M^{\prime}_{\mathrm{exc}}$ is an
isomorphism of exceptional orbits such that $f^{\prime}=i\circ f$. Thus
$\left(M,\omega,\Phi\right)$ and $(M^{\prime},\omega^{\prime},\Phi^{\prime})$
have equivalent paintings.

If $M_{\mathrm{exc}}=\emptyset=M^{\prime}_{\mathrm{exc}}$, then
$\left(M,\omega,\Phi\right)$ and of
$(M^{\prime},\omega^{\prime},\Phi^{\prime})$ also have equivalent paintings
(trivially). Hence, in either case, the result follows by Theorem 2.80. ∎

######  Remark 4.21.

We observe that, by Corollary 2.64, Theorem 1.4 can be restated equivalently
as saying that two compact monotone tall complexity one $T$-spaces are
isomorphic if and only if they have equal Duistermaat-Heckman polytopes.

##  5\. Realizability, extension to toric and finiteness results

###  5.1. Necessary conditions for the realization and a finiteness result

To state the first result of this subsection, we recall that, by Lemma 4.5,
there exists $s\in\mathbb{Z}$ such that, if $v\in\mathcal{F}_{\min}$ is a
vertex and $e$ is the edge of $\Phi(M)$ that comes out of $\mathcal{F}_{\min}$
and is incident to $v$, then the self-intersection of the sphere
$\Phi^{-1}(v)$ in the four-dimensional submanifold $\Phi^{-1}(e)$ is $s$.

######  Proposition 5.1.

Let $\left(M,\omega,\Phi\right)$ be a normalized monotone tall complexity one
Hamiltonian $T$-space, let $s\in\mathbb{Z}$ be as above and let
$k\in\mathbb{Z}$ be the number of connected components of $M_{\mathrm{exc}}$.
The pair $(s,k)\in\mathbb{Z}^{2}$ belongs to the set

$\\{(0,0),(-1,0),(-1,1),(-1,2)\\}.$

###### Proof.

We fix a vertex $v\in\mathcal{F}_{\min}$ and we let $e=e_{n-1}$ be the edge of
$\Phi(M)$ that comes out of $\mathcal{F}_{\min}$ and is incident to $v$. The
normal bundle $N$ to $\Sigma:=\Phi^{-1}(v)$ splits $T$-equivariantly as
$L_{1}\oplus\cdots\oplus L_{n-1}$. By Proposition 4.2, the first Chern number
$c_{1}(L_{j})[\Sigma]$, which agrees with the self-intersection of $\Sigma$ in
$\Phi^{-1}(e_{j})$, equals zero for all $j=1,\ldots,n-2$. Therefore, by Lemma
4.1 and equation (4.2) in its proof,

$0<2+c_{1}(L_{n-1})[\Sigma]=2+s\,.$

Moreover, by (2.33) in Lemma 2.63, $s=c_{1}(L_{n-1})[\Sigma]\leq 0$. Hence, we
conclude that $s\in\\{0,-1\\}$.

Suppose first that $s=0$. By Theorem 4.20, the Duistermaat-Heckman function
$DH\left(M,\omega,\Phi\right):\Phi(M)\to\mathbb{R}$ is given by $w\mapsto
2-k\rho(w)$, where $\rho:\mathfrak{t}^{*}\to\mathbb{R}$ is the function of
(4.25). Since $k\geq 0$, $2-k\rho(w)\leq 2$ for all $w\in\Phi(M)$. Since
$DH\left(M,\omega,\Phi\right)(v)=2$ and since $v\in\mathcal{F}_{\min}$, it
follows that $DH\left(M,\omega,\Phi\right)(w)=2$ for all $w\in\Phi(M)$. By
Proposition 3.18, $\Phi(M)$ is a reflexive (Delzant) polytope. Hence, by Lemma
3.13, $\Phi(M)$ contains the origin in its interior. Thus, by definition of
the function $\rho$, $k=0$.

Suppose that $s=-1$. We must show that $k\leq 2$. To this end, we may assume
that $k>0$. Let $\alpha\in\ell^{*}$ be the isotropy weight of $\Phi^{-1}(v)$
such that the edge $e$ is contained in the half-ray $v+\mathbb{R}_{\geq
0}\langle\alpha\rangle$. By Lemma 3.17, let $t_{\max}\in\mathbb{Z}_{>0}$ be
such that $e=\\{v+t\alpha\mid 0\leq t\leq t_{\max}\\}$. First, we prove that
$t_{\max}\geq 2$. Let $v^{\prime}\in\Phi(M)$ be the other vertex to which $e$
is incident. If $t_{\max}=1$, then $v^{\prime}$ is a vertex of $\Phi(M)$ that
lies on the linear hyperplane $\\{w\in\mathfrak{t}^{*}\mid\langle
w,\nu_{\min}\rangle=0\\}$. By the Convexity Package (Theorem 2.32), and
Proposition 4.6, there are no isolated fixed points in $\Phi^{-1}(e)$. Hence,
by Theorem 4.17, $M_{\mathrm{exc}}=\emptyset$, a contradictionḢence,
$t_{\max}\geq 2$, as desired. As a consequence, $v+2\alpha\in\Phi(M)$. To
conclude the proof, we evaluate $DH\left(M,\omega,\Phi\right)$ at $v+2\alpha$.
By (4.5) and the fact that $v\in\mathcal{F}_{\min}$, we obtain that $\langle
v+2\alpha,\nu_{\min}\rangle=1$. Moreover, since $\left(M,\omega,\Phi\right)$
is tall, $DH\left(M,\omega,\Phi\right)(w)>0$ for all $w\in\Phi(M)$. Therefore,
by (4.24),

$DH\left(M,\omega,\Phi\right)(v+2\alpha)=2+1-k>0.$

Since $k\in\mathbb{Z}$, it follows that $k\leq 2$. ∎

We proceed with the proof of our second main result.

###### Proof of Theorem 1.5.

Suppose that $\Phi(M)=\Delta$ is reflexive Delzant. Since
$\left(M,\omega,\Phi\right)$ is monotone, by Lemma 3.20,
$\left(M,\omega,\Phi\right)$ is normalized monotone. Since there are finitely
many facets of $\Phi(M)$ and since, by Proposition 5.1, there are finitely
many possibilities for $(s,k)$, by Theorem 4.20, there are finitely many
possibilities for $DH\left(M,\omega,\Phi\right)$. Hence, the result follows
from Theorem 1.4. ∎

By Theorem 4.17 and Proposition 5.1, and since there are precisely as many
edges that come out of $\mathcal{F}_{\min}$ as there are vertices of
$\mathcal{F}_{\min}$, we obtain the following bound on the number of isolated
fixed points.

