# The momentum polytope of actions on contact manifolds via toric varieties

Amna Shaddad  Instituto de Ciencias Matemáticas (ICMAT)
CSIC (Madrid<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Due to a previous result which states that contact varieties are isomorphic to
certain varieties, the momentum polytopes of contact manifolds are convex.

###### Key words and phrases:

Momentum polytope, Contact manifold, Toric Varieties.

###### 2010 Mathematics Subject Classification:

53D20; 14L24.

## Institution and mail address

Instituto de Ciencias Matemáticas (ICMAT)
Office 417, Calle Nicolás Cabrera 13-15, 28049, Cantoblanco, Madrid, Spain
<EMAIL_ADDRESS><EMAIL_ADDRESS>
##  1\. Introduction

The introduction of symplectic geometry comes from the investigation of light
rays through a medium, this was extended to every mechanical system. One thing
remained constant - each ray would be described according to two pieces of
information - the position the ray is incident with the medium and the angle
at which the ray hits the medium. According to Hamilton’s paper ‘Caustics’,
the optical system used to study the geometry of rays of light, utilises four
variables to locally specify the rays that enter a system from the left and
exit from the right (the light rays to the left are straight line segments);
two of the variables specify the point of intersection of the line with the
plane perpendicular to the optical axis and two additional angular variables
giving the angle of the line to this plane. Relating the incoming line
segments on the left to the outgoing line segments on the right, is a
transformation from the incoming coordinates to the outgoing ones, called
symplectic diffeomorphisms. Later Hamilton discovered that this same method
applies without modification to mechanics. Replacing the optical axis by the
time axis, light rays for trajectories of the system and the four incoming and
four outgoing variables by the $2n$ incoming and outgoing variables of the
phase space of the mechanical system.

And therefore symplectic geometry was birthed and the associated momentum map
and its image accordingly.

Contact systems exist on $2n-1$ or odd dimensional manifolds. Contact geometry
has been used to describe to many physical phenomena and is related to a lot
of other mathematical structures. Sophus Lie was the first to introduce them
through his work on partial differential equations. Then Gibbs reintroduced
them through thermodynamics, Huygen’s later through geometric optics and
Hamiltonian dynamics. More recently contact structures have been found to
describe heat transport through a medium. But this was only after previous
work relating contact structures to Riemannian geometry, low dimensional
topology and subelleptic operators.

Focussing on cooriented contact structures i.e. hyperplane fields $\xi$ on
oriented smooth $2n-1$-dimensional manifolds which are given by the kernels of
$\alpha\in\Omega^{1}(M)$ so that $\alpha\wedge d\alpha^{n-1}$ is a positive
volume form on said manifold. We establish the momentum polytope for such
structures.

We first introduce some notation concerning toric varieties in particular. We
also talk about remedying singular points of toric varieties. Later we
introduce how these apply to orbits, line bundles and topology. What is
especially significant here is the introduction of $\chi(u)$, the element of
the group algebra of semigroup $S$ defined by the cone in the lattice $N$,
$\mathbb{C}[S]$ corresponding to $u$ in $S$. For the momentum map is defined
according to $\chi(u)$ as a retraction. This is shown to establish a momentum
polytope for actions on contact manifolds by way of example in the final
section. Such a momentum map works independently of dimensional and some
structural restrictions on the manifold acted on.

###  1.1. Main Result

The result of this paper is

######  Theorem 1.1.

For $X$ a smooth projective toric variety of dimension $2n+1$ $(n\geq 1)$,
defined over the field $\mathbb{C}$ of complex numbers and endowed with a
contact structure. Then either $X$ is isomorphic to the complex projective
space $\mathbb{P}^{2n+1}$ or $X$ is isomorphic to the variety
$\mathbb{P}_{\mathbb{P}^{1}\times\cdots\times\mathbb{P}^{1}}(\mathcal{T}_{\mathbb{P}^{1}\times\cdots\times\mathbb{P}^{1}})$.
The momentum map is defined

$\mu(x)=\frac{1}{\sum|\chi^{u}(x)|}\sum\limits_{u\in P\cap M}|\chi^{u}(x)|u$

where $\chi^{u}$ for $u\in P\cap M$ are the sections of
$\varphi:X\rightarrow\mathbb{P}^{2n}$. This map is convex.

