$\mathcal{T}$-space equipped with a flat Lagrangian elliptic connection that
has zero radial residues.

Given an integral affine manifold $(M,\Lambda)$ and a Delzant subspace
$\Delta$, we denote by $\underline{\mathcal{T}_{\textrm{Flat}}}$ the sheaf on
$\Delta$ of $\nabla$-flat sections of $\mathcal{T}_{\Lambda}|_{\Delta}$. This
coincides with the restriction of the sheaf of flat sections of $\mathcal{T}$
to $\Delta$ and, as before, it is a subsheaf of
$\underline{\mathcal{T}_{\textrm{Lag}}}$. Also, the induced map in cohomology
fits in an exact sequence

$\textstyle{\check{H}^{1}(\Delta,\underline{\mathcal{T}_{\textrm{Flat}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\check{H}^{1}(\Delta,\underline{\mathcal{T}_{\textrm{Lag}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{2}(\Omega^{\bullet}_{\partial}(\Delta,T^{*}\Delta),\mathrm{d}^{\nabla}),}$
(4.11)

where the second map is given by (4.6). Therefore, Corollary 3.19 extends. The
same goes for Theorem 3.20.

######  Theorem 4.33.

Fix a Delzant subspace $\Delta$ of an integral affine manifold $(M,\Lambda)$.
There is a canonical 1:1 correspondence

$\left\\{\\\ \hbox{\hbox{\kern
0.0pt\raise-14.44444pt\vbox{\halign{\relax\hfil\txtline@@{#}\hfil\cr\hbox{{\ignorespaces\leavevmode
flat toric Hamiltonian
$\mathcal{T}_{\Lambda}$-spaces}\\\\\hbox{{\ignorespaces$\mu:(S,\omega,\theta)\to
M$}\\\\\hbox{{\ignorespaces with $\mu(S)=\Delta$, up to equivalence
\,}\crcr}}}}\right\\}\ \tilde{\longleftrightarrow}\
\check{H}^{1}(\Delta,\underline{\mathcal{T}_{\textrm{Flat}}})}}}$

The cohomology class corresponding to a flat toric Hamiltonian
$\mathcal{T}$-space is mapped to its Lagrangian Chern class by the first map
in (4.11).

For the proof of this, we use the following observation.

######  Proposition 4.34.

The canonical toric $\mathcal{T}$-space
$\mu_{\Delta}:(S_{\Delta},\omega_{\Delta})\to M$ admits a unique elliptic
connection $\theta_{\Delta}$ that restricts to the canonical flat Lagrangian
connection on $\mathcal{T}_{\Lambda}$ via the canonical open embedding

$\mathcal{T}_{\Lambda}|_{\mathring{\Delta}}\hookrightarrow S_{\Delta}.$ (4.12)

The connection $\theta_{\Delta}$ is flat, Lagrangian and has zero radial
residues.

###### Proof.

Uniqueness holds since the image of (4.12) is dense in $S_{\Delta}$. This also
implies that for existence, it is enough to show that locally the canonical
flat connection on $\mathcal{T}_{\Lambda}|_{\mathring{\Delta}}$ extends to an
elliptic connection $1$-form for $\mu_{\Delta}$ with zero radial residues. So
we can work in the toric coordinates induced by an integral affine chart
$(U,x^{i})$, as in (4.9), over which we also have an induced trivialization of
the symplectic torus bundle $\mathcal{T}_{\Lambda}|_{\mathring{\Delta}}$. The
embedding (4.12) becomes the map
$\mathbb{T}^{n}\times\mathring{\mathbb{R}}^{n}_{k}\to S_{k,n}$ given by

$\left(\theta_{1},...,\theta_{n},x^{1},...,x^{n}\right)\mapsto\left(e^{2\pi
i\theta_{1}}\sqrt{x^{1}},\dots,e^{2\pi
i\theta_{k}}\sqrt{x^{k}},\theta_{k+1},...,\theta_{n},x^{k+1},\dots,x^{n}\right).$

On the other hand, the canonical flat connection on
$\mathcal{T}_{\Lambda}|_{\mathring{\Delta}}$ becomes
$\sum_{i=1}^{n}\mathrm{d}\theta_{i}\otimes\mathrm{d}x^{i}$ so corresponds
under (4.12) with the restriction to $\mathring{S}_{\Delta}$ of the connection
given in Example 4.28. Therefore, the canonical flat connection on
$\mathcal{T}_{\Lambda}|_{\mathring{\Delta}}$ indeed extends to an elliptic
connection $1$-form for $\mu_{\Delta}$ with zero radial residues. Since the
canonical connection on $\mathcal{T}_{\Lambda}$ is flat and Lagrangian and
$\mathring{S}_{\Delta}$ is dense in $S_{\Delta}$, it follows that
$\theta_{\Delta}$ is also flat and Lagrangian. ∎

###### Proof of Theorem 4.33.

The proof is virtually the same as that in the case of Lagrangian fibrations,
where we now take the equivalence class of the flat toric Hamiltonian
$\mathcal{T}_{\Lambda}$-space of the above proposition as the equivalence
class corresponding to the zero element of the group
$\check{H}^{1}(\Delta,\underline{\mathcal{T}_{\textrm{Flat}}})$. ∎

######  Example 4.35.

Consider the Delzant subspace $\Delta=\mathbb{S}^{1}\times[-1,1]$ of the
standard cylinder $(\mathbb{S}^{1}\times\mathbb{R},\Lambda_{\mathrm{st}})$.
Notice that

$\check{H}^{1}(\Delta,\underline{\mathcal{T}_{\textrm{Lag}}})=\check{H}^{2}(\Delta,\mathcal{O}_{\Lambda})=0,$

so every toric $\mathcal{T}_{\Lambda}$-space is isomorphic to the space
$\mu:(\mathbb{T}^{2}\times\mathbb{S}^{2},\omega)\to\mathbb{S}^{1}\times\mathbb{R}$
of Example 2.20.

Since $\Lambda$ has trivial holonomy representation it follows – see Remark
3.21 – that the equivalence classes of flat $\mathcal{T}_{\Lambda}$-toric
fibrations are in 1-1 correspondence with

$\check{H}^{1}(\Delta,\underline{\mathcal{T}_{\textrm{Flat}}})=\check{H}^{1}(\Delta,\mathbb{T}^{2})=\mathbb{T}^{2}.$

For $a,b\in\mathbb{R}$, the classes $(e^{2\pi ia},e^{2\pi
ib})\in\mathbb{T}^{2}$ can be realized by the elliptic connections on
$\mu:\mathbb{T}^{2}\times\mathbb{S}^{2}\to\Delta$ given by the connection
$1$-form

$\theta_{a,b}=\left(-a\mathrm{d}x+\mathrm{d}y\right)\otimes\mathrm{d}x+\left(-b\mathrm{d}h+\mathrm{d}\phi\right)\otimes\mathrm{d}h.$

This corresponds to the Lagrangian foliation on $\mu^{-1}(]-1,1[)$ given by

$D_{a,b}:=\left\langle\partial_{x}+a\partial_{y},\partial_{h}+b\partial_{\phi}\right\rangle.$

######  Example 4.36.

Consider the Delzant subspace $\Delta=\mathbb{S}^{1}\times[-1,1]$ of the non-
standard cylinder
$(\mathbb{S}^{1}\times]-2,\infty[,\mathbb{Z}((h+2)\mathrm{d}x+x\mathrm{d}h)\oplus\mathbb{Z}\mathrm{d}h)$,
as in Example 2.21. We still have
$\check{H}^{1}(\Delta,\underline{\mathcal{T}_{\textrm{Lag}}})=0$ so it follows
that every toric $\mathcal{T}_{\Lambda}$-space is isomorphic to the space
$\mu:(\mathbb{T}^{2}\mathbin{\ooalign{\raise
4.95134pt\hbox{$\scriptstyle\sim$}\cr$\times$\cr}}\mathbb{S}^{2},\omega)\to\mathbb{S}^{1}\times]-2,\infty[$
appearing in Example 2.21.

A computation using the Mayer-Vietoris sequence, shows that

$\check{H}^{1}(\Delta,\underline{\mathcal{T}_{\textrm{Flat}}})=\check{H}^{1}(\Delta,\mathbb{T}^{2})=\mathbb{T}^{2},$

so we conclude that equivalence classes of flat $\mathcal{T}_{\Lambda}$-toric
fibrations are parameterized by $(e^{2\pi ia},e^{2\pi ib})\in\mathbb{T}^{2}$,
with $a,b\in\mathbb{R}$. These can be realized by the elliptic connections on
$\mu:(\mathbb{T}^{2}\mathbin{\ooalign{\raise
4.95134pt\hbox{$\scriptstyle\sim$}\cr$\times$\cr}}\mathbb{S}^{2},\omega)\to\mathbb{S}^{1}\times]-2,\infty[$
given by the connection $1$-form

$\theta_{a,b}=(h+2)\left(-a\mathrm{d}x+\mathrm{d}y\right)\otimes\mathrm{d}x+\left(-b\mathrm{d}h+\mathrm{d}\phi+x\mathrm{d}y\right)\otimes\mathrm{d}h.$

This corresponds to the Lagrangian foliation on $\mu^{-1}(]-1,1[)$ given by

$D_{a,b}:=\left\langle\partial_{x}+a(\partial_{y}-x\partial_{\phi}),\partial_{h}+b\partial_{\phi}\right\rangle.$

##  5\. Delzant construction

The proof of the classification theorem of toric Hamiltonian
$\mathcal{T}$-spaces (Theorem 4.12) requires the construction of a toric
Hamiltonian $\mathcal{T}$-space $\mu:(S,\omega)\to M$ with moment map image
$\mu(S)$ a given Delzant subspace $\Delta\subset(M,\Lambda)$.

In the case of a Delzant polytope
$\Delta\subset(\mathbb{R}^{n},\mathbb{Z}^{n})$ Delzant gave a construction of
a compact symplectic toric manifold with moment map image $\Delta$ by
symplectic reduction of a linear Hamiltonian torus action on
$(\mathbb{C}^{d},\omega_{\mathrm{st}})$, where $d$ is the number of facets.
For a general Delzant subspace, as we recalled in Section 4.5, the
construction of canonical toric $\mathcal{T}$-space uses a different method.
In this section we will give a generalization of Delzant’s construction for a
large class of Delzant subspaces. Although this construction does not apply to
all Delzant subspaces, it does cover many natural examples and has the
advantage of realizing the desired toric space via symplectic reduction of a
rather simple symplectic manifold.

###  5.1. The primitive outward pointing normals

The aim of this section is to show for a Delzant subspace $\Delta$ in a
1-connected integral affine manifold each facet of $\Delta$ admits a naturally
defining integral affine function.

In order to state a precise result, we recall that for a manifold with corners
$\Delta$, given $x\in\partial\Delta$ the cone $C_{x}(\Delta)$ consists of all
inward-pointing tangent vectors at $x$. We further denote by
$\mathcal{S}^{1}(\Delta)$ the collection of codimension $1$ strata of $\Delta$
(i.e., the open facets of $\Delta$).

######  Proposition 5.1.

Let $(M,\Lambda)$ be a $1$-connected integral affine manifold and let $\Delta$
be a Delzant subspace. For each open facet $\Sigma$ of $\Delta$, there is a
unique integral affine function
$\ell=\ell_{\Sigma}\in\mathcal{O}_{\Lambda}(M)$ satisfying the following
properties:

  * (i)

$\ell|_{\Sigma}=0$ and $\ker(\mathrm{d}\ell)|_{\Sigma}=T\Sigma$;

  * (ii)

$\ell$ is outward-pointing, i.e., $\mathrm{d}_{x}\ell(v)\leq 0$ for all $v\in
C_{x}(\Delta)$ and all $x\in\Sigma$;

  * (iii)

for any $f\in\mathcal{O}_{\Lambda}(M)$ with property (i), there is a
$k\in\mathbb{Z}$ such that $f=k\,\ell$.

Moreover, any $x\in\Delta$ admits an open neighbourhood $U$ in $M$ such that

$\Delta\cap U=\bigcap_{\\{\Sigma\in\mathcal{S}^{1}(\Delta)\colon
x\in\overline{\Sigma}\\}}\\{\ell_{\Sigma}\leq 0\\}\cap U.$ (5.1)

The number of open facets of $\Delta$ with $x$ in their closure equals the
depth of $x$ in $\Delta$.

Given a Delzant subspace $\Delta$ of a $1$-connected integral affine manifold
$(M,\Lambda)$ we let

$\mathcal{H}(\Delta):=\\{\ell_{\Sigma}\mid\Sigma\in\mathcal{S}^{1}(\Delta)\\}$
(5.2)

be the collection of integral affine functions $\ell_{\Sigma}$ associated to
the open facets of $\Delta$ given by the previous proposition. We call the
elements of $\mathcal{H}(\Delta)$ primitive boundary defining (integral
affine) functions.

######  Example 5.2.

It can happen that the same integral affine function is associated to
different codimension $1$ strata. An example of this is

$M:=\mathbb{R}^{2}-\bigsqcup_{k\in\mathbb{Z}}\\{k\\}\times[0,\infty[,\quad\Delta=M\cap(\mathbb{R}\times]-\infty,0]),$

where $M$ is equipped with the integral affine structure inherited from
$(\mathbb{R}^{2},\mathbb{Z}^{2})$. Then $\Delta$ in fact has infinitely many
codimension $1$ strata, but $\mathcal{H}(\Delta)$ consist of a single
function.

For the proof of Proposition 5.1, we first recall the following fact.

######  Lemma 5.3.

Let $(V^{*},\Lambda)$ be an integral affine vector space. Suppose that
$H\subset V$ is an hyperplane, which is primitive in the sense that
$\operatorname{span}{R}(H\cap\Lambda^{\vee})=H$. Then there is an
$\alpha\in\Lambda$ such that:

  1. (i)

$H=\ker(\alpha)$;

  2. (ii)

if $\beta\in\Lambda$ and $H=\ker(\beta)$, then $\beta=k\,\alpha$ for some
$k\in\mathbb{Z}$.

Moreover, $\alpha$ is unique up to a sign.

We call $\alpha\in\Lambda$, as in this lemma, a primitive normal covector to
the hyperplane $H$.

######  Lemma 5.4.

Let $(M,\Lambda)$ be an integral affine manifold, let $\Delta$ be a Delzant
subspace and let $\Sigma$ be a codimension $1$ stratum of $\Delta$. For each
$x\in\Sigma$, there is a unique outward-pointing primitive normal covector
$\alpha_{x}\in\Lambda_{x}$ to the hyperplane
$T_{x}\Sigma\subset(T_{x}M,\Lambda^{\vee}_{x})$. The family of all such
outward-pointing primitive normals is holonomy-invariant.

In this lemma, by ’holonomy invariant’ we mean that for any path $\gamma$ in
$\Sigma$, the holonomy representation
$\operatorname{hol}([\gamma]):T_{\gamma(0)}M\to T_{\gamma(1)}M$ pulls
$\alpha_{\gamma(1)}$ back to $\alpha_{\gamma(0)}$.

###### Proof.

The existence and uniqueness of the covectors $\alpha_{x}$ follows from the
previous lemma and the fact that open facets of Delzant subspaces are
primitive codimension $1$ submanifolds of $(M,\Lambda)$. By the uniqueness of
primitive normal covectors, to prove holonomy-invariance it suffices to show
that for any path $\gamma$ in $\Sigma$, the holonomy representation
$\operatorname{hol}([\gamma]):T_{\gamma(0)}M\to T_{\gamma(1)}M$ maps the
tangent cone $C_{\gamma(0)}(\Delta)$ onto the tangent cone
$C_{\gamma(1)}(\Delta)$. It is enough to check this for a path contained in
the domain of an integral affine chart.