######  Corollary 5.2.

Let $\left(M,\omega,\Phi\right)$ be a normalized monotone tall complexity one
$T$-space of dimension $2n$. If $m$ is the number of vertices of a minimal
facet, then there are precisely either zero, $m$ or $2m$ isolated fixed points
in $M$.

The next result gives a combinatorial property of the moment map image of a
normalized monotone tall complexity one $T$-space such that $M_{\mathrm{exc}}$
has two connected components.

######  Corollary 5.3.

Let $\left(M,\omega,\Phi\right)$ be a normalized monotone tall complexity one
$T$-space of dimension $2n$. Let $\mathcal{F}_{\min}$ be a minimal facet of
$\Phi(M)$ supported on the affine hyperplane
$\\{w\in\mathfrak{t}^{*}\mid\langle w,\nu_{\min}\rangle=-1\\}$. If
$M_{\mathrm{exc}}$ has two connected components, then there exists a minimal
facet $\mathcal{F}^{\prime}_{\min}$ of $\Phi(M)$ supported on the affine
hyperplane $\\{w\in\mathfrak{t}^{*}\mid\langle w,-\nu_{\min}\rangle=-1\\}$. In
particular, $\Phi(M)$ is contained in the strip
$\\{w\in\mathfrak{t}^{*}\mid-1\leq\langle w,\nu_{\min}\rangle\leq 1\\}$.

###### Proof.

We fix a vertex $v\in\mathcal{F}_{\min}$. Since the number $k$ of connected
components of $M_{\mathrm{exc}}$ is 2, by Proposition 5.1, it follows that
$s=-1$. In particular, by Theorem 4.20, the Duistermaat-Heckman function
$DH\left(M,\omega,\Phi\right):\Phi(M)\to\mathbb{R}$ of
$\left(M,\omega,\Phi\right)$ is given by

(5.1) $DH\left(M,\omega,\Phi\right)(w)=2+\langle
w,\nu_{\min}\rangle-2\rho(w),$

where $\rho:\Phi(M)\to\mathbb{R}$ is the non-negative function given by
(4.25). Since $v\in\mathcal{F}_{\min}$, $DH\left(M,\omega,\Phi\right)(v)=1$.
Hence, since $\mathcal{F}_{\min}$ is a minimal facet, the minimal value of
$DH\left(M,\omega,\Phi\right)$ equals 1.

Let $e$ be the edge of $\Phi(M)$ that comes out of $\mathcal{F}_{\min}$ and is
incident to $v$, and let $\alpha\in\ell^{*}$ be the isotropy weight of
$\Phi^{-1}(v)$ so that $e$ is contained in the half-ray $v+\mathbb{R}_{\geq
0}\langle\alpha\rangle$. By (4.5) and Proposition 4.2,
$\langle\alpha,\nu_{\min}\rangle=1$ and there exists
$t_{\max}\in\mathbb{Z}_{>0}$ such that $e=\\{v+t\alpha\mid 0\leq t\leq
t_{\max}\\}$. We set $v^{\prime}:=v+t_{\max}\alpha$; this is a vertex of
$\Phi(M)$. Moreover, we observe that $\langle
v^{\prime},\nu_{\min}\rangle=t_{\max}-1$. Since $s=-1$ and $k=2$, arguing as
in the last paragraph of the proof of Proposition 5.1, $t_{\max}\geq 2$. In
particular, by (5.1) and since the minimal value of
$DH\left(M,\omega,\Phi\right)$ is 1,

$DH\left(M,\omega,\Phi\right)(v^{\prime})=3-t_{\max}\geq 1,$

whence $t_{\max}\leq 2$. Hence, $t_{\max}=2$ and
$DH\left(M,\omega,\Phi\right)(v^{\prime})=1$, so that
$DH\left(M,\omega,\Phi\right)$ attains its minimum at $v^{\prime}$. By
Proposition 4.2, $v^{\prime}$ lies on a minimal facet
$\mathcal{F}_{\min}^{\prime}$. In fact, $\mathcal{F}_{\min}^{\prime}$ is
contained in the connected component of the level set
$(DH\left(M,\omega,\Phi\right))^{-1}(1)$ that contains $v^{\prime}$. By (5.1),
the latter is given by the affine hyperplane
$\\{w\in\mathfrak{t}^{*}\mid\langle w,-\nu_{\min}\rangle=-1\\}$. Since the
affine span of a facet is an affine hyperplane, the first statement follows.
The second statement follows at once from the first and the fact that
$\Phi(M)$ is contained in the intersection $\\{w\in\mathfrak{t}^{*}\mid\langle
w,\nu_{\min}\rangle\geq-1\\}\cap\\{w\in\mathfrak{t}^{*}\mid\langle
w,-\nu_{\min}\rangle\geq-1\\}$. ∎

The next result is the most important building block of the main finiteness
result of this paper, Corollary 1.6.

######  Corollary 5.4.

Given a reflexive Delzant polytope $\Delta$ in $\mathfrak{t}^{*}$, there are
finitely many isomorphism classes of normalized monotone tall complexity one
$T$-spaces with moment map image equal to $\Delta$.

###### Proof.

Let $\mathcal{F}$ be a facet of $\Delta$ and let
$(s,k)\in\\{(0,0),(-1,0),(-1,1),(-1,2)\\}$. Since $\Delta$ has finitely many
facets and by Proposition 5.1, it suffices to show that there are finitely
many isomorphism classes of normalized monotone tall complexity one $T$-spaces
with moment map image equal to $\Delta$ such that

  * •

$\mathcal{F}$ is a minimal facet of $\Phi(M)$,

  * •

given any vertex $v\in\mathcal{F}$ and the edge $e$ of $\Phi(M)$ that comes
out of $\mathcal{F}$ and is incident to $v$, the self-intersection of
$\Phi^{-1}(v)$ in $\Phi^{-1}(e)$ equals $s$ (cf. Lemma 4.5), and

  * •

the set of exceptional orbits has precisely $k$ connected components.

If there is no compact normalized monotone tall complexity one $T$-space with
the above properties, there is nothing to prove, so we may assume that there
exists such a space $\left(M,\omega,\Phi\right)$. By Theorem 4.20, the data
$\Delta,\mathcal{F}$ and $(s,k)$ determine uniquely the Duistermaat-Heckman
function of $\left(M,\omega,\Phi\right)$. Hence, by Theorem 1.4, there is
exactly one isomorphism class of compact normalized monotone tall complexity
one $T$-space with the above properties, as desired. ∎

Theorem 1.4 and Corollary 5.4 allow us to prove Corollary 1.6, thus answering
a question posed to us by Yael Karshon. We recall that two Hamiltonian
$T$-spaces $(M_{1},\omega_{1},\Phi_{1})$ and $(M_{2},\omega_{2},\Phi_{2})$ are
equivalent if there exists a symplectomorphism
$\Psi:(M_{1},\omega_{1})\to(M_{2},\omega_{2})$ and an affine transformation
$a\in\mathrm{GL}(\ell^{*})\ltimes\mathfrak{t}^{*}$ such that
$\Phi_{2}\circ\Psi=a\circ\Phi_{1}$. In this case, we write
$(M_{1},\omega_{1},\Phi_{1})\sim(M_{2},\omega_{2},\Phi_{2})$.

######  Remark 5.5.

  * •

In the above notion of equivalence, the reason why we restrict to elements in
$\mathrm{GL}(\ell^{*})\ltimes\mathfrak{t}^{*}$ is the following: Given an
effective Hamiltonian $T$-space $\left(M,\omega,\Phi\right)$ and an affine
transformation $a$ of $\mathfrak{t}^{*}$, the triple $(M,\omega,a\circ\Phi)$
is an effective Hamiltonian $T$-space if and only if
$a\in\mathrm{GL}(\ell^{*})\ltimes\mathfrak{t}^{*}$.

  * •

Isomorphic Hamiltonian $T$-spaces in the sense of Definition 2.1 are
necessarily equivalent, but the converse need not hold.

###### Proof of Corollary 1.6.

We fix $n$ and we denote the set of equivalence classes of compact tall
complexity one $T$-spaces of dimension $2n$ with first Chern class equal to
the class of the symplectic form by $\mathcal{M}_{n}$. By Definition 3.7, any
normalized monotone tall complexity one $T$-space of dimension $2n$ is such
that its first Chern class equals the class of the symplectic form. We define
an auxiliary equivalence relation $\approx$ on the set of normalized monotone
tall complexity one $T$-spaces of dimension $2n$ as follows: Given two such
spaces $(M_{1},\omega_{1},\Phi_{1}),(M_{2},\omega_{2},\Phi_{2})$, we say that
$(M_{1},\omega_{1},\Phi_{1})\approx(M_{2},\omega_{2},\Phi_{2})$ if there
exists a linear transformation $l\in\mathrm{GL}(\ell^{*})$ such that
$\Phi_{2}\circ\Psi=l\circ\Phi_{1}$. We denote the set of $\approx$-equivalence
classes of normalized monotone tall complexity one $T$-spaces of dimension
$2n$ by $\mathcal{NM}_{n}$.

We observe that there is a natural map $\mathcal{NM}_{n}\to\mathcal{M}_{n}$
sending the $\approx$-equivalence class of $\left(M,\omega,\Phi\right)$ to its
$\sim$-equivalence class. Moreover, we claim that this map is a bijection.
First, we show that it is injective. Suppose that
$(M_{1},\omega_{1},\Phi_{1}),(M_{2},\omega_{2},\Phi_{2})$ are normalized
monotone tall complexity one $T$-spaces of dimension $2n$ such that
$(M_{1},\omega_{1},\Phi_{1})\sim(M_{2},\omega_{2},\Phi_{2})$. Then there
exists $a\in\mathrm{GL}(\ell^{*})\ltimes\mathfrak{t}^{*}$ such that
$a(\Phi_{1}(M_{1}))=\Phi_{2}(M_{2})$. We write $a=(l,v)$ for unique
$l\in\mathrm{GL}(\ell^{*})$ and $v\in\mathfrak{t}^{*}$. It suffices to show
that $v=0$. By Proposition 3.18, both $\Phi_{1}(M_{1})$ and $\Phi_{2}(M_{2})$
are reflexive (Delzant) polytopes. In particular, all vertices of
$\Phi_{1}(M_{1})$ and of $\Phi_{2}(M_{2})$ lie in $\ell^{*}$. Since both
polytopes have at least one vertex and since $l\in\mathrm{GL}(\ell^{*})$,
$v\in\ell^{*}$. Moreover, by Lemma 3.13, the origin is the only interior
lattice point in both $\Phi_{1}(M_{1})$ and $\Phi_{2}(M_{2})$. Hence, since
$a(\Phi_{1}(M_{1}))=\Phi_{2}(M_{2})$, the lattice point $v$ lies in the
interior of $\Phi_{2}(M_{2})$. Thus $v=0$, as desired. Next we prove
surjectivity. To this end, let $\left(M,\omega,\Phi\right)$ be a compact tall
complexity one $T$-space of dimension $2n$ with $c_{1}(M)=[\omega]$. By
Proposition 3.5, there exists (a unique) $v\in\mathfrak{t}^{*}$ such that
$(M,\omega,\Phi+v)$ is normalized. Hence,
$\left(M,\omega,\Phi\right)\sim(M,\omega,\Phi+v)$, as desired.

Therefore it suffices to prove that $\mathcal{NM}_{n}$ is finite. To this end,
we denote the orbit space of the standard $\mathrm{GL}(\ell^{*})$-action on
the set of reflexive Delzant polytopes in $\mathfrak{t}^{*}$ by
$\mathcal{RD}_{n}$. By Proposition 3.18, the map
$p:\mathcal{NM}_{n}\to\mathcal{RD}_{n}$ that sends the $\approx$-equivalence
class of $\left(M,\omega,\Phi\right)$ to the $\mathrm{GL}(\ell^{*})$-orbit of
$\Phi(M)$ is surjective. By Corollary 3.14, $\mathcal{RD}_{n}$ is finite.
Hence, it suffices to prove that the fibers of the above map are finite. We
fix a reflexive Delzant polytope $\Delta$ and we consider the map from the set
of isomorphism classes of normalized monotone tall complexity one $T$-spaces
with moment map image equal to $\Delta$ to $p^{-1}([\Delta])$ that sends the
isomorphism class of $\left(M,\omega,\Phi\right)$ to its $\approx$-equivalence
class. This map is surjective: If $\left(M,\omega,\Phi\right)$ is such that
$[\Phi(M)]=[\Delta]$, then there exists $l\in\mathrm{GL}(\ell^{*})$ such that
$\Delta=l(\Phi(M))$. The isomorphism class of $(M,\omega,l\circ\Phi)$ is then
mapped to the $\approx$-equivalence class of $\left(M,\omega,\Phi\right)$. The
result now follows from Corollary 5.4. ∎

###  5.2. Sufficient conditions for the realization and extension to a toric
action

Let $\left(M,\omega,\Phi\right)$ be a normalized monotone tall complexity one
$T$-space. By the results of Section 4, there exists a quadruple
$(\Delta,\mathcal{F},s,k)$ determined by $\left(M,\omega,\Phi\right)$, where
$\Delta=\Phi(M)$, $\mathcal{F}$ is a facet of $\Delta$ that is a minimal facet
of $\left(M,\omega,\Phi\right)$ (see Definition 4.3, $s\in\mathbb{Z}$ is the
the self intersection of the sphere $\Phi^{-1}(v)$ in $\Phi^{-1}(e)$, where
$v\in\mathcal{F}$ is any vertex and $e$ is an edge of $\Delta$ that comes out
of $\mathcal{F}$ and is incident to $v$, and $k\in\mathbb{Z}$ is the number of
connected components of $M_{\mathrm{exc}}$. The overall aim of this section is
to determine which quadruples arise in this fashion (see Corollary 5.15). So
far, we have established the following necessary conditions:

  * (i)

$\Delta$ is a full-dimensional reflexive Delzant polytope in
$\mathfrak{t}^{*}$ (Propositions 3.18 and 2.75).