Let $(U,\phi=(x^{1},\dots,x^{m}))$ be an integral affine chart adapted to
$\Delta$ in the sense that $\phi(U\cap\Delta)$ is an open in
$[0,\infty[\times\mathbb{R}^{m-1}$ around the origin. Then for any point
$x\in\phi^{-1}(\\{0\\}\times\mathbb{R}^{m-1})$ we have

$C_{x}(\Delta)=\textrm{Cone}\left(\partial_{x^{1}},\pm\partial_{x^{2}}\dots,\pm\partial_{x^{m}}\right).$

If $\gamma$ is a path lying in an open facet $\Sigma\cap
U\subset\phi^{-1}(\\{0\\}\times\mathbb{R}^{m-1})$, then
$\operatorname{hol}([\gamma])$ maps $C_{\gamma(0)}(\Delta)$ onto
$C_{\gamma(1)}(\Delta)$, since the frames induced by integral affine
coordinate charts are flat and the coordinate vector fields $\partial_{x^{i}}$
are invariant under the action of $\operatorname{hol}([\gamma])$. ∎

###### Proof of Proposition 5.1.

Let $\Sigma$ be a codimension $1$ stratum of $\Delta$. Since $M$ is
$1$-connected, for any $x_{0}\in\Sigma$ there is a unique parallel $1$-form
$\alpha\in\Omega^{1}(M)$ that evaluates at $x_{0}$ to the the outward-pointing
primitive normal covector. By the holonomy-invariance in Lemma 5.4, for any
$x\in\Sigma$ the value $\alpha_{x}$ is the outward-pointing primitive normal
covector at $x$. As $M$ is $1$-connected, there is a unique function
$\ell=\ell_{\Sigma}\in C^{\infty}(M)$ such that $\mathrm{d}\ell=\alpha$ and
$\ell|_{\Sigma}=0$. Since $\alpha$ is parallel and
$\alpha_{x_{0}}\in\Lambda_{x_{0}}$, $\alpha$ takes values in $\Lambda$. Hence,
$\ell$ is an integral affine function. It satisfies the required properties
(i)–(iii) and is the unique such integral affine function, due to the
properties of the covectors in Lemma 5.4 and the fact that the differential of
an integral affine function is a parallel $1$-form.

Next, fix $x\in\Delta$ of depth $k$. Pick an integral affine chart
$(U,\phi=(x^{1},\dots,x^{n}))$ centered at $x$ such that
$W:=\phi(U\cap\Delta)$ is a convex open set in
$[0,\infty[^{k}\times\mathbb{R}^{n-k}$. The set of points in $U\cap\Delta$ of
depth $1$ corresponds to the disjoint union of the convex subsets

$W_{i}:=\left\\{(x^{1},\dots,x^{m})\in W\mid x^{i}=0\textrm{ and
}x^{j}>0\textrm{ for }1\leq j\leq k,\text{ }j\neq i\right\\},$

for $i=1,\dots,k$. Since each $W_{i}$ is connected, $\phi^{-1}(W_{i})$ is
contained in a single open facet $\Sigma_{i}$ of $\Delta$, with defining
function satisfying $\ell_{\Sigma_{i}}|_{U}=-x^{i}$. The latter shows that
$\Sigma_{i}\neq\Sigma_{j}$ if $i\neq j$. Therefore, the number of open facets
with $x$ in their closure is indeed equal to $k$. It is now also clear that
(5.1) holds for this choice of $U$. ∎

###  5.2. Delzant subspaces of finite type

Let $(M,\Lambda)$ be a connected integral affine manifold and let $\Delta$ be
a connected Delzant subspace. Consider the canonical map

$(i_{\Delta})_{*}:\widetilde{\Delta}\to\widetilde{M},$

from the universal cover of $\Delta$ into that of $M$, relative to some choice
of base-point. The image
$\widehat{\Delta}:=(i_{\Delta})_{*}(\widetilde{\Delta})$ is a Delzant subspace
of $\widetilde{M}$. For the Delzant subspaces to which our construction
applies, the collection $\mathcal{H}(\widehat{\Delta})$ of primitive boundary
defining functions is finite and determines the Delzant subspace in the
following sense.

######  Definition 5.5.

A connected Delzant subspace $\Delta$ is of finite type if
$\mathcal{H}(\widehat{\Delta})$ is finite and

$\widehat{\Delta}=\bigcap_{f\in\mathcal{H}(\widehat{\Delta})}\\{f\leq 0\\}.$
(5.3)

Below we give two examples illustrating how each of these conditions may fail.

######  Example 5.6.

Let

$M:=\mathbb{R}^{2}-\\{(0,y)\mid y\geq 0\\}$

with the standard integral affine structure
$\Lambda=\mathbb{Z}\mathrm{d}x\oplus\mathbb{Z}\mathrm{d}y$. Consider the
Delzant subspace

$\Delta=\left\\{(x,y)\in M\mid y\leq|x|\right\\}.$

This is closed in $M$ and has two open facets, with primitive boundary
defining functions $\ell_{\pm}(x,y)=\pm(x-y)$. However, $\Delta$ does not
satisfy (5.3).

######  Example 5.7.

Consider the non-standard integral affine cylinder

$(M,\Lambda):=(\mathbb{S}^{1}\times\mathbb{R},\mathbb{Z}\mathrm{d}x\oplus\mathbb{Z}(\mathrm{d}h-x\mathrm{d}x))$

and the Delzant subspace

$\Delta=\left\\{(x,h)\in\mathbb{S}^{1}\times\mathbb{R}\mid
h\leq\tfrac{1}{2}x^{2}-\tfrac{1}{8},\ -\tfrac{1}{2}\leq
x\leq\tfrac{1}{2}\right\\}.$

The lift $\widehat{\Delta}\subset\mathbb{R}^{2}=\widetilde{M}$ is the region
bounded from above by the curves

$\left\\{h=\tfrac{1}{2}(x+n)^{2}-\tfrac{1}{8}\right\\}\subset\mathbb{R}^{2},\quad
n\in\mathbb{Z}.$

In this example, even though $\Delta$ has only one open facet, the collection
$\mathcal{H}(\widehat{\Delta})$ is infinite, as it consists of the functions

$\ell_{n}(x,h)=h-\tfrac{1}{2}(x+n)^{2}+\tfrac{1}{8},\quad n\in\mathbb{Z}.$

In both of these examples the Delzant subspace is not compact. We do not know
of an example of a compact Delzant subspace that is not of finite type.

In the coming subsection, the following property of Delzant subspaces of
finite type will be important.

######  Proposition 5.8.

Let $(M,\Lambda)$ be a $1$-connected integral affine manifold, $\Delta$ a
Delzant subspace of finite type and $x\in\Delta$. Consider the primitive
boundary defining functions $\ell\in\mathcal{H}(\Delta)$ for which
$\ell(x)=0$.

  1. (i)

The number of such functions equals the depth of $x$.

  2. (ii)

Their differentials at $x$ can be extended to a basis of $\Lambda_{x}$ and
span $(T_{x}F)^{0}$.

###### Proof of Proposition 5.8.

Let $x\in\Delta$. By Proposition 5.1 the number of
$\ell\in\mathcal{H}(\Delta)$ for which $\ell(x)=0$ is at least
$k:=0pt_{\Delta}(x)$. To show that these are in fact equal, given
$\ell_{0}\in\mathcal{H}(\Delta)$ such that $\ell_{0}(x)=0$, we need to prove
that $\ell_{0}$ is one of the functions $\ell_{i}$ associated to one of the
$k$ open facets $\Sigma_{i}$ with $x$ in their closure. Choose an integral
affine chart $(U,\phi=(x^{1},\dots,x^{n}))$ centered at $x$, as in the second
part of the proof of Proposition 5.1. Since $\ell_{0}$ is affine and vanishes
at $x$, its restriction to $U$ is of the form:

$\ell_{0}|_{U}=-\sum_{i=1}^{n}t_{i}x^{i},\quad t_{i}\in\mathbb{R}.$

In view of (5.3), $\ell_{0}$ is non-positive on $\Delta$ and so $t_{i}\geq 0$
for all $i=1,\dots,k$ and $t_{i}=0$ for all $i=k+1,\dots,n$. As in the proof
of Proposition 5.1, $\ell_{i}|_{U}=-x^{i}$ for each $i=1,..,k$. So,

$\ell_{0}=\sum_{i=1}^{k}t_{i}\ell_{i},\quad t_{i}\geq 0,$

since this equality holds on $U$ and the functions on both sides are affine.
It follows from property (iii) in Proposition 5.1 that, if
$\ell_{0}\neq\ell_{i}$ for all $i=1,\dots,k$, then there are at least two
distinct $i_{1},i_{2}\in\\{1,\dots,k\\}$ for which $t_{i_{1}},t_{i_{2}}\neq
0$. In that case, since $\Delta$ satisfies (5.3), $\Sigma_{0}$ must be
contained in the codimension $2$ submanifold
$\\{\ell_{i_{1}}=\ell_{i_{2}}=0\\}$, which is a contradiction. So, our first
claim follows. Moreover, the fact that $\ell_{i}|_{U}=-x^{i}$ for all
$i=1,...,k$ shows that the differentials of these functions at $x$ extend to a
basis of $\Lambda_{x}$ and span $(T_{x}F)^{0}$. ∎

###  5.3. The 1-connected case

In this section we give a Delzant type construction for a finite type Delzant
subspace of a $1$-connected integral affine manifold. Treating the 1-connected
case first is convenient for the proof of the general case, and moreover it is
interesting on its own, as it already covers a class of non-compact toric
symplectic manifolds (cf. Section 4.3).

Let $(M,\Lambda)$ be a 1-connected integral affine manifold and let $\Delta$
be a Delzant subspace of finite type so

$\Delta=\bigcap_{i=1}^{d}\\{\ell_{i}\leq 0\\},$

where $\ell_{1},\dots,\ell_{d}$ are the primitive boundary defining functions.
We consider the map:

$\phi:=(\ell_{1},...,\ell_{d}):(M,\Lambda)\to(\mathbb{R}^{d},\mathbb{Z}^{d}).$

Since this map is integral affine, it induces a morphism of symplectic torus
bundles

$(\phi,\underline{\phi}^{*}):(\mathcal{T}_{\Lambda},\Omega_{\Lambda})\dashrightarrow(\mathbb{T}^{d}\times\mathbb{R}^{d},\sum_{i=1}^{d}\mathrm{d}\theta_{i}\wedge\mathrm{d}x^{i})$

as in Proposition 2.16. This, in turn, induces a Hamiltonian
$\mathbb{T}^{d}$-action on $(\mathcal{T}_{\Lambda},\Omega_{\Lambda})$ with
moment map the composition of $\phi$ with the bundle projection $\pi$ and
action given by

$(\theta_{1},...,\theta_{d})\cdot[\alpha]=\Big{[}\alpha+\sum_{i=1}^{d}\theta_{i}\,\mathrm{d}\ell_{i}\Big{]}.$

######  Theorem 5.9.

Consider the diagonal Hamiltonian $\mathbb{T}^{d}$-action with moment map

$\mu_{\phi}:=\phi+(|z_{1}|^{2},...,|z_{d}|^{2}):(\mathcal{T}_{\Lambda},\Omega_{\Lambda})\times(\mathbb{C}^{d},\omega_{\textrm{st}})\to\mathbb{R}^{d}.$

The $\mathbb{T}^{d}$-action is free on $\mu_{\phi}^{-1}(0)$ and the symplectic
quotient

$((\mathcal{T}_{\Lambda}\times\mathbb{C}^{d})\mathbin{/\mkern-6.0mu/}\mathbb{T}^{d},\omega_{\mathrm{red}})$

is a toric Hamiltonian $\mathcal{T}_{\Lambda}$-space for the action induced by
translations on the first factor. Moreover, this space is equivalent to the
canonical toric Hamiltonian $\mathcal{T}_{\Lambda}$-space
$\mu_{\Delta}:(S_{\Delta},\omega_{\Delta})\to M$.

###### Proof.

We start by checking the diagonal $\mathbb{T}^{d}$-action on
$\mu_{\phi}^{-1}(0)$ is free, so the symplectic quotient is well-defined. Let
$([\alpha_{x}],z)\in\mu_{\phi}^{-1}(0)$ and
$\theta=(\theta_{1},...,\theta_{d})\in\mathbb{T}^{d}$ be such that $\theta$
fixes $([\alpha_{x}],z)$. Since $\theta$ fixes $z$, it follows that
$\theta_{i}=0\mod\mathbb{Z}$ whenever $z_{i}\neq 0$. On the other hand,
because $\ell_{i}(x)=-|z_{i}|^{2}$, the remaining $i\in\\{1,...,d\\}$, say
$i_{1},...,i_{k}$, are those for which $\ell_{i}(x)=0$. So, by Proposition
5.8, $(\mathrm{d}\ell_{i_{1}})_{x},\dots,(\mathrm{d}\ell_{i_{k}})_{x}$ extends
to a basis of $\Lambda_{x}$. Hence, the fact that $\theta$ fixes
$[\alpha_{x}]$ implies that
$\theta_{i_{1}}=...=\theta_{i_{k}}=0\mod\mathbb{Z}$ as well.

Next, observe that the action of $\mathcal{T}_{\Lambda}$ on itself by
translation descends to a Hamiltonian $\mathcal{T}_{\Lambda}$-action along

$\mu:\big{(}(\mathcal{T}_{\Lambda}\times\mathbb{C}^{d})\mathbin{/\mkern-6.0mu/}\mathbb{T}^{d},\omega_{\mathrm{red}}\big{)}\to
M,\quad[\alpha,z]\mapsto\pi(\alpha),$

which has moment map image

$\bigcap_{i=1}^{d}\\{\ell_{i}\leq 0\\}=\Delta.$

Finally, we claim that we have an equivalence of Hamiltonian
$\mathcal{T}_{\Lambda}$-spaces

---
$\textstyle{(S_{\Delta},\omega_{\Delta})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{\Delta}}$$\scriptstyle{\sim}$$\scriptstyle{\Psi}$$\textstyle{((\mathcal{T}_{\Lambda}\times\mathbb{C}^{d})\mathbin{/\mkern-6.0mu/}\mathbb{T}^{d},\omega_{\mathrm{red}}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\textstyle{M}$

given by

$[\alpha_{x}]\longmapsto[\alpha_{x},(\sqrt{-\ell_{1}(x)},\dots,\sqrt{-\ell_{d}(x)})].$

This map is a $\mathcal{T}_{\Lambda}$-equivariant bijection with inverse
induced by

$(\alpha_{x},z)\longmapsto[t^{-1}\cdot\alpha_{x}],$

where $t=(t_{1},\dots,t_{d})$ is given by

$t_{i}=t_{i}(z,x):=\begin{cases}\tfrac{z_{i}}{|z_{i}|},\quad\text{if
$\ell_{i}(x)\neq 0$},\\\ \\\ \ 0,\quad\text{if $\ell_{i}(x)=0$}.\end{cases}$

To see that $\Psi$ is a smooth map, we note that around a point $x_{0}$ of
depth $k$ its expression in the chart (4.9) is

$(z_{1},\dots,z_{k},\theta_{k+1},\dots,\theta_{n},x^{k+1},\dots,x^{n})\mapsto\big{[}\sum_{i=k+1}^{n}\theta_{i}\mathrm{d}x^{i},(w_{1},\dots,w_{d})\big{]},$

where

$w_{i}:=\begin{cases}\quad z_{i},\quad\text{if $\ell_{i}(x_{0})=0$},\\\ \\\
\sqrt{-\ell_{i}(x)},\quad\text{if $\ell_{i}(x_{0})\neq 0$}.\end{cases}$

Finally, we observe that this map is a symplectomorphism. Indeed, over
$\mathring{\Delta}$, where $k=0$, it is given by the symplectic map

$(\theta_{1},\dots,\theta_{n},x^{1},\dots,x^{n})\mapsto\big{[}\sum_{i=1}^{n}\theta_{i}\mathrm{d}x^{i},(\sqrt{-\ell_{1}(x)},\dots,\sqrt{-\ell_{d}(x)})\big{]}.$

By density, we conclude that $\Phi$ is a symplectic map. Since it is a
bijection between manifolds of the same dimension, it is a symplectomorphism.
∎

######  Example 5.10 (The standard Delzant construction).

One can recover the usual Delzant construction of a compact symplectic
manifold from the construction in Theorem 5.9 via reduction by stages as
follows.