  * (ii)

$\mathcal{F}$ is a facet of $\Delta$ that is supported on the affine
hyperplane $\\{w\in\mathfrak{t}^{*}\mid\langle w,\nu\rangle=-1\\}$.

  * (iii)

The pair $(s,k)$ belongs to the set $\\{(0,0),(-1,0),(-1,1),(-1,2)\\}$
(Proposition 5.1).

  * (iv)

If there is a vertex of $\Phi(M)$ on the linear hyperplane
$\\{w\in\mathfrak{t}^{*}\mid\langle w,\nu\rangle=0\\}$, then $k=0$ (Corollary
4.14).

  * (v)

If $k=2$ then there exists a facet $\mathcal{F}^{\prime}$ supported on the
hyperplane $\\{w\in\mathfrak{t}^{*}\mid\langle w,-\nu\rangle=-1\\}$ (Corollary
5.3).

The first step towards proving which quadruples are associated to a normalized
monotone tall complexity one $T$-space and whether the $T$ action extends to a
toric action (Corollary 5.15) is establishing its combinatorial analogue,
namely Theorem 5.12. To this end, we introduce the following terminology.

######  Definition 5.6.

We say that a quadruple $(\Delta,\mathcal{F},s,k)$ consisting of a polytope
$\Delta$ in $\mathfrak{t}^{*}$, a facet $\mathcal{F}\subset\Delta$, and
integers $s,k$, is admissible if it satisfies conditions (i)–(v) above.

######  Remark 5.7.

If $(\Delta,\mathcal{F},-1,2)$ is admissible, then, by the proof of Corollary
5.3, the polytope $\Delta$ is contained in the strip
$\\{w\in\mathfrak{t}^{*}\mid-1\leq\langle w,\nu\rangle\leq 1\\}$.

######  Lemma 5.8.

Let $k\in\\{1,2\\}$. If $(\Delta,\mathcal{F},-1,k)$ is admissible, then for
any edge $e$ of $\Delta$ that intersects the linear hyperplane
$\\{w\in\mathfrak{t}^{*}\mid\langle w,\nu\rangle=0\\}$, there exists a vertex
$v\in\mathcal{F}$ and a weight $\alpha_{v}\in\ell^{*}$ at $v$ such that

$e\cap\\{w\in\mathfrak{t}^{*}\mid\langle
w,\nu\rangle=0\\}=\\{v+\alpha_{v}\\}.$

In particular, $e\cap\\{w\in\mathfrak{t}^{*}\mid\langle w,\nu\rangle=0\\}$ is
contained in $\ell^{*}$.

###### Proof.

Fix such an edge $e$. Since $(\Delta,\mathcal{F},-1,k)$ is admissible and
$k>0$, $\Delta$ has no vertex on the linear hyperplane
$\\{w\in\mathfrak{t}^{*}\mid\langle w,\nu\rangle=0\\}$. Hence, $e$ intersects
$\\{w\in\mathfrak{t}^{*}\mid\langle w,\nu\rangle=0\\}$ in the relative
interior of $e$, so that the intersection
$e\cap\\{w\in\mathfrak{t}^{*}\mid\langle w,\nu\rangle=0\\}$ consists of one
element, which is not a vertex since $(\Delta,\mathcal{F},-1,k)$ is
admissible. Let $v\in\Delta$ be the vertex that is incident to $e$ and
satisfies $\langle v,\nu\rangle<0$. Since $\Delta$ is integral and is
contained in the upper-half plane $\\{w\in\mathfrak{t}^{*}\mid\langle
w,\nu\rangle\geq-1\\}$, and since $\nu\in\ell^{*}$ is primitive, $\langle
v,\nu\rangle=-1$, i.e., $v\in\mathcal{F}$. Moreover, $e$ is the edge that
comes out of $\mathcal{F}$ that is incident to $v$. Let
$\alpha_{v}\in\ell^{*}$ be the weight of $v$ such that $e$ is contained in the
half-ray $v+\mathbb{R}_{\geq 0}\langle\alpha_{v}\rangle$. By Lemma 3.17,
$\langle v+\alpha_{v},\nu\rangle=0$, whence
$e\cap\\{w\in\mathfrak{t}^{*}\mid\langle
w,\nu\rangle=0\\}=\\{v+\alpha_{v}\\}$. Since $v,\alpha_{v}\in\ell^{*}$, the
result follows. ∎

Admissible quadruples encode the abstract analogs of the functions given by
(4.24) in Theorem 4.20.

######  Definition 5.9.

Let $(\Delta,\mathcal{F},s,k)$ be an admissible quadruple. The abstract
Duistermaat-Heckman function determined by $(\Delta,\mathcal{F},s,k)$ is the
map $\Delta\to\mathbb{R}$ that sends $w\in\Delta$ to

(5.2) $DH(w)=2-s\langle w,\nu\rangle-k\rho(w),$

where $\rho:\mathfrak{t}^{*}\to\mathbb{R}$ is given by

$w\mapsto\begin{cases}0&\text{if }\langle w,\nu\rangle\leq 0\\\ \langle
w,\nu\rangle&\text{if }\langle w,\nu\rangle\geq 0.\end{cases}$

######  Remark 5.10.

Let $(\Delta,\mathcal{F},s,k)$ and
$(\Delta^{\prime},\mathcal{F}^{\prime},s^{\prime},k^{\prime})$ be admissible
quadruples. By the arguments in the proof of Theorem 1.4 (see page 4.5), if
the abstract Duistermaat-Heckman functions determined by
$(\Delta,\mathcal{F},s,k)$ and
$(\Delta^{\prime},\mathcal{F}^{\prime},s^{\prime},k^{\prime})$ are equal, then
$(\Delta,\mathcal{F},s,k)=(\Delta^{\prime},\mathcal{F}^{\prime},s^{\prime},k^{\prime})$.

In what follows, we fix the map
$\mathrm{pr}:\mathfrak{t}^{*}\times\mathbb{R}\to\mathfrak{t}^{*}$ given by
projection to the first component and we denote the Lebesgue measure on
$\mathbb{R}$ by $dy$. Given a polytope $\Delta^{\prime}$ in
$\mathfrak{t}^{*}\times\mathbb{R}$, the projection
$\mathrm{pr}(\Delta^{\prime})$ is a polytope in $\mathfrak{t}^{*}$. On such a
projection we define the combinatorial analog of the function constructed in
Example 2.52 (the terminology used below is not standard).

######  Definition 5.11.

Let $\Delta^{\prime}$ be a polytope in $\mathfrak{t}^{*}\times\mathbb{R}$. The
height function of $\Delta:=\mathrm{pr}(\Delta^{\prime})$ is the map
$\Delta\to\mathbb{R}$ that sends $w\in\Delta$ to

$\mathrm{Length}(\Delta^{\prime}_{w}):=\int\limits_{\Delta^{\prime}_{w}}dy,$

where $\Delta^{\prime}_{w}:=\mathrm{pr}^{-1}(w)\cap\Delta$.

The combinatorial realizability result is as follows.

######  Theorem 5.12.

For each admissible quadruple $(\Delta,\mathcal{F},s,k)$, there exists a
reflexive Delzant polytope $\Delta^{\prime}$ in
$\mathfrak{t}^{*}\times\mathbb{R}$ such that
$\mathrm{pr}(\Delta^{\prime})=\Delta$ and the height function of $\Delta$
equals the abstract Duistermaat-Heckman function determined by
$(\Delta,\mathcal{F},s,k)$.

In Figure 1.2 we provide the complete list of reflexive Delzant polytopes
$\Delta^{\prime}$ such that the projection is the reflexive square $\Delta$ of
Figure 1.1. Before turning to the proof of Theorem 5.12, following [44,
Section 2.4], we introduce an important construction on smooth polytopes.

######  Definition 5.13.

Let $\Delta$ be a Delzant polytope in $\mathfrak{t}^{*}$ given by
$\Delta=\bigcap_{i=1}^{l}\,\\{w\in\mathfrak{t}^{*}\mid\langle
w,\nu_{i}\rangle\geq c_{i}\\}$, let $\mathcal{F}$ be a face of $\Delta$ of
codimension at least two, and let $I\subset\\{1,\ldots,l\\}$ be the subset of
those indices corresponding to the facets containing $\mathcal{F}$. We set
$\nu_{0}:=\sum\limits_{i\in I}\nu_{i}$ and, given $\epsilon>0$, we also set
$c_{0}:=\epsilon+\sum\limits_{i\in I}c_{i}$. For any $\epsilon>0$ such that
any vertex $v$ of $\Delta$ not lying on $\mathcal{F}$ satisfies $\langle
v,\nu_{0}\rangle>c_{0}$, we define the blow-up of $\Delta$ along $\mathcal{F}$
of size $\epsilon$ to be the polytope

$\Delta\cap\\{w\in\mathfrak{t}^{*}\mid\langle w,\nu_{0}\rangle\geq c_{0}\\}.$

######  Remark 5.14.

As remarked in [44, Section 2.4], any blow-up of a Delzant polytope (along any
face and of any size) is a Delzant polytope (so long as the face has
codimension at least two and the size satisfies the condition stated in
Definition 5.13).

###### Proof of Theorem 5.12.

We fix an admissible quadruple $(\Delta,\mathcal{F},s,k)$. We split the proof
in two cases, depending depending on whether $k=0$ or not.

Case 1: $k=0$. The abstract Duistermaat-Heckman function is $DH(w)=2-s\langle
w,\nu\rangle$ for all $w\in\Delta$. We set

(5.3)
$\Delta_{(s,0)}^{\prime}:=\\{(w,y)\in\mathfrak{t}^{*}\times\mathbb{R}\mid
w\in\Delta\,,\,-1\leq y\leq DH(w)-1\\},$

where $DH:\Delta\to\mathbb{R}$ is the abstract Duistermaat-Heckman function
determined by $(\Delta,\mathcal{F},s,k)$ – see Figure 5.1. First, we claim
that $\mathrm{pr}(\Delta_{(s,0)}^{\prime})=\Delta$. To this end, it suffices
to prove that $DH(w)\geq 1$ for all $w\in\Delta$. This follows immediately
from the fact that, by Definition 5.6, $s\in\\{0,-1\\}$ and $\Delta$ is
contained in the upper half-space of $\mathfrak{t}^{*}$ given by
$\\{w\in\mathfrak{t}^{*}\mid\langle w,\nu\rangle\geq-1\\}$. By construction,
the height function of $\Delta_{(s,0)}^{\prime}$ equals $DH$, so it remains to
show that $\Delta_{(s,0)}^{\prime}$ is a reflexive Delzant polytope. To this
end, we write $\Delta$ in its minimal representation (see (1.1))

$\Delta=\bigcap_{i=1}^{l}\,\\{w\in\mathfrak{t}^{*}\mid\langle
w,\nu_{i}\rangle\geq-1\\},$

where $\nu_{i}\in\ell^{*}$ is primitive for all $i=1,\ldots,l$ and, without
loss of generality, the hyperplane supporting $\mathcal{F}$ is
$\\{w\in\mathfrak{t}^{*}\mid\langle w,\nu_{1}\rangle\geq-1\\}$, i.e.,
$\nu_{1}=\nu$. By (5.3),

(5.4)
$\begin{split}\Delta_{(s,0)}^{\prime}&=\\{(w,y)\in\mathfrak{t}^{*}\times\mathbb{R}\mid\langle(w,y),(0,1)\rangle\geq-1\\}\\\
&\cap\\{(w,y)\in\mathfrak{t}^{*}\times\mathbb{R}\mid\langle(w,y),(-s\nu,-1)\rangle\geq-1\\}\\\
&\cap\bigcap_{i=1}^{l}\,\\{(w,y)\in\mathfrak{t}^{*}\times\mathbb{R}\mid\langle(w,y),(\nu_{i},0)\rangle\geq-1\\},\end{split}$

where, by a slight abuse of notation, we denote the natural pairing between
$\mathfrak{t}^{*}\times\mathbb{R}$ and $\mathfrak{t}\times\mathbb{R}$ also by
$\langle\cdot,\cdot\rangle$. Therefore, $\Delta_{(s,0)}^{\prime}$ is a
polytope (see Section 1.3). Moreover, the vertices of
$\Delta_{(s,0)}^{\prime}$ are precisely the elements of the set

(5.5) $\\{(v,-1)\in\mathfrak{t}^{*}\times\mathbb{R}\mid v\in\Delta\text{
vertex}\\}\,\cup\,\\{(v,1-s\langle
v,\nu\rangle)\in\mathfrak{t}^{*}\times\mathbb{R}\mid v\in\Delta\text{
vertex}\\}.$