Fix a Delzant polytope $\Delta\subset\mathbb{R}^{n}$ with $d$ facets. In this
case, Theorem 5.9 gives a symplectic toric manifold

$\mu:\big{(}(\mathbb{T}^{n}\times\mathbb{R}^{n}\times\mathbb{C}^{d})\mathbin{/\mkern-6.0mu/}\mathbb{T}^{d},\omega_{\mathrm{red}}\big{)}\to\mathbb{R}^{n}.$
(5.4)

If $\ell_{1},\dots,\ell_{d}$ are the primitive boundary defining functions,
one has a linear map

$\mathbb{R}^{d}\to\mathbb{R}^{n},\quad e_{a}\mapsto\mathrm{d}\ell_{a},$

where $\\{e_{a}\\}$ is the standard basis of $\mathbb{R}^{d}$. The kernel of
this map is a Lie subalgebra $\mathfrak{n}\subset\mathbb{R}^{n}$, which
integrates to a subtorus $N\subset\mathbb{T}^{d}$. Choosing a subtorus
$\mathbb{T}^{n}\subset\mathbb{T}^{d}$ complementary to $N$, one can first
perform symplectic reduction by this subtorus. The resulting reduced space is
canonically isomorphic to the $d$-dimensional complex vector space with its
canonical symplectic form via the map

$(\mathbb{T}^{n}\times\mathbb{R}^{n}\times\mathbb{C}^{d})\mathbin{/\mkern-6.0mu/}\mathbb{T}^{d},\omega_{\mathrm{red}})\to(\mathbb{C}^{d},\omega_{\mathrm{st}}),\quad(x,\theta,z)\mapsto(-\theta)\cdot
z.$

Therefore, the symplectic toric manifold (5.4) is canonically isomorphic to

$\mu:(\mathbb{C}^{d}\mathbin{/\mkern-6.0mu/}N,\omega_{\mathrm{red}})\to\mathbb{R}^{n}.$
(5.5)

This is the compact toric symplectic manifold constructed by Delzant [15].

###  5.4. The general construction

Let $(M,\Lambda)$ be a connected integral affine manifold and let $\Delta$ be
a Delzant subspace of finite type. Consider the canonical map

$(i_{\Delta})_{*}:\widetilde{\Delta}\to\widetilde{M},$

from the universal cover of $\Delta$ into that of $M$, relative to a fixed
base-point in $\Delta$. The image
$\widehat{\Delta}:=(i_{\Delta})_{*}(\widetilde{\Delta})$ is a Delzant subspace
of $\widetilde{M}$. Let $\Gamma$ be the image of the group homomorphism
$(i_{\Delta})_{*}:\pi_{1}(\Delta)\to\pi_{1}(M)$. Note that the canonical
$\Gamma$-action on $\widetilde{M}$ is by integral affine isomorphisms and
preserves $\widehat{\Delta}$. Therefore, it induces an action on the set of
primitive boundary defining functions $\mathcal{H}(\widehat{\Delta})$. Fixing
an ordering $\mathcal{H}(\Delta)=\\{\ell_{1},...,\ell_{d}\\}$, this induces a
homomorphism $\Gamma\to\text{Sym}(1,\dots,d)$ and, therefore, actions on
$\mathbb{C}^{d}$ and $\mathbb{T}^{d}$ by coordinate permutations. The group
$\Gamma$ also acts on $\mathcal{T}_{\widetilde{\Lambda}}$ via the cotangent
lift of the action on $\widetilde{M}$. All together, we obtain a Hamiltonian
action of the semi-direct product $\Gamma\ltimes\mathbb{T}^{d}$, with moment
map

$\mu_{\phi}:=\phi+(|z_{1}|^{2},...,|z_{d}|^{2}):(\mathcal{T}_{\widetilde{\Lambda}},\Omega_{\widetilde{\Lambda}})\times(\mathbb{C}^{d},\omega_{\textrm{st}})\to\mathbb{R}^{d},$

where $(\mathcal{T}_{\widetilde{\Lambda}},\Omega_{\widetilde{\Lambda}})$ is
the symplectic torus bundle over $(\widetilde{M},\widetilde{\Lambda})$.

######  Theorem 5.11.

The $\Gamma\ltimes\mathbb{T}^{d}$-action is free on $\mu_{\phi}^{-1}(0)$ and
the symplectic quotient

$((\mathcal{T}_{\widetilde{\Lambda}}\times\mathbb{C}^{d})\mathbin{/\mkern-6.0mu/}(\Gamma\ltimes\mathbb{T}^{d}),\,\omega_{\mathrm{red}})$

is a toric Hamiltonian $\mathcal{T}_{\Lambda}$-space, which is equivalent to
the canonical toric Hamiltonian $\mathcal{T}_{\Lambda}$-space
$\mu_{\Delta}:(S_{\Delta},\omega_{\Delta})\to M$.

###### Proof.

The fact that the $\Gamma\ltimes\mathbb{T}^{d}$-action on $\mu^{-1}_{\phi}(0)$
is free and proper, follows using that the $\mathbb{T}^{d}$-action on this set
is free and the $\Gamma$-action on $\widetilde{M}$ is free and proper.

Next, note that the symplectic torus bundle
$(\mathcal{T}_{\widetilde{\Lambda}},\Omega_{\widetilde{\Lambda}})$ is
canonical isomorphic to the pull-back of
$(\mathcal{T}_{\Lambda},\Omega_{\Lambda})$ along the covering map
$\widetilde{M}\to M$. The action of
$(\mathcal{T}_{\widetilde{\Lambda}},\Omega_{\widetilde{\Lambda}})$ on the
first factor in $\mathcal{T}_{\widetilde{\Lambda}}\times\mathbb{C}^{d}$
descends to a Hamiltonian $\mathcal{T}_{\Lambda}$-action on the symplectic
quotient. This makes it a toric $\mathcal{T}_{\Lambda}$-space.

For the last statement, we use the following lemma.

######  Lemma 5.12.

Let $(M,\Lambda)$ be a connected integral affine manifold, let $\Delta$ be a
connected Delzant subspace and consider $\widehat{\Delta}$ and $\Gamma$ as
above. The $\Gamma$-action on $\widehat{\Delta}$ lifts to a symplectic action
on $(S_{\widehat{\Delta}},\omega_{\widehat{\Delta}})$ and there is an
equivalence of Hamiltonian $\mathcal{T}_{\Lambda}$-spaces

---
$\textstyle{(S_{\widehat{\Delta}},\omega_{\widehat{\Delta}})/\Gamma\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\underline{\mu_{\widetilde{\Delta}}}}$$\textstyle{(S_{\Delta},\omega_{\Delta})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{\Delta}}$$\textstyle{M}$

where $\underline{\mu_{\widetilde{\Delta}}}$ is the composition of the map
$S_{\widehat{\Delta}}/\Gamma\to\widetilde{M}/\Gamma$ induced by
$\mu_{\widetilde{\Delta}}$ with the quotient map $\widetilde{M}/\Gamma\to M$.

###### Proof.

This readily follows from the naturality of the construction of the canonical
toric space associated to a Delzant subspace (see [28]). ∎

Since $\mathbb{T}^{d}$ is a normal subgroup of $\Gamma\ltimes\mathbb{T}^{d}$,
there is an induced $\Gamma$-action on the symplectic quotient
$((\mathcal{T}_{\widetilde{\Lambda}}\times\mathbb{C}^{d})\mathbin{/\mkern-6.0mu/}\mathbb{T}^{d},\,\omega_{\mathrm{red}})$.
The equivalence between $\mathcal{T}_{\widetilde{\Lambda}}$-spaces

$(S_{\widehat{\Delta}},\omega_{\widehat{\Delta}})\to(\mathcal{T}_{\widetilde{\Lambda}}\times\mathbb{C}^{d})\mathbin{/\mkern-6.0mu/}\mathbb{T}^{d},\,\omega_{\mathrm{red}})$

given by Theorem 5.9 is $\Gamma$-equivariant. Passing to the quotients, the
equivalence in the statement follows from the previous lemma. ∎

######  Remark 5.13 (Realizing non-trivial Lagrangian-Chern classes).

Theorem 5.11 gives a Delzant type construction for toric $\mathcal{T}$-spaces
with trivial Lagrangian-Chern class. This construction can be extended to the
case of non-trivial Lagrangian-Chern class. After possible shrinking $M$, one
can assume that any class in
$\check{H}^{1}(\Delta,\underline{\mathcal{T}_{\textrm{Lag}}})$ can be realized
as the restriction of the Lagrangian-Chern class of a principal
$\mathcal{T}_{\Lambda}$-space $\mu_{P}:(P,\omega_{P})\to M$. Then a toric
$\mathcal{T}$-space with a non-trivial class, can be realized as a symplectic
quotient of the form

$((q^{*}P\times\mathbb{C}^{d})\mathbin{/\mkern-6.0mu/}(\Gamma\ltimes\mathbb{T}^{d}),\,\omega_{\mathrm{red}})$

where $q:\widetilde{M}\to M$ is the covering map.

##  6\. Invariant Kähler metrics I: Lagrangian fibrations

In this section we study $\mathcal{T}$-invariant Kähler structures on
_principal_ Hamiltonian $\mathcal{T}$-spaces, i.e., Lagrangian fibrations. In
the next section we will consider toric Hamiltonian $\mathcal{T}$-spaces.

######  Definition 6.1.

Given a Hamiltonian $\mathcal{T}$-space $\mu:(S,\omega)\to M$, a
$\mathcal{T}$-invariant Kähler structure is a $\mathcal{T}$-invariant Kähler
metric $G$ on $S$ with Kähler form $\omega$.

By Proposition 2.28, the $\mathcal{T}$-invariance of the Kähler structure can
be expressed as

$\mathscr{L}_{\mathbscr{a}(\alpha)}G=0\quad\text{ if }\quad\nabla\alpha=0.$
(6.1)

Since $\omega$ is $\mathcal{T}$-invariant, this is equivalent to the
$\mathcal{T}$-invariance of the complex structure $J:TS\to TS$, i.e.,

$\mathscr{L}_{\mathbscr{a}(\alpha)}J=0\quad\text{ if }\quad\nabla\alpha=0.$
(6.2)

Given a Lagrangian fibration $\mu:(S,\omega)\to M$ we will call a compatible
Kähler metric $g$ on $S$ invariant if it is $\mathcal{T}_{\Lambda}$-invariant,
where $\Lambda\subset T^{*}M$ is the induced integral affine structure on the
base. Invariant Kähler metrics on a Lagrangian fibration are intimately
related to Hessian metrics on the base of the fibration. We recall that a
Hessian metric on an (integral) affine manifold $M$ is a Riemannian metric $g$
satisfying

$\mathrm{d}^{\nabla}g^{\flat}=0.$ (6.3)

where $g^{\flat}:TM\to T^{*}M$ denotes the $T^{*}M$-valued form on $M$ given
by contraction with $g$. This condition is equivalent to the existence of a
potential for the metric in each (integral) affine chart $(U,x^{i})$ with
contractible domain, i.e., a function $\phi\in C^{\infty}(U)$ such that

$g|_{U}=\sum_{i,j}\frac{\partial^{2}\phi}{\partial x^{i}\partial
x^{j}}\mathrm{d}x^{i}\otimes\mathrm{d}x^{j}.$

We refer to [33] for details on Hessian metrics.

The main theorem of this section is the following.

######  Theorem 6.2.

Let $\mu:(S,\omega)\to M$ be a a Lagrangian fibration. A choice of invariant
Kähler metric $G$ on $S$ makes $\mu:S\to M$ into a Riemannian submersion, with
the following properties:

  1. (i)

$(\ker\mathrm{d}\mu)^{\perp}$ is a flat Lagrangian connection;

  2. (ii)

The metric $g$ induced on the base $M$ is a Hessian metric.

Conversely, given a flat Lagrangian connection $D$ and a Hessian metric $g$ on
$M$, there is unique invariant Kähler metric $G$ on $S$ inducing $g$ such that
$D=(\ker\mathrm{d}\mu)^{\perp}$.

######  Remark 6.3.

This theorem shows that the existence problem for invariant Kähler metric on a
Lagrangrian fibration reduces to two independent obstruction problems. As
discussed in Section 3.4, the obstruction to the existence of a flat
Lagrangian connection for a Lagrangian fibration can be expressed in terms of
its Lagrangian Chern class. The obstruction to the existence of a Hessian
metric on an (integral) affine manifold are more subtle and have been studied
by several authors (see, e.g., [33] and the references therein).

######  Example 6.4.

Consider the symplectic torus bundle in Example 2.8. Its base
$(\mathbb{T}^{2},\mathbb{Z}\mathrm{d}x^{1}\oplus\mathbb{Z}(\mathrm{d}x^{2}-x^{1}\mathrm{d}x^{1}))$
is an integral affine manifold which does not admit a Hessian metric. Indeed,
this being a symplectic torus bundle, it admits a flat Lagrangian connection.
However, its total space $(\mathcal{T},\Omega)$ does not admit any Kähler
structure, as it is the Kodaira-Thurston symplectic manifold.

Notice that the symplectic manifold $(\mathcal{T},\Omega)$ admits another
Lagrangian fibration

$\mu:\mathcal{T}\to\mathbb{T}^{2},\quad[\theta_{1},\theta_{2},x^{1},x^{2}]\mapsto[\theta_{2},x^{1}].$

The induced integral affine structure on the base $\mathbb{T}^{2}$ is now the
standard one. This fibration does not admit a Lagrangian section. In fact, due
to Theorem 6.2, it does not even admit a flat Lagrangian connection, since its
base $(\mathbb{T}^{2},\Lambda_{\mathrm{st}})$ admits a Hessian metric, but its
total space is not Kähler.

Two equivalent Kähler Lagrangian fibrations induce the same metric on their
base. Moreover, given two Kähler Lagrangian fibrations that induce the same
Hessian metric on $M$, an equivalence of the underlying Lagrangian fibrations
intertwines the Kähler structures if and only if it intertwines the orthogonal
complements to the fibers. Therefore, we conclude the following from Theorems
3.20 and 6.2.