We observe that, since $\Delta$ is reflexive, any vertex of $\Delta$ lies in
$\ell^{*}$; since $\nu\in\ell$, it follows that, if $v\in\Delta$ is a vertex,
then $1-s\langle v,\nu\rangle\in\mathbb{Z}$. Hence, by (5.5), any vertex of
$\Delta_{(s,0)}^{\prime}$ lies in $\ell^{*}\times\mathbb{Z}$, i.e.,
$\Delta_{(s,0)}^{\prime}$ is integral. Since $\nu_{i}\in\ell^{*}$ is
primitive, $(\nu_{i},0)\in\ell^{*}\times\mathbb{Z}$ is primitive. Moreover,
since $\nu=\nu_{1}$ and since $s\in\\{0,-1\\}$,
$(-s\nu,-1)\in\ell^{*}\times\mathbb{Z}$ is also primitive. Hence, by (5.4),
$\Delta_{(s,0)}^{\prime}$ is reflexive. Finally, to see that
$\Delta_{(s,0)}^{\prime}$ is Delzant, we fix a vertex $v\in\Delta$. By (5.4),
the set of inward normals of the facets of $\Delta_{(s,0)}^{\prime}$ that
contain $(v,-1)$ (respectively $(v,1-s\langle v,\nu\rangle)$) consists of
$(0,1)$ (respectively $(-s\nu,-1)$), and of $\\{(\nu_{v},0)\\}$, where
$\\{\nu_{v}\\}$ is the set of inward normals of the facets of $\Delta$ that
contain $v$. Since $\Delta$ is Delzant, it follows that
$\Delta_{(s,0)}^{\prime}$ is smooth at $(v,-1)$ (respectively $(v,1-s\langle
v,\nu\rangle)$). Since any vertex of $\Delta_{(s,0)}^{\prime}$ is equal to
$(v,-1)$ or $(v,1-s\langle v,\nu\rangle)$ for some vertex $v$ of $\Delta$,
$\Delta_{(s,0)}^{\prime}$ is Delzant, thus completing the proof in this case.

Figure 5.1. A schematic construction of $\Delta_{(0,0)}^{\prime}$ (left) and
of $\Delta_{(-1,0)}^{\prime}$ (right).

Case 2: $k\neq 0$. Since $(\Delta,\mathcal{F},s,k)$ is admissible,
$(s,k)\in\\{(-1,1),(-1,2)\\}$. Moreover, by Definition 5.6, $\Delta$ has no
vertices on the linear hyperplane $\\{w\in\mathfrak{t}^{*}\mid\langle
w,\nu\rangle=0\\}$ and the quadruple $(\Delta,\mathcal{F},0,0)$ is also
admissible. Hence, by Case 1, there exists a reflexive Delzant polytope
$\Delta_{(0,0)}^{\prime}$ satisfying the conclusions of the statement for the
admissible quadruple $(\Delta,\mathcal{F},0,0)$. We deal with the cases
$(s,k)=(-1,1)$ and $(s,k)=(-1,2)$ separately.

$\bullet$ Suppose that $(s,k)=(-1,1)$. By (5.4), the reflexive Delzant
polytope $\Delta_{(0,0)}^{\prime}$ has a codimension two face
$\tilde{\mathcal{F}}$ given by the intersection of the facets supported by the
affine hyperplanes
$\\{(w,y)\in\mathfrak{t}^{*}\times\mathbb{R}\mid\langle(w,y),(0,-1)\rangle=-1\\}$
and
$\\{(w,y)\in\mathfrak{t}^{*}\times\mathbb{R}\mid\langle(w,y),(\nu,0)\rangle=-1\\}$.
This is a copy of $\mathcal{F}$ on the affine hyperplane
$\\{(w,y)\in\mathfrak{t}^{*}\times\mathbb{R}\mid y=1\\}$. We wish to perform
the blow-up of $\Delta^{\prime}_{(0,0)}$ along $\tilde{\mathcal{F}}$ of size
$1$ (see Figure 5.2). To this end, with the notation in Definition 5.13,
$\nu_{0}=(\nu,-1)$ and $c_{0}=-1$. By (5.5) and since $s=-1$, a vertex of
$\Delta^{\prime}_{(0,0)}$ that does not lie on $\tilde{\mathcal{F}}$ is either
of the form $(v,-1)$ for some vertex $v$ of $\Delta$ or of the form $(v,1)$
for some vertex $v$ of $\Delta$ that does not lie on $\mathcal{F}$. Since
$\langle w,\nu\rangle\geq-1$ for any $w\in\Delta$, if $v$ is a vertex of
$\Delta$, then

(5.6) $\langle(v,-1),(\nu,-1)\rangle\geq-1+1=0>-1=c_{0}.$

On the other hand, since there are no vertices of $\Delta$ lying on the linear
hyperplane $\\{w\in\mathfrak{t}^{*}\mid\langle w,\nu\rangle=0\\}$ and since
$\Delta$ is integral, if $v$ is a vertex of $\Delta$ that does not lie on
$\mathcal{F}$, then $\langle v,\nu\rangle\geq 1$. Hence, in this case,

(5.7) $\langle(v,-1),(\nu,1)\rangle\geq 1-1>-1=c_{0}.$

By (5.6) and (5.7), we can perform the the blow-up of
$\Delta^{\prime}_{(0,0)}$ along $\tilde{\mathcal{F}}$ of size $1$ that we
denote by $\Delta^{\prime}_{(-1,1)}$, i.e.,

(5.8)
$\Delta^{\prime}_{(-1,1)}=\Delta^{\prime}_{(0,0)}\cap\\{(w,y)\in\mathfrak{t}^{*}\times\mathbb{R}\mid\langle(w,y),(\nu,-1)\rangle\geq-1\\}.$

By (5.3) and (5.8), and since $s=-1$,

(5.9)
$\Delta^{\prime}_{(-1,1)}=\\{(w,y)\in\mathfrak{t}^{*}\times\mathbb{R}\mid
w\in\Delta\,,\,-1\leq y\leq\min(1,1+\langle w,\nu\rangle)\\}.$

Since $\Delta^{\prime}_{(0,0)}$ is Delzant, by Remark 5.14,
$\Delta^{\prime}_{(-1,1)}$ is also Delzant. By (5.9), it can be checked
directly that a vertex of $\Delta^{\prime}_{(-1,1)}$ is one of the following
three types:

  * •

$(v,\min(1,1+\langle v,\nu\rangle))$ for some vertex $v$ of $\Delta$,

  * •

$(w,1)$, where $w$ lies on an edge of $\Delta$ and satisfies $\langle
w,\nu\rangle=0$, or

  * •

$(v,-1)$ for some vertex $v$ of $\Delta$.

Since $\Delta$ is integral and since $\nu\in\ell^{*}$, if $v$ is a vertex of
$\Delta$, then $(v,\min(1,1+\langle v,\nu\rangle))$ and $(v,-1)$ belong to
$\ell^{*}\times\mathbb{Z}$. Moreover, by Lemma 5.8, if $w$ lies on an edge of
$\Delta$ and satisfies $\langle w,\nu\rangle=0$, then $w\in\ell^{*}$. Hence,
$(w,1)\in\ell^{*}\times\mathbb{Z}$, so that $\Delta^{\prime}_{(-1,1)}$ is
integral. Moreover, by (5.4) and (5.8), $\Delta^{\prime}_{(-1,1)}$ is
reflexive. Since $(s,k)=(-1,1)$, it follows that the map $\Delta\to\mathbb{R}$
that sends $w$ to $\min(1,1+\langle w,\nu\rangle)$ equals $DH-1$, where
$DH:\Delta\to\mathbb{R}$ is the abstract Duistermaat-Heckman function
determined by $(\Delta,\mathcal{F},-1,1)$. This implies both that
$\mathrm{pr}(\Delta^{\prime}_{(-1,1)})=\Delta$ (since the minimal value of the
above map on $\Delta$ is $0$), and that the height function of $\Delta$ equals
the abstract Duistermaat-Heckman function determined by
$(\Delta,\mathcal{F},-1,1)$, as desired.

Figure 5.2. A schematic representation of $\Delta_{(-1,1)}^{\prime}$ as a
blow-up of $\Delta_{(0,0)}^{\prime}$ along $\tilde{\mathcal{F}}$ of size $1$.

$\bullet$ Suppose that $(s,k)=(-1,2)$: Since $(\Delta,\mathcal{F},-1,2)$ is
admissible, there exists a facet $\mathcal{F}^{\prime}$ of $\Delta$ supported
on the affine hyperplane $\\{w\in\mathfrak{t}^{*}\mid\langle
w,-\nu\rangle=-1\\}$ and the quadruple $(\Delta,\mathcal{F},-1,1)$ is also
admissible. Let $\Delta^{\prime}_{(-1,1)}$ be the reflexive Delzant polytope
constructed from $(\Delta,\mathcal{F},-1,1)$ as above. Hence, by (5.4) and
(5.8), the reflexive Delzant polytope $\Delta^{\prime}_{(-1,1)}$ has a
codimension two face $\tilde{\mathcal{F}}^{\prime}$ given by the intersection
of the facets supported by the affine hyperplanes
$\\{(w,y)\in\mathfrak{t}^{*}\times\mathbb{R}\mid\langle(w,y),(0,1)\rangle=-1\\}$
and
$\\{(w,y)\in\mathfrak{t}^{*}\times\mathbb{R}\mid\langle(w,y),(-\nu,0)\rangle=-1\\}$.
This is a copy of $\mathcal{F}^{\prime}$ on the affine hyperplane
$\\{(w,y)\in\mathfrak{t}^{*}\times\mathbb{R}\mid y=-1\\}$. We wish to perform
the blow-up of $\Delta^{\prime}_{(-1,1)}$ along $\tilde{\mathcal{F}}^{\prime}$
of size $1$ (see Figure 5.3). To this end, with the notation in Definition
5.13, $\nu_{0}=(-\nu,1)$ and $c_{0}=-1$. A vertex of
$\Delta^{\prime}_{(-1,1)}$ that does not lie on $\tilde{\mathcal{F}}^{\prime}$
is of one of three types:

  * •

$(v,\min(1,1+\langle v,\nu\rangle))$ for some vertex $v$ of $\Delta$,

  * •

$(w,1)$, where $w\in\Delta$ lies on an edge of $\Delta$ and satisfies $\langle
w,\nu\rangle=0$, or

  * •

$(v,-1)$ for some vertex $v$ of $\Delta$ that does not lie on
$\mathcal{F}^{\prime}$,

(see the proof in the case $(s,k)=(-1,1)$.) In the first case, we have that

(5.10) $\langle(v,\min(1,1+\langle
v,\nu\rangle)),(-\nu,1)\rangle=\min(1-\langle v,\nu\rangle,1)>-1=c_{0},$

where the inequality follows from the fact that $\Delta$ is contained in the
strip $\\{w\in\mathfrak{t}^{*}\mid-1\leq\langle w,\nu\rangle\leq 1\\}$ (see
Remark 5.7). In the second case, we have that

(5.11) $\langle(w,1),(-\nu,1)\rangle=1>-1=c_{0}.$

As in the case $(s,k)=(-1,1)$, if $v$ is a vertex of $\Delta$, then $\langle
v,-\nu\rangle\geq 1$. Hence, in the third case, we have that

(5.12) $\langle(v,-1),(-\nu,1)\rangle\geq 0>-1=c_{0}.$

By (5.10), (5.11) and (5.12), we can perform the the blow-up of
$\Delta^{\prime}_{(-1,1)}$ along $\tilde{\mathcal{F}}^{\prime}$ of size $1$
that we denote by $\Delta^{\prime}_{(-1,2)}$, i.e.,

(5.13)
$\Delta^{\prime}_{(-1,2)}=\Delta^{\prime}_{(-1,1)}\cap\\{(w,y)\in\mathfrak{t}^{*}\times\mathbb{R}\mid\langle(w,y),(-\nu,1)\rangle\geq-1\\}.$

By (5.9), we have that

(5.14)
$\Delta^{\prime}_{(-1,2)}=\\{(w,y)\in\mathfrak{t}^{*}\times\mathbb{R}\mid
w\in\Delta\,,\,\max(-1,-1+\langle w,\nu\rangle)\leq y\leq\min(1,1+\langle
w,\nu\rangle)\\}.$

Since $\Delta^{\prime}_{(-1,1)}$ is smooth, by Remark 5.14,
$\Delta^{\prime}_{(-1,2)}$ is also smooth. Moreover, by (5.14), it can be
checked directly that a vertex of $\Delta^{\prime}_{(-1,2)}$ is one of the
following three types:

  * •

$(v,\min(1,1+\langle v,\nu\rangle))$ for some vertex $v$ of $\Delta$,

  * •

$(w,\pm 1)$, where $w$ lies on an edge of $\Delta$ and satisfies $\langle
w,\nu\rangle=0$, or

  * •

$(v,\max(-1,-1+\langle v,\nu\rangle)$ for some vertex $v$ of $\Delta$.