######  Corollary 6.5.

Given an integral affine manifold $(M,\Lambda)$, there is a canonical 1:1
correspondence

$\left\\{\\\ \hbox{\hbox{\kern
0.0pt\raise-13.94444pt\vbox{\halign{\relax\hfil\txtline@@{#}\hfil\cr\hbox{{\ignorespaces\leavevmode
K\"{a}hler Lagrangian fibrations}\\\\\hbox{{\ignorespaces$\mu:(S,\omega,G)\to
M$ }\\\\\hbox{{\ignorespaces up to equivalence \,}\crcr}}}}\right\\}\
\tilde{\longleftrightarrow}\ \left\\{\\\ \hbox{\hbox{\kern
0.0pt\raise-7.80554pt\vbox{\halign{\relax\hfil\txtline@@{#}\hfil\cr\hbox{{\ignorespaces\leavevmode
Hessian metrics }\\\\\hbox{{\ignorespaces$g$ on $M$\,}\crcr}}}}\right\\}\
\times\check{H}^{1}(M,\underline{\mathcal{T}_{\textrm{Flat}}})}}}}}$

Choosing an integral affine chart $(U,x^{i})$ with contractible domain, there
exist action-angle coordinates $(\mu^{-1}(U),x^{i},\theta_{j})$ (see Remark
3.3) in which the connection takes the form

$\theta=\sum_{i=1}^{n}\mathrm{d}\theta_{i}\otimes\mathrm{d}x^{i}.$

In such coordinates the invariant Kähler metric $G$ can be written as

$G=\sum_{i,j=1}^{n}g_{ij}\,\mathrm{d}x^{i}\otimes\mathrm{d}x^{j}+\sum_{i,j=1}^{n}g^{ij}\,\mathrm{d}\theta_{i}\otimes\mathrm{d}\theta_{j},$
(6.4)

where

$g_{ij}=\frac{\partial^{2}\phi}{\partial x^{i}\partial x^{j}},$

for some function $\phi\in C^{\infty}(U)$, and $(g^{ij})$ denotes the inverse
matrix of $(g_{ij})$. The corresponding complex structure takes the form

$J=\sum_{i,j=1}^{n}g_{ij}\,\mathrm{d}x^{i}\otimes\partial_{\theta_{j}}-\sum_{i,j=1}^{n}g^{ij}\mathrm{d}\theta_{i}\otimes\partial_{x^{j}}.$

###  6.1. Proof of Theorem 6.2

Let $\Lambda$ be the induced integral affine structure on the base of
$\mu:(S,\omega)\to M$. Denote by $G$ a $\mathcal{T}_{\Lambda}$-invariant
Kähler metric on $S$. The distribution orthogonal to the fibers

$D:=(\ker\mathrm{d}\mu)^{\perp}$

is $\mathcal{T}_{\Lambda}$-invariant and the restriction of $G$ to $D$ is also
$\mathcal{T}_{\Lambda}$-invariant. It follows that there is a unique
Riemannian metric $g$ on $M$ such that $\mathrm{d}_{x}\mu:D_{x}\to T_{p}M$ is
an isometry for all $x\in\mu^{-1}(p)$, i.e., $\mu:(S,G)\to(M,g)$ is a
Riemannian submersion.

Now observe that we can express $D$ has:

$D=J(\ker\mathrm{d}\mu),$

where $J:TS\to TS$ denotes the complex structure. Since $\ker\mathrm{d}\mu$ is
Lagrangian, we conclude that $D$ is Lagrangian. It follows also that the
vector fields $J\mathbscr{a}(\alpha)$, with $\alpha$ flat 1-forms, generate
$D$. Now, $J$ is $\mathcal{T}_{\Lambda}$-invariant, so by (6.2) we find that:

$[\mathbscr{a}(\alpha),J(\mathbscr{a}(\beta))]=(\mathscr{L}_{\mathbscr{a}(\alpha)}J)+J([\mathbscr{a}(\alpha),\mathbscr{a}(\beta)])=0,$

for any flat 1-forms $\alpha$ and $\beta$. Then the vanishing of the Nijenhuis
torsion of $J$ gives:

$[J(\mathbscr{a}(\alpha)),J(\mathbscr{a}(\beta))]=J([J(\mathbscr{a}(\alpha)),\mathbscr{a}(\beta)]+[\mathbscr{a}(\alpha),J(\mathbscr{a}(\beta))])+[\mathbscr{a}(\alpha),\mathbscr{a}(\beta)]=0.$

This shows that $D$ is flat, so (i) holds.

To prove that $g$ is an Hessian metric we will check that (6.3) holds:

$\mathrm{d}^{\nabla}g^{\flat}(X,Y)=\nabla_{X}g^{\flat}(Y)-\nabla_{Y}g^{\flat}(X)-g^{\flat}([X,Y])=0,\quad(X,Y\in\mathfrak{X}(M)).$

To prove this it is enough to consider vector fields of the form
$X=g^{\sharp}(\alpha)$ and $Y=g^{\sharp}(\beta)$, with $\alpha$ and $\beta$
flat, since these locally generate all vector fields. Therefore, we are left
to check that:

$[g^{\sharp}(\alpha),g^{\sharp}(\beta)]=0\quad\text{ if
}\quad\nabla\alpha=\nabla\beta=0.$

Now observe that the definition of the metric $h$ yields:

$\mu_{*}(J(\mathbscr{a}(\alpha)))=g^{\sharp}(\alpha).$

Hence, if $\alpha$ and $\beta$ are flat, this last relation and the
computation above shows that:

$[g^{\sharp}(\alpha),g^{\sharp}(\beta)]=\mu_{*}([J(\mathbscr{a}(\alpha)),J(\mathbscr{a}(\beta))]=0.$

This shows that (ii) holds.

To prove the converse, note that given a flat Lagrangian connection $D$ we can
define a unique Riemannian metric $G$ on $S$ by requiring the following
properties to hold:

  1. (a)

$\ker\mathrm{d}\mu$ and $D$ are orthogonal;

  2. (b)

On $D$ the metric is the unique one that gives isometries

$\mathrm{d}_{p}\mu:(D_{p},G|_{D_{p}})\to(T_{\mu(p)}M,g_{\mu(p)});$

  3. (c)

On $\ker\mathrm{d}\mu$ the metric is the unique one that gives isometries

$(T^{*}_{\mu(p)}M,g^{-1}_{\mu(p)})\to(\ker\mathrm{d}_{p}\mu,G|_{\ker\mathrm{d}_{p}\mu}),\quad\alpha\mapsto\mathbscr{a}(\alpha)|_{p}.$

The resulting Riemannian metric is clearly invariant and we are left to prove
that it is a Kähler metric with Kähler form $\omega$.

Given $p\in S$ and $\alpha\in T_{\mu(p)}M$ there is a unique horizontal vector
$\widetilde{g^{\sharp}(\alpha)}\in D_{p}$ such that

$\mathrm{d}_{p}\mu(\widetilde{g^{\sharp}(\alpha)})=g^{\sharp}(\alpha).$

Since both sides of the previous equation are linear in $\alpha$, it follows
that there is a unique vector bundle map $J:\ker\mathrm{d}\mu\to D$ such that

$J(\mathbscr{a}(\alpha)):=\widetilde{g^{\sharp}(\alpha)}.$ (6.5)

Since $TS=\ker\mathrm{d}\mu\oplus D$, we obtain a vector bundle map $J:TS\to
TS$ by requiring $J^{2}=-I$. Notice that $J$ is an invariant almost complex
structure

$\mathscr{L}_{\mathbscr{a}(\alpha)}J=0,\quad\text{ if }\quad\nabla\alpha=0,$

and claim that

$G(X,Y)=\omega(X,JY),\quad(X,Y\in TS).$

To see this, we check that the bilinear form $\omega(\cdot,J\cdot)$ satisfy
properties (a)-(c) above:

  1. (a)

Since $J(\ker\mathrm{d}\mu)=D$ and $J(D)=\ker\mathrm{d}\mu$, and both
distributions are Lagrangian, it follows that they are orthogonal relative to
the bilinear form $\omega(\cdot,J\cdot)$.

  2. (b)

By (6.5), it follows that

$\displaystyle\omega(\widetilde{g^{\sharp}(\alpha)},J\widetilde{g^{\sharp}(\beta)})$
$\displaystyle=-\omega(\widetilde{g^{\sharp}(\alpha)},\mathbscr{a}(\beta))$
$\displaystyle=(\mu^{*}\beta)(\widetilde{g^{\sharp}(\alpha)})$
$\displaystyle=\beta(g^{\sharp}(\alpha))=h(g^{\sharp}(\alpha),g^{\sharp}(\beta)).$

Hence, the restriction of $\mathrm{d}\mu$ to $D$ maps the bilinear form
$\omega(\cdot,J\cdot)$ to $g$.

  3. (c)

Again by (6.5), it follows that

$\displaystyle\omega(\mathbscr{a}(\alpha),J(\mathbscr{a}(\beta)))$
$\displaystyle=\omega(\mathbscr{a}(\alpha),\widetilde{g^{\sharp}(\beta)}))$
$\displaystyle=(\mu^{*}\alpha)(\widetilde{g^{\sharp}(\beta)})$
$\displaystyle=\alpha(g^{\sharp}(\beta))=g^{-1}(\alpha,\beta).$

Hence, the last item is also satisfied.

It remains to show that $J$ is integrable, i.e., that its Nijenhuis torsion
vanishes. For this, note that:

$[\mathbscr{a}(\alpha),J(\mathbscr{a}(\beta))]=(\mathscr{L}_{\mathbscr{a}(\alpha)}J)(\mathbscr{a}(\beta))+J([\mathbscr{a}(\alpha),\mathbscr{a}(\beta)])=0,\quad\text{
if }\quad\nabla\alpha=\nabla\beta=0,$ (6.6)

since $J$ is invariant and $\mathbscr{a}(\alpha)$ and $\mathbscr{a}(\beta)$
commute. On the other hand, since $g$ is Hessian, we have for flat $\alpha$
and $\beta$:

$\displaystyle g^{\flat}([g^{\sharp}(\alpha),g^{\sharp}(\beta)])$
$\displaystyle=g^{\flat}([g^{\sharp}(\alpha),g^{\sharp}(\beta)])-\nabla_{g^{\sharp}(\alpha)}\beta+\nabla_{g^{\sharp}(\beta)}\alpha$
$\displaystyle=-\mathrm{d}^{\nabla}g^{\flat}(g^{\sharp}(\alpha),g^{\sharp}(\beta))=0.$

It follows that $[g^{\sharp}(\alpha),g^{\sharp}(\beta)]=0$, and using (6.5),
we conclude that:

$[J(\mathbscr{a}(\alpha)),J(\mathbscr{a}(\beta))]=0,\quad\text{ if
}\quad\nabla\alpha=\nabla\beta=0.$ (6.7)

From (6.6) and (6.7), we find that the Nijenhuis torsion of $J$ satisfies:

$N_{J}(\mathbscr{a}(\alpha),\mathbscr{a}(\beta))=N_{J}(\mathbscr{a}(\alpha),J(\mathbscr{a}(\beta)))=N_{J}(J(\mathbscr{a}(\alpha)),J(\mathbscr{a}(\beta)))=0,$

for any flat sections $\alpha$ and $\beta$. Since the vector fields
$\mathbscr{a}(\alpha)$ and $J(\mathbscr{a}(\alpha))$ locally generate all
vector fields, we conclude that $N_{J}\equiv 0$. This concludes the proof. ∎

###  6.2. Abreu’s equation

The scalar curvature of an invariant Kähler structure on a Lagrangian
fibration is an invariant smooth function on the total space, and hence
corresponds to a smooth function on the base of the fibration. In his work on
classical symplectic toric manifolds [1], Abreu gave an explicit expression
for this function in terms of integral affine coordinates and observed that
the Kähler metric is extremal in the sense of Calabi (see, e.g., [7]) if and
only if this function is affine. This generalizes to toric
$\mathcal{T}$-spaces. In this section we consider the case of Lagrangian
fibrations.

Recall that for an affine manifold $(M,\nabla)$ with a Hessian metric:

  * •

the first Koszul form $\alpha$ is the $1$-form on $M$ defined by:

$\alpha(X)\nu=\nabla_{X}\nu,\quad X\in\mathfrak{X}(M),$

with $\nu$ the volume density of the Hessian metric,

  * •

the second Koszul form $\beta:=\nabla\alpha$ is the symmetric $(0,2)$-tensor
given by:

$(\nabla\alpha)(X,Y)=(\nabla_{X}\alpha)(Y).$

As will be clear from the proof below, the second Koszul form is an
incarnation of the Ricci tensor in Hessian geometry. Further, recall that any
integral affine manifold $(M,\Lambda)$ carries a canonical volume density
$\nu_{\Lambda}$, which in any integral affine chart $(U,x^{i})$ is given by

$\nu_{\Lambda}|_{U}=|\mathrm{d}x^{1}\wedge\cdots\wedge\mathrm{d}x^{n}|.$

######  Theorem 6.6.

Let $\mu:(S,\omega,G)\to M$ be a Kähler Lagrangian fibration, inducing an
integral affine structure $\Lambda$ on $M$ and a Hessian metric $g$ on
$(M,\Lambda)$. Let $\alpha$ and $\beta$ be the first and second Koszul forms
of $(M,\Lambda,g)$, $\nu_{\Lambda}$ the volume density of $\Lambda$ and
$\nabla^{g}$ the Levi-Civita connection of $g$. The scalar curvature of $G$ is
given by

$S_{G}=\mathrm{Tr}_{g}(2\nabla^{g}\alpha-\beta)\circ\mu=\mathrm{div}_{\nu_{\Lambda}}(g^{\sharp}(\alpha))\circ\mu.$
(6.8)

Moreover, $G$ is an extremal metric if and only if $S_{G}$ is affine, viewed
as function on the integral affine manifold $(M,\Lambda)$.

######  Remark 6.7.

  1. (i)

Given a compact complex manifold $(S,J)$ and a Kähler class, the Calabi
functional is defined on the space of $J$-compatible Kähler metrics $G$ in the
given Kähler class and is given by

$G\mapsto\int_{M}S_{G}^{2}\,\mathrm{d}v_{G},$ (6.9)

where $v_{G}$ is the Riemannian volume form. A Kähler metric $G$ is called
extremal if it is a critical point of this functional [7]. Using the
corresponding Euler-Lagrange equations one finds (see, e.g., [34]) that a
Kähler metric $G$ is extremal if and only if

$(\operatorname{grad}S_{G})^{(1,0)}\textrm{ is a holomorphic vector field.}$
(E)

We adopt condition (E) as the definition of extremal Kähler metric, whether
$S$ is a compact manifold or not.

  2. (ii)

In our approach, where we fix a symplectic manifold $(S,\omega)$, one has the
following equivalent approach. The functional (6.9) is now defined on the
space of $\omega$-compatible Kähler metrics $G$ whose complex structure belong
to a fixed symplectomorphism class. The critical points are still the Kähler
metrics satisfying (E), which in symplectic terms can be reformulated as

$X_{S_{G}}\textrm{ is a Killing vector field.}$ (E’)

  3. (iii)

In an integral affine chart $(U,x^{1},\dots,x^{n})$ for $(M,\Lambda)$, the
first Koszul form is given by

$\alpha|_{U}=\frac{1}{2}\sum_{j=1}^{n}\frac{\partial\log(\det(g))}{\partial
x^{j}}$

and (6.8) yields

$S_{G}|_{U}=-\frac{1}{2}\sum_{j,k=1}^{n}\frac{\partial^{2}g^{jk}}{\partial
x^{j}\partial x^{k}},$

where $g^{-1}=\sum_{j,k=1}^{n}g^{jk}\partial_{x^{j}}\partial_{x^{k}}$. The
condition that the function $S_{G}$ is affine then becomes a 4th order PDE for
the potential of the metric $g$, known as Abreu’s equation.