As in the case $(s,k)=(-1,1)$, it follows that $\Delta^{\prime}_{(-1,2)}$ is
integral. Moreover, since $\Delta^{\prime}_{(-1,1)}$ is reflexive, by (5.8)
$\Delta^{\prime}_{(-1,1)}$ is reflexive. Since $\Delta$ is contained in the
strip $\\{w\in\mathfrak{t}^{*}\mid-1\leq\langle w,\nu\rangle\leq 1\\}$, the
maximal (respectively minimal) value of the map $\Delta\to\mathbb{R}$ that
takes $w$ to $\max(-1,-1+\langle w,\nu\rangle)$ (respectively
$\min(1,1+\langle w,\nu\rangle)$) is zero. Hence,
$\mathrm{pr}(\Delta^{\prime}_{(-1,2)})=\Delta$. Moreover, the height function
of $\Delta$ is the map $\Delta\to\mathbb{R}$ that sends $w\in\Delta$ to

$\min(1,1+\langle w,\nu\rangle)-\max(-1,-1+\langle
w,\nu\rangle)=\min(2-\langle w,\nu\rangle,2+\langle w,\nu\rangle).$

Since $(s,k)=(-1,2)$, the above map equals the abstract Duistermaat-Heckman
function determined by $(\Delta,\mathcal{F},-1,2)$, as desired.

Figure 5.3. A schematic representation of $\Delta_{(-1,2)}^{\prime}$ as a
blow-up of $\Delta_{(-1,1)}^{\prime}$ along $\tilde{\mathcal{F}^{\prime}}$ of
size $1$.

∎

Theorem 5.12 and Delzant’s classification of compact symplectic toric
manifolds [9] yield the following geometric realizability and extension
result.

######  Corollary 5.15.

If $(\Delta,\mathcal{F},s,k)$ is an admissible quadruple, then there exists a
normalized monotone tall complexity one $T$-space $\left(M,\omega,\Phi\right)$
such that

  * •

$\Phi(M)=\Delta$ and the Duistermaat-Heckman function of
$\left(M,\omega,\Phi\right)$ equals the abstract Duistermaat-Heckman function
determined by $(\Delta,\mathcal{F},s,k)$, and

  * •

the Hamiltonian $T$-action extends to an effective Hamiltonian $(T\times
S^{1})$-action.

###### Proof.

By Theorem 5.12, there exists a reflexive Delzant polytope $\Delta^{\prime}$
in $\mathfrak{t}^{*}\times\mathbb{R}$ such that
$\mathrm{pr}(\Delta^{\prime})=\Delta$ and the height function of $\Delta$
equals the abstract Duistermaat-Heckman function determined by
$(\Delta,\mathcal{F},s,k)$. By [9], there exists a compact complexity zero
$(T\times S^{1})$-space $(M,\omega,\tilde{\Phi}=(\Phi,\Psi))$ such that the
moment map image $\tilde{\Phi}(M)=\Delta^{\prime}$, where we identify
$(\mathfrak{t}\times\mathbb{R})^{*}$ with $\mathfrak{t}^{*}\times\mathbb{R}$.
We claim that $\left(M,\omega,\Phi\right)$ satisfies the desired properties.
To see this, we observe that, by construction, it is tall and has complexity
one, and the $T$-action extends to an effective Hamiltonian $(T\times
S^{1})$-action. Moreover, since $\Delta^{\prime}$ is a reflexive Delzant
polytope, by Proposition 3.19, $(M,\omega,\tilde{\Phi})$ is normalized
monotone so that, in particular, $(M,\omega)$ satisfies $c_{1}=[\omega]$.
Since $\Delta$ is reflexive Delzant and since
$\mathrm{pr}(\Delta^{\prime})=\Delta$, $\Phi(M)=\Delta$, so that $\Phi$
satisfies the weight sum formula. Hence, $\left(M,\omega,\Phi\right)$ is
normalized monotone. Finally, by Example 2.52, the Duistermaat-Heckman
function of $\left(M,\omega,\Phi\right)$ equals the height function of
$\Delta$. Since the latter equals the abstract Duistermaat-Heckman function
determined by $(\Delta,\mathcal{F},s,k)$, the result follows. ∎

######  Remark 5.16.

In fact, the constructions in the proof of Theorem 5.12 have geometric
counterparts that allow to give an explicit geometric description of
$\left(M,\omega,\Phi\right)$ in Corollary 5.15. For instance, the case $k=0$
is described explicitly in [30, Example 1.6], while it is well-known that the
combinatorial blow-up of a polytope along a face corresponds to an equivariant
symplectic blow-up (see [44]).

We can prove another important result of this paper.

###### Proof of Theorem 1.7.

By Corollary 3.9, we may assume that $\left(M,\omega,\Phi\right)$ is
normalized monotone. Hence, $\Delta:=\Phi(M)$ is a reflexive Delzant polytope
by Proposition 3.18. Let $\mathcal{F}_{\min}\subset\Phi(M)$ be a minimal facet
and let $(s,k)$ be as in the statement of Proposition 5.1. By construction,
the quadruple $(\Delta,\mathcal{F}_{\min},s,k)$ is admissible. Hence, by
Corollary 5.15, there exists a normalized monotone tall complexity one
$T$-space $(M^{\prime},\omega^{\prime},\Phi^{\prime})$ such that

  * •

its Duistermaat-Heckman function equals the abstract Duistermaat-Heckman
function associated to $(\Delta,\mathcal{F}_{\min},s,k)$, and

  * •

the Hamiltonian $T$-action extends to an effective Hamiltonian $(T\times
S^{1})$-action.

By construction, $\left(M,\omega,\Phi\right)$ and
$(M^{\prime},\omega^{\prime},\Phi^{\prime})$ have equal Duistermaat-Heckman
functions. Hence, by Theorem 1.4, they are isomorphic and the result follows.
∎

###  5.3. Compact monotone tall complexity one spaces are equivariantly Fano

In this section, we prove the last main result of our paper, Theorem 1.8. To
this end, we recall that a compact complex manifold $(Y,J)$ is Fano if and
only if there exists a Kähler form $\sigma\in\Omega^{1,1}(Y)$ such that
$c_{1}(Y)=[\omega]$.

###### Proof of Theorem 1.8.

By Corollary 3.9, there is no loss of generality in assuming that
$\left(M,\omega,\Phi\right)$ is normalized monotone. By Theorem 1.7, the
Hamiltonian $T$-action extends to an effective Hamiltonian $(T\times
S^{1})$-action. We denote the corresponding normalized monotone symplectic
toric manifold by $(M,\omega,\tilde{\Phi}=(\Phi,\Psi))$. By the classification
of compact symplectic toric manifolds in [9] that there exists an integrable
almost complex structure $J$ on $M$ that is compatible with $\omega$ and
$(T\times S^{1})$-invariant; moreover, $\omega$ equals the Kähler form of
$(M,J)$. By Proposition 3.19, $[\omega]=c_{1}(M)>0$, so that the Kähler
manifold $(M,J)$ is Fano. Finally, by [19, Theorems 4.4 and 4.5], the
$T$-action extends to an effective holomorphic $T_{\mathbb{C}}$-action, as
desired. ∎

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