###### Proof.

The proof is essentially as in [1]. In view of Corollary 6.5 and Remark 3.21,
$M$ can be covered by opens $U$ over which the given Kähler Lagrangian
fibration is equivalent to the Lagrangian fibration
$(\mathcal{T}_{\Lambda},\Omega_{\Lambda})\to M$ equipped with the Kähler
structure induced as in Theorem 6.2 by $g$ and the canonical horizontal
distribution. So, we can assume that
$(S,\omega)=(\mathcal{T}_{\Lambda},\Omega_{\Lambda})$ and $\mu$ is the bundle
projection.

Given an integral affine manifold $(M,\Lambda)$ with a Hessian metric $g$,
there is another integral affine structure on $M$ defined by the lattice dual
to

$\Lambda^{\sharp}:=g^{\sharp}(\Lambda)\subset TM.$

The metric $g$ is Hessian with respect to $\Lambda^{\sharp}$ as well. This
defines the dual Hessian structure, in the terminology of [33]. The second
Koszul form of $(M,\Lambda^{\sharp},g)$ is (see [33, Lemma 3.1])

$\beta_{\sharp}=\beta-2\nabla^{g}\alpha.$ (6.10)

Like $\mathcal{T}_{\Lambda}=T^{*}M/\Lambda$, the torus bundle
$\pi:TM/{\Lambda^{\sharp}}\to M$ comes with a canonical horizontal
distribution and vector bundle isomorphism
$\mathbscr{a}:\pi^{*}TM\xrightarrow{\sim}\ker(\mathrm{d}\pi)$. Moreover,
$TM/{\Lambda^{\sharp}}$ comes with a natural complex structure $J$ given by

$J(\mathbscr{a}(X))=\widetilde{X},\quad J(\widetilde{X})=-\mathbscr{a}(X),$

and the metric $g$ induces a metric $G_{\sharp}$ on $TM/{\Lambda^{\sharp}}$
given by

$G_{\sharp}(\mathbscr{a}(X_{1})+\widetilde{Y_{1}},\mathbscr{a}(X_{2})+\widetilde{Y_{2}})=g(X_{1},X_{2})+g(Y_{1},Y_{2}),$

where $\widetilde{X}$ denotes the horizontal lift of $X\in TM$. These define a
Kähler structure, which corresponds to the above Kähler structure on
$\mathcal{T}_{\Lambda}$ via the torus bundle isomorphism:

$TM/{\Lambda^{\sharp}}\xrightarrow{\sim}\mathcal{T}_{\Lambda}$ (6.11)

induced by $g$ (see, e.g., [5]). It follows from [33, Proposition 3.5] that
the Ricci tensor $R_{\sharp}$ of $G_{\sharp}$ is given by:

$R_{\sharp}(\mathbscr{a}(X_{1})+\widetilde{Y_{1}},\mathbscr{a}(X_{2})+\widetilde{Y_{2}})=-\left(\beta_{\sharp}(X_{1},X_{2})+\beta_{\sharp}(Y_{1},Y_{2})\right),$

with $\beta_{\sharp}$ the second Koszul form of $(M,\Lambda^{\sharp},g)$. From
this it readily follows that the scalar curvature $S_{G}$ (which equals that
of $G_{\sharp}$, since (6.11) is an isometry) is equal to
$-\textrm{Tr}_{g}(\beta_{\sharp})\circ\mu$, which proves the first equality in
(6.8) in view of (6.10). The second equality follows from Remark 6.7.

A straightforward computation using action-angle coordinates and expression
(6.4) for $G$ shows that for a function $f$, depending only on the action
coordinates, the Hamiltonian vector field $X_{f}$ is a Killing vector field
for $G$ if and only if $f$ is affine. This yields the equivalence of the
extremal condition (E’) with the condition for $S_{G}$. ∎

###  6.3. Examples

We will now illustrate how one can use Theorems 6.2 and 6.6 to construct
extremal Kähler metrics on Lagrangian fibrations.

######  Example 6.8.

Any metric on $(\mathbb{S}^{1},b\mathbb{Z}\mathrm{d}x)$, with $b>0$, is
Hessian (for dimensional reasons) and of the form $f\mathrm{d}x^{2}$, for
$f\in C^{\infty}(\mathbb{S}^{1})$ strictly positive. It follows from (6.8)
that the scalar curvature of any invariant Kähler metric $G$ on the complex
torus that induces this Hessian metric is given by

$S_{G}=\frac{f^{\prime\prime}}{2f^{2}}-\frac{(f^{\prime})^{2}}{f^{3}}.$ (6.12)

Since $\mathbb{S}^{1}$ is compact, if $S_{G}$ is affine, then it is in fact
constant. Since $f$ is strictly positive and periodic, it follows that this
constant must then be zero. The only solutions to the ODE

$ff^{\prime\prime}-2(f^{\prime})^{2}=0,$

are the constant functions on $\mathbb{S}^{1}$. Therefore, we conclude that
the only extremal metrics are those for which $f$ is constant.

As we saw in Example 3.23, any Lagrangian fibration inducing $(M,\Lambda)$
takes the form

$\operatorname{pr}_{2}:(\mathbb{S}^{1}\times\mathbb{S}^{1},b\,\mathrm{d}y\wedge\mathrm{d}x)\to\mathbb{S}^{1}.$

If we fix the Hessian metric $g=b^{2}\,\mathrm{d}x^{2}$ and choose any of the
Lagrangian connections $D_{a}$ in that example, the resulting extremal Kähler
metric on $S=\mathbb{T}^{2}$ is the flat metric

$G=(a^{2}+b^{2})\,\mathrm{d}x^{2}+2a\,\mathrm{d}x\,\mathrm{d}y+\mathrm{d}y^{2}.$

The $2$-torus equipped with corresponding complex structure is biholomorphic
to the complex torus $\mathbb{C}/\langle 1,\tau\rangle$, where $\tau=a+ib$.

######  Example 6.9.

Consider the Lagrangian fibration
$\operatorname{pr}_{1}:(\mathbb{T}^{2}\times\mathbb{T}^{2},\omega_{a})\to\mathbb{T}^{2}$
of Examples 3.7 and 3.24, where $a\in\mathbb{R}$ and
$\omega_{a}=\omega_{\mathrm{can}}+a\mathrm{d}x^{1}\wedge\mathrm{d}x^{2}$. This
admits the flat Lagrangian connection

$D=\langle\partial_{x_{1}},\partial_{x_{2}}+a\partial_{\theta_{1}}\rangle.$

If we consider the flat Hessian metric
$g=\mathrm{d}x^{1}\mathrm{d}x^{1}+\mathrm{d}x^{2}\mathrm{d}x^{2}$ on
$\mathbb{T}^{2}$, we obtain from Theorem 6.2 the $\omega_{a}$-compatible
extremal Kähler metric

$G=\mathrm{d}x^{1}\mathrm{d}x^{1}+(1+a^{2})\mathrm{d}x^{2}\mathrm{d}x^{2}-2a\mathrm{d}x^{1}\mathrm{d}\theta_{2}+\mathrm{d}\theta_{1}\mathrm{d}\theta_{1}+\mathrm{d}\theta_{2}\mathrm{d}\theta_{2}.$

Notice that, when one varies $a\in[0,1[$, these give non-isomorphic Kähler
metrics.

######  Example 6.10.

Consider the standard integral affine cylinder

$(M,\Lambda)=(\mathbb{S}^{1}\times\mathbb{R},\mathbb{Z}\mathrm{d}x\oplus\mathbb{Z}\mathrm{d}h)$

of Example 2.20. Given a metric

$g=A\mathrm{d}h^{2}+2B\mathrm{d}h\mathrm{d}x+C\mathrm{d}x^{2}$

on this cylinder, We find that

$\mathrm{d}^{\nabla}g^{\flat}(\partial_{h},\partial_{x})=\left(\partial_{h}B-\partial_{x}A\right)\mathrm{d}h+\left(\partial_{h}C-\partial_{x}B\right)\mathrm{d}x.$

Therefore, we have a family of Hessian metrics

$g=\frac{1}{\tau(h)}\mathrm{d}h^{2}+\frac{1}{\epsilon(x)}\mathrm{d}x^{2},$

where both $\tau(h)$ and $\epsilon(x)$ are strictly positive.

Note that
$H^{1}(M,\underline{\mathcal{T}_{\textrm{Lag}}})=H^{2}(M,\mathcal{O}_{\Lambda})=0$,
so that any toric $\mathcal{T}_{\Lambda}$-space is isomorphic to the
symplectic torus bundle over $(M,\Lambda)$ (see Example 2.20). Applying (6.8),
it follows that the scalar curvature of the lifted Kähler metric $G$ on this
symplectic torus bundle is given by

$S_{G}=-\frac{1}{2}(\tau^{\prime\prime}(h)+\epsilon^{\prime\prime}(x)),$

for any choice of flat Lagrangian connection. The condition for $G$ to be
extremal, i.e., for $S_{G}$ to be affine, amounts to the condition that
$\epsilon(x)$ is constant and $\tau(h)$ is a cubic polynomial in $h$. Since
$\tau(h)$ is strictly positive, this leads to the following two possibilities
for $G$ to be extremal:

  1. (i)

$\tau(h)=bh^{2}+ch+d$ with $b>0$ and $c^{2}-4bd<0$, in which case $G$ has
constant negative scalar curvature $-\tfrac{b}{2}$;

  2. (ii)

$\tau(h)$ is constant, in which case $G$ has zero scalar curvature.

######  Example 6.11.

Consider the non-standard integral affine cylinder of Example 2.21. Given a
general metric on $\mathbb{S}^{1}\times]-2,\infty[$, written in the form

$g=A\,\mathrm{d}h^{2}+2B\,\mathrm{d}h\mathrm{d}x+C\,\mathrm{d}x^{2}.$

We find that

$\mathrm{d}^{\nabla}g^{\flat}(\partial_{h},\partial_{x})=\left(\partial_{h}B-\partial_{x}A+\tfrac{B}{h+2}\right)\mathrm{d}h+\left(\partial_{h}C-\partial_{x}B-\tfrac{C}{h+2}\right)\mathrm{d}x.$

Therefore, we have a family of Hessian metrics

$g=\frac{1}{\tau(h)}\mathrm{d}h^{2}+(h+2)\epsilon\,\mathrm{d}x^{2},$

where $\epsilon>0$ and $\tau(h)$ is a strictly positive smooth function on
$]-2,\infty[$.

Like in the previous example,
$H^{1}(M,\underline{\mathcal{T}_{\textrm{Lag}}})=H^{2}(M,\mathcal{O}_{\Lambda})=0$,
so any toric $\mathcal{T}_{\Lambda}$-space is isomorphic to the symplectic
torus bundle over $(M,\Lambda)$ (see Example 2.21). Applying (6.8), it follows
that the scalar curvature of the lifted Kähler metric $G$ on this symplectic
torus bundle is given by

$S_{G}=-\frac{1}{2}\tau^{\prime\prime}(h)-\frac{1}{h+2}\tau^{\prime}(h),$

for any choice of flat Lagrangian connection. Such a metric is extremal if

$\frac{1}{2}\tau^{\prime\prime}(h)+\frac{1}{h+2}\tau^{\prime}(h)=c_{1}(h+2)+c_{0}$

for some constants $c_{1}$ and $c_{0}$. One finds that

$\tau(h)=\frac{c_{1}}{6}(h+2)^{3}+\frac{c_{0}}{3}(h+2)^{2}+\frac{c_{2}}{h+2}+c_{3}$

where $c_{2}$ and $c_{3}$ are constants of integration. Therefore, choosing
the constants such that $\tau(h)>0$, we obtain extremal Kähler metrics on the
symplectic torus bundle. These include examples both with constant and non-
constant scalar curvature.

##  7\. Invariant Kähler metrics II: toric Lagrangian fibrations

In this section we will generalize the characterization of compatible
invariant Kähler metrics on Lagrangian fibrations given in Theorem 6.2 to
toric $(\mathcal{T},\Omega)$-spaces.

Below, in Section 7.1, we will introduce the notion of hybrid $b$-metric for
manifolds with corners. These are metrics admitting a specific type of
singularity at the boundary and with a well-defined notion of residue at the
open facets. Using this notion, we can state the main result of this section
as follows.

######  Theorem 7.1.

Let $(M,\Lambda)$ be an integral affine manifold and let $\mu:(S,\omega)\to M$
be a toric $(\mathcal{T}_{\Lambda},\Omega_{\Lambda})$-space, with Delzant
subspace $\Delta:=\mu(S)$. Denote by $\mathring{\Delta}$ the interior of
$\Delta$ and set $\mathring{S}:=\mu^{-1}(\mathring{\Delta})$. For any
invariant Kähler metric $G$ on $S$ compatible with $\omega$, the following
hold:

  1. (i)

$(\ker\mathrm{d}\mu)^{\perp}|_{\mathring{S}}$ extends to an elliptic
connection with zero radial residue over the open facets;

  2. (ii)

The Hessian metric $g$ on $\mathring{\Delta}$, induced via Theorem 6.2,
extends to a hybrid $b$-metric on $\Delta$ with the residue at the open facets
given by the primitive outward-pointing normals multiplied by
$\tfrac{1}{4\pi}$.

Conversely, given a flat Lagrangian elliptic connection $\theta$ with zero
radial residues and a hybrid $b$-metric $g$ on $\Delta$ with residues as in
(ii), there is a unique invariant compatible Kähler metric $G$ on $S$ inducing
$g$ and such that
$\ker\theta|_{\mathring{S}}=(\ker\mathrm{d}\mu)^{\perp}|_{\mathring{S}}$.

######  Remark 7.2.

The scalar curvature $S_{G}$ of an invariant Kähler metric $G$ descends to a
smooth function on $\Delta$, because it is $\mathcal{T}$-invariant. It follows
from the case of smooth Lagrangian fibrations (by restricting to
$\mathring{\Delta}$) that $G$ is an extremal Kähler metric if and only if
$S_{G}:\Delta\to\mathbb{R}$ is an affine function.

By the same reasoning as for Corollary 6.5, we conclude the following from
Theorems 4.33 and 7.1.

######  Corollary 7.3.

Given an integral affine manifold $(M,\Lambda)$ and a Delzant subspace
$\Delta$, there is a canonical 1:1 correspondence

$\left\\{\\\ \hbox{\hbox{\kern
0.0pt\raise-13.94444pt\vbox{\halign{\relax\hfil\txtline@@{#}\hfil\cr\hbox{{\ignorespaces\leavevmode
K\"{a}hler toric
$\mathcal{T}$-spaces}\\\\\hbox{{\ignorespaces$\mu:(S,\omega,G)\to M$ with
$\mu(S)=\Delta$}\\\\\hbox{{\ignorespaces up to equivalence
\,}\crcr}}}}\right\\}\ \tilde{\longleftrightarrow}\ \left\\{\\\
\hbox{\hbox{\kern
0.0pt\raise-8.88889pt\vbox{\halign{\relax\hfil\txtline@@{#}\hfil\cr\hbox{{\ignorespaces\leavevmode
Hessian hybrid }\\\\\hbox{{\ignorespaces\,$b$-metrics $g$ on
$\Delta$\,}\crcr}}}}\right\\}\
\times\check{H}^{1}(\Delta,\underline{\mathcal{T}_{\textrm{Flat}}})}}}}}$

###  7.1. Hybrid $b$-metrics

Let $\Delta$ be a manifold with corners. Given a smooth section

$g\in\Gamma^{\infty}(\tensor[^{b}]{T^{*}\Delta}{}\otimes T^{*}\Delta),$ (7.1)

we let $g^{\flat}:\tensor[^{b}]{T\Delta}{}\to T^{*}\Delta$ denote the induced
vector bundle map. We also denote by

$\widehat{g}\in\Gamma^{\infty}(\tensor[^{b}]{T^{*}\Delta}{}\otimes\tensor[^{b}]{T^{*}\Delta}{})$

the section induced by $g$ via the anchor map of $\tensor[^{b}]{T\Delta}{}$,
so that $\widehat{g}^{\flat}=\rho^{*}\circ g^{\flat}$.

######  Definition 7.4.

A smooth section $g$ of $\tensor[^{b}]{T^{*}\Delta}{}\otimes T^{*}\Delta$ will
be called:

  1. (i)

non-degenerate if $g^{\flat}_{x}$ is an isomorphism for each $x\in\Delta$;

  2. (ii)

symmetric if $\widehat{g}_{x}$ is symmetric for all $x\in\Delta$
(equivalently, for all $x\in\mathring{\Delta}$);

  3. (iii)

positive semi-definite if $\widehat{g}_{x}$ is positive semi-definite for all
$x\in\Delta$ (equivalently, for all $x\in\mathring{\Delta}$).

We will call $g$ a hybrid $b$-metric on $\Delta$ if it satisfies all these
three properties.

A hybrid $b$-metric on $\Delta$ can be thought of as a smooth Riemannian
metric on $\mathring{\Delta}$ with specified singularities at the boundary. In
local coordinates, these singularities are of the same type as those of the
singular metrics in [1]. Assume $\Delta$ is an open in $\mathbb{R}^{n}_{k}$. A
smooth section $g$ of $\tensor[^{b}]{T^{*}\Delta}{}\otimes T^{*}\Delta$ is
symmetric if and only if it is of the form

$g=\sum_{i=1}^{k}\frac{1}{x^{i}}g_{ii}\mathrm{d}x^{i}\otimes\mathrm{d}x^{i}+\sum_{(i,j)\in
I_{k}}g_{ij}\mathrm{d}x^{i}\otimes\mathrm{d}x^{j},$ (7.2)

for some $g_{ij}\in C^{\infty}(\Delta)$ such that $g_{ij}=g_{ji}$ for all
$i,j\in\\{1,\dots,n\\}$, where we denote

$I_{k}:=\\{(i,j)\in\\{1,\dots,n\\}^{2}\mid i\neq j\text{ or }i=j>k\\}.$

If $g$ is symmetric, then

  * •

positive semi-definiteness of $g$ means that for each $x\in\mathring{\Delta}$
the matrix $\mathring{g}(x)$ given by

$\mathring{g}(x)_{ij}:=\begin{cases}\frac{1}{x^{i}}g_{ii}(x)\quad&\text{ if
}i=j\leq k,\\\ g_{ij}(x)\quad&\text{ if }(i,j)\in I_{k},\end{cases}$ (7.3)

is positive semi-definite.

  * •

non-degeneracy of $g$ at a point in $\Delta$ means that

$(x^{1}\cdot\dots\cdot x^{k})\cdot\det(\mathring{g})\in C^{\infty}(\Delta)$
(7.4)

is non-zero at that point.

Therefore, if $g$ is symmetric and positive semi-definite, then it is non-
degenerate if and only if (7.4) is strictly positive on all of $\Delta$. The
latter condition is as in [1].

Assume now that $\Delta$ is an affine manifold with corners, i.e., $\Delta$
has an atlas for which the transition functions are restrictions of affine
transformations. An Hessian hybrid $b$-metric $g$ on $\Delta$ is a hybrid
$b$-metric satisfying

$\mathrm{d}^{\nabla}g^{\flat}=0,$

where $\nabla$ is the associated flat $\tensor[^{b}]{T\Delta}{}$-connection on
$T^{*}\Delta$. Such a $b$-metric $g$ admits local potentials around points in
the boundary of $\Delta$ which are of a very particular form.

######  Proposition 7.5.

Let $\Delta$ is an affine manifold with corners and $g$ a Hessian hybrid
$b$-metric on $\Delta$. Around a point $x\in\partial\Delta$ of depth $k$ there
exists a chart $(U,x^{i})$ centered at $x$ for which

$g=\sum_{i,j=1}^{n}\frac{\partial^{2}\phi}{\partial x^{i}\partial
x^{j}}\mathrm{d}x^{i}\otimes\mathrm{d}x^{j}$

where $\phi$ is a smooth function on $\mathring{\Delta}\cap U$ of the form

$\phi=\sum_{i=1}^{k}c_{i}x^{i}\log(x^{i})+f,\quad
c_{i}:=g_{ii}(0)\in\mathbb{R},\quad f\in C^{\infty}(U).$ (7.5)

###### Proof.

Fix an affine chart for $\Delta$ centered at $x$ onto an open
$[0,\delta[^{k}\times]-\delta,\delta[^{n-k}$. The fact that $g$ is Hessian
means that the coefficients (7.3) of the Riemannian metric on
$\mathring{\Delta}$ underlying $g$ satisfy the usual set of differential
equations

$\frac{\partial\mathring{g}^{ij}}{\partial
x^{l}}=\frac{\partial{\mathring{g}^{il}}}{\partial x^{j}},\quad 1\leq
i,j,l\leq n.$ (7.6)

In terms of the functions $g_{ij}\in C^{\infty}(U)$ as in (7.2) this means
that

$\displaystyle\frac{\partial g_{ii}}{\partial x^{j}}=x^{i}\frac{\partial
g_{ij}}{\partial x^{i}}$ $\displaystyle\quad\text{if }i\leq k,\text{ }i\neq
j,$ (7.7) $\displaystyle\frac{\partial g_{ij}}{\partial x^{l}}=\frac{\partial
g_{il}}{\partial x^{j}}$ $\displaystyle\quad\text{if }(i,j),(i,l)\in I_{k}.$
(7.8)

Since (7.7) vanishes if $x^{i}=0$, for $i\in\\{1,\dots,k\\}$ the function

$x\mapsto g_{ii}(x^{1},\dots,x^{i-1},0,x^{i+1},\dots,x^{n})$

is constant on $U$ with value $c_{i}$. Since $g_{ii}(x)-c_{i}$ vanishes if
$x^{i}=0$, $g_{ii}-c_{i}$ is of the form $x^{i}h_{i}$ for a smooth function
$h_{i}\in C^{\infty}(U)$. The equations (7.7) and (7.8) then express the fact
that the $1$-forms $\alpha_{i}$ on $U$ given by

$\alpha_{i}=\begin{cases}h_{i}\mathrm{d}x^{i}+\sum_{j\neq
i}g_{ij}\mathrm{d}x^{j}\quad&\text{if }1\leq i\leq k,\\\ \\\
\sum_{j=1}^{n}g_{ij}\mathrm{d}x^{j},\quad&\text{if }k<i\leq n,\end{cases}$

are closed. By the Poincaré lemma for manifolds with corners, each
$\alpha_{i}$ must in fact be exact. So, there are $\phi_{i}\in C^{\infty}(U)$
such that $\mathrm{d}\phi_{i}=\alpha_{i}$. Applying the Poincaré lemma once
more, this time to the closed $1$-form $\sum_{i=1}^{n}\phi_{i}\mathrm{d}x^{i}$
on $U$, we find an $f\in C^{\infty}(U)$ such that:

$\frac{\partial^{2}f}{\partial x^{i}\partial
x^{j}}=\begin{cases}h_{i}\quad&\text{if }i=j<k,\\\ \\\ g_{ij}\quad&\text{if
}(i,j)\in I_{k},\end{cases}$

with $I_{k}$ as in (7.3). For this choice of $f$, the function $\phi$ defined
by (7.5) is indeed a potential for $g$ on $\mathring{\Delta}\cap U$. ∎

Next, we introduce the notion of residue appearing in Theorem 7.1.

######  Definition 7.6.

The residue of a hybrid $b$-metric $g$ at an open facet of $\Delta$ is the
residue of the vector valued 1-form
$g^{\flat}\in\Omega^{1}(\tensor[^{b}]{T\Delta}{},T^{*}\Delta)$ at the facet
(see Definition 4.21).

######  Example 7.7.

Consider a Delzant polytope

$\Delta:=\bigcap_{i=1}^{d}\\{\ell_{i}\leq 0\\}\subset\mathbb{R}^{n}.$

Then $\mathring{\Delta}$ is contractible, so any Hessian hybrid $b$-metric on
$\Delta$ admits a global potential $\phi\in C^{\infty}(\mathring{\Delta})$. By
applying successively Proposition 7.5, one concludes that for a Delzant
polytope a Hessian hybrid $b$-metric as in Theorem 7.1 (ii) has potential of
the form

$\phi(x)=-\tfrac{1}{4\pi}\sum_{i=1}^{d}\ell_{i}(x)\log\ell_{i}(x)+f(x),$

where $f$ is smooth on $\Delta$ (not just on $\mathring{\Delta}$). In view of
this and (7.4), we recover the type of singular Hessian metrics on Delzant
polytopes appearing in the works of Guillemin [24] and Abreu [1].

######  Remark 7.8.

In [1] Abreu also observed that the difference between any two of his singular
hessian metrics on a Delzant polytope extends smoothly over the boundary. This
holds more generally for any two hybrid $b$-metrics on a manifold with corners
$\Delta$ that have the same residues.

For the proof of Theorem 7.1, it will be useful to also consider the notion
dual to that of the singular metrics above. These will be smooth sections

$\delta\in\Gamma^{\infty}(T\Delta\otimes\tensor[^{b}]{T\Delta}{}).$

Given such a section $\delta$, we denote by
$\delta^{\sharp}:T^{*}\Delta\to\tensor[^{b}]{T\Delta}{}$ the induced vector
bundle map. We also denote by

$\widehat{\delta}\in\Gamma^{\infty}(T\Delta\otimes T\Delta)$

the section induced by $\delta$ via the anchor map of
$\tensor[^{b}]{T\Delta}{}$, so that
$\widehat{\delta}^{\sharp}=\rho\circ\delta^{\sharp}$.

######  Definition 7.9.

A smooth section $\delta$ of $T\Delta\otimes\tensor[^{b}]{T\Delta}{}$ will be
called:

  1. (i)

non-degenerate if $\delta^{\sharp}_{x}$ is an isomorphism for each
$x\in\Delta$;

  2. (ii)

symmetric if $\widehat{\delta}_{x}$ is symmetric for all $x\in\Delta$
(equivalently, for all $x\in\mathring{\Delta}$);

  3. (iii)

positive semi-definite if $\widehat{\delta}_{x}$ is positive semi-definite for
all $x\in\Delta$ (equivalently, for all $x\in\mathring{\Delta}$).

We will call $\delta$ a hybrid $b^{*}$-metric on $\Delta$ if it satisfies all
these three properties.

A hybrid $b^{*}$-metric has an underlying metric on $T^{*}\mathring{\Delta}$
that extends smoothly to the tensor $\widehat{\delta}$ on all of $\Delta$.
Note, however, that $\widehat{\delta}$ is degenerate at the boundary. To see
how, suppose that $\Delta$ is an open around the origin in
$\mathbb{R}^{n}_{k}$. Then a smooth section $\delta$ of
$T\Delta\otimes\tensor[^{b}]{T\Delta}{}$ is symmetric if and only if it is of
the form

$\displaystyle\delta=\sum_{i,j=1,j\neq
i}^{k}x^{i}x^{j}\delta^{ij}{\partial}_{x^{i}}\otimes{\partial}_{x^{j}}$
$\displaystyle+\sum_{i=1}^{k}x^{i}\delta^{ii}{\partial}_{x^{i}}\otimes{\partial}_{x^{i}}+\sum_{i,j=k+1}^{n}\delta^{ij}{\partial}_{x^{i}}\otimes{\partial}_{x^{j}}$
(7.9)
$\displaystyle+\sum_{i=1}^{k}\sum_{j=k+1}^{n}x^{i}\delta^{ij}\left({\partial}_{x^{i}}\otimes{\partial}_{x^{j}}+{\partial}_{x^{j}}\otimes{\partial}_{x^{i}}\right),$

for $\delta^{ij}\in C^{\infty}(\Delta)$ such that $\delta^{ij}=\delta^{ji}$
for all $1\leq i,j\leq n$. For positive semi-definiteness and non-degeracy,
there are criteria analogous to those for hybrid $b$-metrics.

On a given manifold with corners, hybrid $b$-metrics are in bijective
correspondence with hybrid $b^{*}$-metrics, in which a hybrid $b$-metric $g$
and its corresponding hybrid $b^{*}$-metric $\delta$ are related by the fact
that $g^{\flat}$ and $\delta^{\sharp}$ are inverse to each other.

###  7.2. Proof of Theorem 7.1

In this section we prove the two directions of the statement of Theorem 7.1.
We start by proving the following local result concerning invariant metrics in
the local model discussed in Section 4.1.

######  Lemma 7.10.

Let $(S_{k,n},\omega_{k,n})$ be a standard local model with the
$\mathbb{T}^{n}$-action (4.1), and let $U$ be an open in
$\Delta:=\mathbb{R}^{n}_{k}$. Then:

  1. (i)

A smooth $\mathbb{T}^{n}$-invariant symmetric tensor:

$G:\mu_{k,n}^{-1}(U\cap\mathring{\Delta})\to T^{*}S_{k,n}\otimes T^{*}S_{k,n}$

extends smoothly to all of $\mu_{k,n}^{-1}(U)$ if and only if it takes the
following form

$\displaystyle G$
$\displaystyle=\sum_{j=1}^{k}\left(g_{r_{j},r_{j}}(\mathrm{d}r_{j})^{2}+r_{j}^{3}g_{r_{j},\phi_{j}}\mathrm{d}r_{j}\mathrm{d}\phi_{j}+r_{j}^{2}g_{\phi_{j},\phi_{j}}(\mathrm{d}\phi_{j})^{2}\right)$
$\displaystyle+\sum_{j,l=1,j\not=l}^{k}\left(r_{j}r_{l}g_{r_{j},r_{l}}\mathrm{d}r_{j}\mathrm{d}r_{l}+r_{j}r_{l}^{2}g_{r_{j},\phi_{l}}\mathrm{d}r_{j}\mathrm{d}\phi_{l}+r_{j}^{2}r_{l}^{2}g_{\phi_{j},\phi_{l}}\mathrm{d}\phi_{j}\mathrm{d}\phi_{l}\right)$
$\displaystyle\quad+\sum_{j,l=1}^{k,n-k}\left(r_{j}g_{r_{j},x^{l}}\mathrm{d}r_{j}\mathrm{d}x^{l}+r_{j}g_{r_{j},\theta_{l}}\mathrm{d}r_{j}\mathrm{d}\theta_{l}+r_{j}^{2}g_{\phi_{j},x^{l}}\mathrm{d}\phi_{j}\mathrm{d}x^{l}+r_{j}^{2}g_{\phi_{j},\theta_{l}}\mathrm{d}\phi_{j}\mathrm{d}\theta_{l}\right)$
$\displaystyle\quad\quad+\sum_{j,l=1}^{n-k}\left(g_{\theta_{j},\theta_{l}}\mathrm{d}\theta_{j}\mathrm{d}\theta_{l}+g_{x^{j},\theta_{l}}\mathrm{d}x^{j}\mathrm{d}\theta_{l}+g_{x^{j},x^{l}}\mathrm{d}x^{j}\mathrm{d}x^{l}\right)$

where the coefficients functions are $\mathbb{T}^{n}$-invariant smooth
functions on $\mu_{k,n}^{-1}(U)$ and
$g_{\phi_{j},\phi_{j}}=(2\pi)^{2}g_{r_{j},r_{j}}$ over $U\cap\\{x^{j}=0\\}$.

  2. (ii)

Dually, a smooth $\mathbb{T}^{n}$-invariant symmetric tensor

$G^{*}:\mu_{k,n}^{-1}(U\cap\mathring{\Delta})\to TS_{k,n}\otimes TS_{k,n}$

extends smoothly to all of $\mu_{k,n}^{-1}(U)$ if and only if its coefficients
with respect to the coframe on $\mathring{S}$

$\mathrm{d}\phi_{1},\dots,\mathrm{d}\phi_{k},\frac{1}{r_{1}}\mathrm{d}r_{1},\dots,\frac{1}{r_{k}}\mathrm{d}r_{k},\mathrm{d}\theta_{1},\dots,\mathrm{d}\theta_{n-k},\mathrm{d}x^{1},\dots,\mathrm{d}x^{n-k}$

are $\mathbb{T}^{n}$-invariant smooth functions on $\mu_{k,n}^{-1}(U)$, with
the exception of the following coefficients, which take the form

$G^{*}\left(\frac{1}{r_{j}}\mathrm{d}r_{j},\frac{1}{r_{j}}\mathrm{d}r_{j}\right)=\frac{1}{r_{j}^{2}}g^{*}_{r_{j},r_{j}},\quad
G^{*}\left(\mathrm{d}\phi_{j},\mathrm{d}\phi_{j}\right)=\frac{1}{r_{j}^{2}}g^{*}_{\phi_{j},\phi_{j}},$

with $g^{*}_{r_{j},r_{j}}$ and $g^{*}_{\phi_{j},\phi_{j}}$
$\mathbb{T}^{n}$-invariant smooth functions on $\mu_{k,n}^{-1}(U)$ such that
$g^{*}_{\phi_{j},\phi_{j}}=(2\pi)^{2}g^{*}_{r_{j},r_{j}}$ over
$U\cap\\{x^{j}=0\\}$.

###### Proof.

The backward implication in (i) readily follows by expressing the coefficients
of $G$ with respect to the frame:

$\partial_{u_{1}},\partial_{v_{1}},\dots,\partial_{u_{k}},\partial_{v_{k}},\partial_{\theta_{1}},\dots,\partial_{\theta_{n-k}},\partial_{x^{1}},\dots,\partial_{x^{n-k}}$

on $U$ in terms of the coefficients with respect to the frame (4.5) on
$\mathring{U}$. For the forward implication, suppose that $G$ extends smoothly
to all of $\mu^{-1}_{k,n}(U)$. By continuity this extension is $T$-invariant
as well, hence so are its coefficient functions with respect to the frame
(4.5). Applying Proposition 4.6 to these, it follows that the coefficients in
the last 3 lines of the formula for $G$ are of the desired form. Moreover, it
follows that there are $\mathbb{T}^{n}$-invariant smooth functions
$g_{\phi_{j},\phi_{j}},g_{r_{j},r_{j}},\widetilde{g}_{\phi_{j},r_{j}}$ on
$\mu^{-1}_{k,n}(U)$ such that:

$\displaystyle G\left(\partial_{\phi_{j}},\partial_{\phi_{j}}\right)$
$\displaystyle=r_{j}^{2}g_{\phi_{j},\phi_{j}},$ $\displaystyle
G\left(r_{j}\partial_{r_{j}},r_{j}\partial_{r_{j}}\right)$
$\displaystyle=r_{j}^{2}g_{r_{j},r_{j}},$ $\displaystyle
G\left(\partial_{\phi_{j}},r_{j}\partial_{r_{j}}\right)$
$\displaystyle=r_{j}^{2}\widetilde{g}_{\phi_{j},r_{j}}.$

Denote the smooth extension of $G$ to $\mu^{-1}_{k,n}(U)$ by $\widehat{G}$.
Comparing the limits of the function
$\widehat{G}(\partial_{u_{j}},\partial_{v_{j}})$ along the two paths:

$\displaystyle\mathbb{R}\ni\varepsilon\mapsto(z_{1},\dots,z_{j-1},\varepsilon,z_{j+1},\dots,z_{k},t,x)$
$\displaystyle\mathbb{R}\ni\varepsilon\mapsto(z_{1},\dots,z_{j-1},i\varepsilon,z_{j+1},\dots,z_{k},t,x)$

as $\varepsilon$ tends to zero (for any given $j\in\\{1,\dots,k\\}$ and
$(z,t,x)\in U\cap\\{z_{j}=0\\}$) leads to the conclusion that
$\widetilde{g}_{\phi_{j},r_{j}}$ vanishes over $U\cap\\{x^{j}=0\\}$, so
$\widetilde{g}_{\phi_{j},r_{j}}=r_{j}^{2}{g}_{\phi_{j},r_{j}}$. Computing the
same limits for the function $\widehat{G}(\partial_{u_{j}},\partial_{u_{j}})$
shows that $g_{\phi_{j},\phi_{j}}$ and $(2\pi)^{2}\cdot g_{r_{j},r_{j}}$
coincide over $U\cap\\{x^{j}=0\\}$, which completes the proof of (i). Part
(ii) follows from the same type of arguments. ∎

####  7.2.1. From invariant (Kähler) metrics to hybrid $b$-metrics

Let $\mu:(S,\omega)\to M$ be a toric
$(\mathcal{T}_{\Lambda},\Omega_{\Lambda})$-space with Delzant subspace
$\Delta:=\mu(S)$. Further, let $G$ be a $\omega$-compatible invariant Kähler
metric on $S$. As in the proof of Theorem 6.2, the $\mathcal{T}$-invariance of
$G$ implies that there is a unique symmetric, positive semi-definite section
$\widehat{\delta}\in\Gamma^{\infty}(T\Delta\otimes T\Delta)$ making the
following diagrams commute

$\textstyle{TS\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{G^{\flat}}$$\textstyle{T^{*}S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a^{*}}$$\textstyle{T^{*}S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{(G^{\flat})}^{-1}}$$\textstyle{TS\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{d}\mu}$$\textstyle{\mu^{*}(T^{*}\Delta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\scriptstyle{\widehat{\delta}^{\sharp}}$$\textstyle{\mu^{*}(T\Delta)}$$\textstyle{\mu^{*}(T^{*}\Delta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{d}\mu^{*}}$$\scriptstyle{\widehat{\delta}^{\sharp}}$$\textstyle{\mu^{*}(T\Delta)}$
(7.10)

The tensor $\widehat{\delta}$ extends the metric on $T^{*}\mathring{\Delta}$
dual to the unique Riemannian metric on $g$ on $\mathring{\Delta}$ making
$\mu:(\mathring{S},G)\to(\mathring{\Delta},g)$ a Riemannian submersion. For
each locally defined $1$-form $\alpha$ on $\Delta$ the vector field
$(G^{\flat})^{-1}(\mu^{*}\alpha)$ is $\mathcal{T}$-invariant, since both
$\mu^{*}\alpha$ and $G$ are. Therefore,
$(G^{\flat})^{-1}\circ(\mathrm{d}\mu)^{*}$ lifts to a unique vector bundle
map:

.$\textstyle{\tensor[^{\mu}]{TS}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\textstyle{\mu^{*}(T^{*}\Delta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(G^{\flat})^{-1}\circ(\mathrm{d}\mu)^{*}}$$\scriptstyle{h}$$\textstyle{TS}$
(7.11)

From this and the commutativity of the right-hand square in (7.10), it follows
that $\widehat{\delta}^{\sharp}$ takes $1$-forms on $\Delta$ to $b$-vector
fields. Therefore, $\delta$ lifts to a symmetric and positive semi-definite
section $\delta\in\Gamma^{\infty}(T\Delta\otimes\tensor[^{b}]{T\Delta}{})$
making the following digrams commute

$\textstyle{\tensor[^{b}]{T\Delta}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\textstyle{\tensor[^{\mu}]{TS}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{*}}$$\textstyle{T^{*}\Delta\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta^{\sharp}}$$\scriptstyle{\widehat{\delta}^{\sharp}}$$\textstyle{T\Delta}$$\textstyle{\mu^{*}(T^{*}\Delta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\scriptstyle{\delta^{\sharp}}$$\textstyle{\mu^{*}(\tensor[^{b}]{T\Delta}{})}$
(7.12)

with $\mu_{*}$ as in Proposition 4.22. In what follows, we will show that
$\delta$ is non-degenerate, so that it is a hybrid $b^{*}$-metric on $\Delta$.
Its dual will then be the hybrid $b$-metric on $\Delta$ extending the
Riemannian metric $g$, as in (i). Moreover, by commutativity of the right-hand
triangle in (7.12), the composite

$\textstyle{\mu^{*}(\tensor[^{b}]{T\Delta}{})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\delta^{\sharp})^{-1}}$$\textstyle{\mu^{*}(T^{*}\Delta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\textstyle{\tensor[^{\mu}]{TS}{}}$
(7.13)

will be a section of $\mu_{*}$, which gives the extension of
$(\ker\mathrm{d}\mu)^{\perp}|_{\mathring{S}}$ to the elliptic connection
$\theta$ in (i).

It remains to prove that:

  1. (i)

$\delta$ is non-degenerate at each $x\in\partial\Delta$;

  2. (ii)

the residue of $\delta$ at each open facet is the primitive outward-pointing
normal multiplied by $\frac{1}{4\pi}$;

  3. (iii)

$\theta$ has zero radial residue.

Let us fix $x\in\partial\Delta$ and standard toric coordinates
$(z_{i},\theta_{j},x^{j})$ centered at $x$ (see Section 4.1). In these
coordinates, the smooth section $\delta$ of
$T\Delta\otimes\tensor[^{b}]{T\Delta}{}$ takes the form (7.9), so we can write
the map $\delta^{\sharp}_{x}:T_{x}^{*}\Delta\to\tensor[^{b}]{T_{x}\Delta}{}$
as a matrix of the form

$\begin{pmatrix}D_{k}&\vline&D_{n-k,k}\\\ \hline\cr
0&\vline&D_{n-k}\end{pmatrix},$ (7.14)

with respect to standard bases of $T^{*}_{x}\Delta$ and
$\tensor[^{b}]{T_{x}\Delta}{}$ for $\Delta=\mathbb{R}^{n}_{k}$, where $D_{k}$
is the diagonal $k$ by $k$ matrix with $i^{\textrm{th}}$ diagonal entry
$\delta^{ii}(x)$, $D_{n-k}$ is the $n-k$ by $n-k$ matrix with $(i,j)$-entry
$\delta^{k+i,k+j}(x)$, and similarly for $D_{n-k,k}$. Since

$\partial_{\phi_{1}},\dots,\partial_{\phi_{k}},\partial_{\theta_{1}},\dots,\partial_{\theta_{n-k}},$

are the vector fields associated to standard coframe on $\mathbb{R}^{n}$ by
the infinitesimal action $\mathbscr{a}$, it follows from part (i) of Lemma
7.10 and the left diagram in (7.10) that

$\delta^{ij}=\begin{cases}g_{\phi_{i},\phi_{j}}\quad&\text{ if }i,j\leq k,\\\
g_{\phi_{i},\theta_{j-k}}\quad&\text{ if }i\leq k,\text{ }j\geq k+1,\\\
g_{\phi_{j},\theta_{i-k}}\quad&\text{ if }i\geq k+1,\text{ }j\leq k,\\\
g_{\theta_{i-k},\theta_{j-k}}\quad&\text{ if }i\geq k+1,\text{ }j\geq
k+1.\end{cases}$ (7.15)

From this and the fact that $G$ is positive definite, it follows that the
functions $\delta^{ii}$ are strictly positive on $\Delta$ and the matrix
(7.14) is invertible. This proves item (i).

For items (ii) and (iii) we can further assume that the depth of $x$ is $1$,
i.e., $k=1$. We are left to show that $\delta^{11}(x)=4\pi$. Since

$2r\mathrm{d}r,\mathrm{d}x^{1},\dots,\mathrm{d}x^{n-1},$

are the $1$-forms obtained by pulling back the standard coframe on
$\mathbb{R}^{n}$ along $\mu$, it follows from part (ii) of Lemma 7.10 and the
right diagram in (7.10) that

$\delta^{ij}=\begin{cases}4\,g^{*}_{r,r},\\\ 2\,g^{*}_{r,x^{j-1}}\quad&\text{
if }\text{ }j\geq 2,\\\ 2\,g^{*}_{r,x^{i-1}}\quad&\text{ if }i\geq 2,\\\
g^{*}_{x^{i-1},x^{j-1}}\quad&\text{ if }i\geq 2,\text{ }j\geq 2.\end{cases}$
(7.16)

By considering the coefficient functions of $g$ with respect to the frame

$\frac{1}{r}\partial_{\phi},\partial_{r},\partial_{\theta_{1}},\dots,\partial_{\theta_{n-1}},\partial_{x^{1}},\dots,\partial_{x^{n-1}}$

over $\mathring{S}$ and those of $g^{*}$ with respect to the dual coframe

$r\mathrm{d}\phi,\mathrm{d}r,\mathrm{d}\theta_{1},\dots,\mathrm{d}\theta_{n-1},\mathrm{d}x^{1},\dots,\mathrm{d}x^{n-1}$

over $\mathring{S}$, we obtain mutually inverse matrices of functions that
converge as $z$ tends to zero. Comparing the two limits as $z$ tends to zero
leads to the conclusion that

$g^{*}_{r,r}(x)\,g_{r,r}(x)=1.$

Combined with the fact that $g_{\phi,\phi}(x)=(2\pi)^{2}\,g_{r,r}(x)$ by Lemma
7.10, and with (7.15) and (7.16), it follows that $\delta^{11}(x)=4\pi$.

To prove item (iii), we ought to show that, for any $b$-vector field $\nu$
that is Euler-like with respect to a open facet $F$, the lift of $2\,\nu$
along (7.13) is Euler-like with respect to $S_{F}$. In view of Remark 4.20, we
can assume that $\nu=x^{1}\partial_{x^{1}}$. The lift of the $b$-vector field
$2x^{1}\partial_{x^{1}}$ is of the form

$\widetilde{2x^{1}\partial_{x^{1}}}=r\partial_{r}+f^{\phi}\partial_{\phi}+\sum_{j=1}^{n-1}f^{\theta}_{j}\partial_{\theta_{j}},$

where $f^{\phi}$ and $f^{\theta}_{j}$ are $\mathbb{T}^{n}$-invariant smooth
functions on $\mu_{1,n}^{-1}(U)$. By Remark 4.24, it is enough to show that
$f^{\phi}(x)=f^{\theta}_{1}(x)=\cdots=f^{\theta}_{n-1}(x)=0$ whenever
$x^{1}=0$. Note that over $\mathring{S}$ one has

$\displaystyle 0$
$\displaystyle=g\left(2\widetilde{x^{1}\partial_{x^{1}}},\partial_{\theta_{l}}\right)=r^{2}\,g_{r,\theta_{l}}+g_{\phi,\theta_{l}}\,f^{\phi}+\sum_{j=1}^{n-1}g_{\theta_{j},\theta_{l}}\,f^{\theta}_{j},$
$\displaystyle 0$
$\displaystyle=\frac{1}{r^{2}}\,g\left(2\widetilde{x^{1}\partial_{x^{1}}},\partial_{\phi}\right)=r^{2}\,g_{\phi,r}+g_{\phi,\phi}\,f^{\phi}+\sum_{j=1}^{n-1}g_{\phi,\theta_{j}}\,f^{\theta}_{j},$

where we first we used that $\ker(\mathrm{d}\mu)$ and $\mathcal{L}$ are
orthogonal over $\mathring{S}$, and then we applied Lemma 7.10. In view of
(7.15), letting $z$ tend to zero it follows that
$(f^{\phi}(x),f_{1}^{\theta}(x),\dots,f_{n-1}^{\theta}(x))$ belongs to the
kernel of the invertible matrix (7.14) whenever $x^{1}=0$. So,
$f^{\phi}(x)=f^{\theta}_{1}(x)=\dots=f^{\theta}_{n-1}(x)=0$ if $x^{1}=0$.

This concludes the proof of one direction of Theorem 7.1.

####  7.2.2. From hybrid $b$-metrics to invariant (Kähler) metrics

Let $\mu:(S,\omega)\to M$ be a toric
$(\mathcal{T}_{\Lambda},\Omega_{\Lambda})$-space with Delzant subspace
$\Delta:=\mu(S)$. Further, let $\theta$ be an elliptic connection and let $g$
be a hybrid $b$-metric on $\Delta$ as in the statement of the theorem. We use
$\delta$ to denote the corresponding hybrid $b^{*}$-metric and we let $G$ be
the unique invariant $\omega$-compatible Kähler metric on $\mathring{S}$ such
that $\mu:(\mathring{S},G)\to(\mathring{\Delta},g)$ is a Riemannian submersion
and $\ker\theta|_{\mathring{S}}=(\ker\mathrm{d}\mu)^{\perp}|_{\mathring{S}}$.
By the proof of Theorem 6.2, this data satisfies (7.10) on $\mathring{S}$. We
ought to show that $G$ extends to a Riemannian metric on all of $S$. This
extension is then automatically invariant and compatible with $\omega$, due to
density of $\mathring{S}$.

First, we will show that $G$ extends smoothly to $S$ as a tensor. Since this
is a local property, we can verify this in standard toric coordinates
$(z,\theta,x)$ by showing that the coefficients of $G$ are of the form in part
(i) of Lemma 7.10. Because the left square in (7.10) commutes, the
coefficients of $G$ involving only

$\partial_{\phi_{1}},\dots,\partial_{\phi_{k}},\partial_{\theta_{1}},\dots,\partial_{\theta_{n-k}}$

are of this form, with the required $\mathbb{T}^{n}$-invariant smooth
functions on $\mu_{k,n}^{-1}(U)$ given by the pull-backs of the corresponding
coefficients $\delta^{ij}$ of $\delta$, as in (7.9). For the coefficients of
$G$ involving $r_{i}\partial_{r_{i}}$, notice that the horizontal lift of
$2x^{i}{\partial}_{x^{i}}$ must be of the form

$\widetilde{2x^{i}{\partial}_{x^{i}}}=r_{i}\partial_{r_{i}}+r_{i}^{2}\,\Big{(}\sum_{j=1}^{k}f^{\phi}_{i,j}\partial_{\phi_{j}}+\sum_{j=1}^{n-k}f_{i,j}^{\theta}\partial_{\theta_{j}}\Big{)},$
(7.17)

where $f^{\phi}_{j}$ and $f^{\theta}_{j}$ are $\mathbb{T}^{n}$-invariant
smooth functions on $\mu_{k,n}^{-1}(U)$. Indeed, this follows from the
condition in (ii) in Theorem 7.1, Remarks 4.20 and 4.24, and Proposition 4.6.
Since the horizontal lift is orthogonal to $\ker\mathrm{d}\mu$, it follows
that the coefficients
$G\left(r_{i}\partial_{r_{i}},\partial_{\phi_{l}}\right)$ and
$G\left(r_{i}\partial_{r_{i}},\partial_{\theta_{l}}\right)$ are of the
required form, and that

$G\left(\widetilde{2x^{i}{\partial}_{x^{i}}},\widetilde{2x^{l}\partial_{x^{l}}}\right)=G\left(\widetilde{2x^{i}{\partial}_{x^{i}}},r_{l}\partial_{r_{l}}\right).$
(7.18)

for $i,l\leq k$. Since $\mu$ is a Riemannian submersion, it holds that:

$G\left(\widetilde{2x^{i}{\partial}_{x^{i}}},\widetilde{2x^{l}\partial_{x^{l}}}\right)=\begin{cases}r_{i}^{2}r_{l}^{2}\mu^{*}(g_{il})&\text{
if }i\neq l,\\\ r_{i}^{2}\mu^{*}(g_{ii})&\text{ if }i=l,\end{cases}$

with $g_{il}\in\mathcal{C}^{\infty}(U)$ as in (7.2). So, it follows from
(7.18) that the coefficients
$G\left(r_{i}\partial_{r_{i}},r_{l}\partial_{r_{l}}\right)$ are also of the
required form, and that

$g_{\phi_{i},\phi_{i}}=\mu_{k,n}^{*}(\delta^{ii})\quad\text{and}\quad
g_{r_{i},r_{i}}=\mu^{*}_{k,n}(g_{ii})\quad\text{over}\quad x^{i}=0.$

Since $g^{\flat}$ and $\delta^{\sharp}$ are inverse to each other, it holds
that $g_{ii}(x)\delta^{ii}(x)=1$ when $x^{i}=0$. By the assumption on the
residue of $g$, one has that $g_{ii}(x)=\frac{1}{4\pi}$ when $x^{i}=0$. Hence,
it follows that

$g_{\phi_{i},\phi_{i}}=(2\pi)^{2}g_{r_{i},r_{i}}\quad\text{over}\quad
x^{i}=0.$

By similar arguments, the other coefficients are of the required form as well.
So, the tensor $G$ indeed extends smoothly to all of $S$. This extension is
symmetric and positive semi-definite, since it is so on the dense subset
$\mathring{S}$. So, to conclude that it is a Riemannian metric on $S$, it
suffices to show that the dual metric $G^{*}$ on $\mathring{S}$ extends
smoothly, as a tensor, to all of $S$. This follows from an argument along the
same lines as that for $G$, by instead using part (ii) of Lemma 7.10 and the
coframe over $\mathring{S}\cap\mu^{-1}(U)$ dual to the frame:

$\widetilde{2x^{1}\partial_{x^{1}}},\dots,\widetilde{2x^{k}\partial_{x^{k}}},\widetilde{\partial_{x^{k+1}}},\dots,\widetilde{\partial_{x^{n}}},\partial_{\phi_{1}},\dots,\partial_{\phi_{k}},\partial_{\theta_{1}},\dots,\partial_{\theta_{n-k}}.$

This concludes the proof of Theorem 7.1.

###  7.3. Examples

######  Example 7.11.

Consider the integral affine cylinder
$(M,\Lambda)=(\mathbb{S}^{1}\times\mathbb{R},\mathbb{Z}\mathrm{d}x\oplus\mathbb{Z}\mathrm{d}h)$
and the Delzant subspace $\Delta=\mathbb{S}^{1}\times[-1,1]$ as in Example
4.35. Further, consider the corresponding canonical toric
$\mathcal{T}_{\Lambda}$-space
$\mu:(\mathbb{T}^{2}\times\mathbb{S}^{2},\omega)\to\mathbb{S}^{1}\times\mathbb{R}$
described in Example 2.20. As in Example 6.10, there are Hessian hybrid
$b$-metrics of the form

$g=\left(\frac{1}{4\pi(1-h^{2})}+f(h)\right)\mathrm{d}h^{2}+c(x)\,\mathrm{d}x^{2},$

where now the singular term is so that the residue at both facets is
$\tfrac{1}{4\pi}$. By a computation as in Example 6.10, we find that the
extremal condition holds if and only if $f=0$ and $c(x)$ is constant. Using
the flat Lagrangian elliptic connection $D_{a,b}$ of Example 4.35, Theorem 7.1
gives the family of $\omega$-compatible extremal Kähler metrics

$G=\left(\frac{1}{\tau(h)}+b^{2}\tau(h)\right)\mathrm{d}h^{2}+\tau(h)\mathrm{d}\phi(\mathrm{d}\phi-2b\mathrm{d}h)+g_{\mathbb{T}^{2}},$
(7.19)

where $\tau(h)=4\pi(1-h^{2})$ and $g_{\mathbb{T}^{2}}$ is the flat metric on
the torus given by

$g_{\mathbb{T}^{2}}=\frac{1}{c}\big{(}(a^{2}+c^{2})\mathrm{d}x^{2}-2a\mathrm{d}x\mathrm{d}y+\mathrm{d}y^{2}\big{)}.$

Note that when $a=b=0$ and $c=1$ we recover the product of the round metric on
$\mathbb{S}^{2}$ with the standard flat metric on $\mathbb{T}^{2}$.

######  Example 7.12.

Let
$(M,\Lambda)=(\mathbb{S}^{1}\times]-2,\infty[,\mathbb{Z}((h+2)\mathrm{d}x+x\mathrm{d}h)\oplus\mathbb{Z}\mathrm{d}h)$
be the non-standard integral affine cylinder and consider the Delzant subspace
$\Delta=\mathbb{S}^{1}\times[-1,1]$ (see Example 4.36). Further, we consider
the corresponding canonical toric $\mathcal{T}_{\Lambda}$-space
$\mu:(\mathbb{T}^{2}\mathbin{\ooalign{\raise
4.95134pt\hbox{$\scriptstyle\sim$}\cr$\times$\cr}}\mathbb{S}^{2},\omega)\to\mathbb{S}^{1}\times]-2,\infty[$,
which is described in Example 2.21. Similar to Example 6.11, there are Hessian
hybrid $b$-metrics of the form

$g=\left(\frac{1}{4\pi(1-h^{2})}+f(h)\right)\mathrm{d}h^{2}+(h+2)c\,\mathrm{d}x^{2},$

with $c>0$ a constant and $f\in C^{\infty}(]-2,\infty[)$. Let us define a
function $\tau(h)$ by

$\frac{1}{\tau(h)}:=\frac{1}{4\pi(1-h^{2})}+f(h).$

One finds the boundary conditions

$\tau(\pm 1)=0,\quad\tau^{\prime}(\pm 1)=\mp 8\pi.$

Applying formula (6.8), one finds that

$S_{G}=-\frac{1}{2}\tau^{\prime\prime}(h)-\frac{\tau^{\prime}(h)}{h+2}.$

Proceeding as in Example 6.11, taking into account the boundary conditions,
one finds that $S_{G}$ is an affine function if and only if

$\tau(h)=-\frac{4\pi}{11}\left(2(h+2)^{3}-5(h+2)^{2}-15+\frac{18}{h+2}\right).$

This gives the affine function

$S_{G}=\frac{4\pi}{11}(12h+9).$

The corresponding hybrid Hessian $b$-metric takes the form

$g=\left(\frac{1}{4\pi(1-h^{2})}+\frac{1}{2\pi(2h^{2}+11h+20)}\right)\mathrm{d}h^{2}+(h+2)c\,\mathrm{d}x^{2}.$

We can now use the family of elliptic Lagrangian connections $D_{a,b}$ on the
$\mathcal{T}_{\Lambda}$-space $\mu:(\mathbb{T}^{2}\mathbin{\ooalign{\raise
4.95134pt\hbox{$\scriptstyle\sim$}\cr$\times$\cr}}\mathbb{S}^{2},\omega)\to\mathbb{S}^{1}\times]-2,\infty[$
discussed in Example 4.36 to lift $g$ to the family of $\omega$-compatible
extremal Kähler metrics

$G=\left(\frac{1}{\tau(h)}+b^{2}\tau(h)\right)\mathrm{d}h^{2}+\tau(h)\theta(\theta-2b\mathrm{d}h)+(h+2)g_{\mathbb{T}^{2}},$
(7.20)

where $\theta:=(x\mathrm{d}y+\mathrm{d}\phi)$ and $g_{\mathbb{T}^{2}}$ is the
flat metric on the torus given by

$g_{\mathbb{T}^{2}}=\frac{1}{c}\big{(}(a^{2}+c^{2})\mathrm{d}x^{2}-2a\mathrm{d}x\mathrm{d}y+\mathrm{d}y^{2}\big{)}.$

Notice the similarity between formulas (7.19) and (7.20). When $b=0$, the
metric (7.20) appears as a special case of the metrics considered by Apostolov
et al. [2] (see the formula in Theorem 3, loc. cit.). The precise relation
between the approach followed here and the results in [2] will be discussed in
a sequel to this paper.

###  7.4. Invariant Kähler metrics via Kähler reduction

The Delzant-type construction from Section 5 (see Theorem 5.11) can be used to
construct invariant Kähler metrics via Kähler reduction, much like in [24].

Let $(M,\Lambda)$ be a connected integral affine manifold and $\Delta\subset
M$ a Delzant subspace of finite type. Assume that $M$ admits a Hessian metric
$g$ and let $\widetilde{g}$ be its lift to the universal covering space
$(\widetilde{M},\widetilde{\Lambda})$. Let $G_{\widetilde{\Lambda}}$ be the
$\Omega_{\widetilde{\Lambda}}$-compatible invariant Kähler metric on
$\mathcal{T}_{\widetilde{\Lambda}}$ obtained by lifting $\widetilde{g}$ via
the canonical Lagrangian connection.

######  Proposition 7.13.

Let $(M,\Lambda,g)$ be a connected integral affine Hessian manifold and let
$\Delta$ be a Delzant subspace of finite type. Consider the product metric
$G_{\widetilde{\Lambda}}\times g_{\mathrm{st}}$ on
$\mathcal{T}_{\widetilde{\Lambda}}\times\mathbb{C}^{d}$. The Hessian hybrid
$b$-metric $g_{\Delta}$ on $\Delta$ induced by the reduced Kähler metric on
the symplectic quotient

$((\mathcal{T}_{\widetilde{\Lambda}}\times\mathbb{C}^{d})\mathbin{/\mkern-6.0mu/}(\Gamma\ltimes\mathbb{T}^{d}),\,\omega_{\mathrm{red}})$

takes the form

$g_{\Delta}=g+\mathrm{Hess}_{\Lambda}(\phi),$

where $\phi$ is the smooth function on $\mathring{\Delta}$ corresponding to
the function on $\widetilde{M}$ given by

$\widetilde{\phi}=-\frac{1}{4\pi}\sum_{i=1}^{d}\ell_{i}\log|\ell_{i}|.$ (7.21)

The proof follows the usual Kähler reduction recipe: one restricts the metric
to the zero level, obtaining a $(\Gamma\ltimes\mathbb{T}^{d})$-invariant
metric, that then descends to the symplectic quotient, yielding the expression
in the theorem. Notice that, although each term in (7.21) is a function on
$\widetilde{M}$ which may fail to descend to $\mathring{\Delta}$, the sum does
descend to a smooth function, since the $\Gamma$-action permutes the primitive
boundary defining functions.

######  Remark 7.14.

For a Delzant polytope, Guillemin showed in [24] that, if one applies Kähler
reduction to the standard Delzant construction, one obtains an invariant
Kähler metric on the symplectic toric manifold which induces a Hessian metric
on $\mathring{\Delta}$ with potential given by (7.21). Our Delzant type
construction is different from the standard one (see Remark 5.10) and results
in the presence of the original Hessian metric on $M$ as an additional term.
This is unavoidable since for a general Delzant subspace of finite type there
may not exist a global potential. Moreover, the Hessian of (7.21) may not even
define a metric in $\mathring{\Delta}$. This is the case, for instance, in
Examples 7.11 and 7.12. Also, the Hessian hybrid $b$-metrics constructed in
those examples show that $\Delta$ may carry Hessian hybrid $b$-metrics with
residues as in Theorem 7.1 which are not obtained by Kähler reduction as in
Proposition 7.13.

######  Corollary 7.15.

Any finite-type Delzant subspace of a Hessian integral affine manifold admits
a Hessian hybrid $b$-metric with residues as in Theorem 7.1.

## References

  * [1] M. Abreu, Kähler geometry of toric varieties and extremal metrics, _Internat. J. Math._ 9 (1998), 641–651.
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