###  1.2. Literature Review

A hyperplane field $\xi$ on a manifold $M$ is a codimension one sub-bundle of
the tangent bundle $TM$. Hyperplane fields can always locally be described as
the kernel of a 1-form. This means that for every point in $M$, there exists a
neighbourhood $U$ and a 1-form $\alpha$ defined on $U$ whose kernel of the
linear map $\alpha_{x}:T_{x}M\rightarrow\mathbb{R}$ is $\xi_{x}$ for all $x\in
U$. $\alpha$ is the local defining form for $\xi$ and a contact structure on a
$(2n+1)$-dimensional manifold $M$ is a ’maximally non-integrable hyperplane
field’ $\xi$. If for any locally defining 1-form $\alpha$, $\alpha\wedge
d\alpha\neq 0$ then the hyperplane field $\xi$ is said to be maximally non-
integrable. Essentially what this says is that the form is pointwise never
equal to zero. This non-integrability of $\xi$ geometrically means that no
hypersurface in $M$ can be tangent to $\xi$ along an open subset of the
hypersurface. In other words, the hyperplane ’twists too much’ to be tangent
to the hypersurface. The pair $(M,\xi)$ is named a contact manifold and
$\alpha$, the defining form for $\xi$ is the contact form for $\xi$.

Let’s look at the relations that exist between contact geometry and symplectic
geometry. For symplectic manifold $(S,\omega)$, $L_{v}\omega=\omega$ for
vector field $v$, where $L_{v}\omega$ is the Lie derivative of $\omega$ in the
direction of $v$, and this is called a symplectic dilation. In $(S,\omega)$,
compact hypersurface $M$ is said to have contact type if there exists a
symplectic dilation $v$ in a neighbourhood of $M$ that is transverse to $M$.
The characteristic line field $LM$ in the tangent bundle of given hypersurface
$M$ in $(S,\omega)$, is the symplectic complement of $TM$ in $TS$. $M$ is
therefore coisotropic since it is codimension one, and therefore the
symplectic complement is contained in $TM$ and is of dimension one.

For $M$ a compact hypersurface in a symplectic manifold $(S,\omega)$, with
inclusion map $i:M\rightarrow M$, then $M$ has contact type if and only if a
1-form $\alpha$ exists on $M$ so that $d\alpha=i^{*}\omega$ and $\alpha$ is
never zero on the characteristic line field.

So if $M$ is a hypersurface of contact type, then $\alpha$ is found by
contracting the symplectic dilation $v$ into the symplectic form i.e.
$\alpha=\iota_{v}\omega$. It is easy to prove that the 1-form $\alpha$ is a
contact form on $M$ and therefore a hypersurface of contact type in a
symplectic manifold has a co-oriented contact structure.

So provided a co-orientable contact manifold $(M,\xi)$, a symplectisation
$\text{Symp}(M,\xi)=(S,\omega)$ can be formulated. Meaning that for manifold
$S=M\times(0,\infty)$, given $\alpha$ a global contact form for $\xi$, then
$\omega=d(t\alpha)$ where $t$ is the coordinate on $\mathbb{R}$. one may also
symplectise as $(M\times\mathbb{R},d(e^{t}\alpha))$. The symplectisation is
independent of the choice of $\alpha$.

So for a co-oriented contact manifold $(M\xi)$, a symplectic manifold
$\text{Symp}(M,\xi)$ exists in which $M$ sits as a hypersurface of contact
type. and in fact, contact form $\alpha$ for $\xi$ provides an embedding of
$M$ into this symplectic manifold that recognises $M$ as a hypersurface of
contact type. Given $M$, a compact hypersurface of contact type in a
$(S,\omega)$, with symplectic dilation $v$. Then there if a neighbourhood of
$M$ in $S$ symplectomorphic to a neighbourhood of $M\times\\{1\\}$ in
$\text{Symp}(M,\xi)$ where the symplectisation is simply $M\times(0,\infty)$
using $\alpha=\iota_{v}\omega|_{M}$ and $\text{ker}\alpha=\xi$.

Let $(M,\xi)$ be a contact manifold. Associated to $\alpha$ to $\xi$ is the
Reeb vector field $v_{\alpha}$ which is a unique vector field that satisfies
$\iota_{v_{\alpha}}\alpha=1$ and $\iota_{v_{\alpha}}d\alpha=0$. $v_{\alpha}$
is certainly transverse to the contact hyperplanes and $\xi$ is preserved by
the flow of $v_{\alpha}$. These two conditions characterise Reeb vector
fields. Meaning that a vector field $v$ is the Reeb vector field for some
contact form for $\xi$ if and only if it is transverse to $\xi$ and its flow
preserves $\xi$.

Yi Lin and Reyer Sjamaar’s result in [1] applies, to Elisa Prato’s quasifolds
and to Hamiltonians on contact manifolds and cosymplectic manifolds. In their
paper they define a presymplectic manifold $(\mathcal{X},\omega)$ as a
paracompact $C^{\infty}$-manifold that has a closed 2-form of constant rank,
and a Hamiltonian action on a presymplectic manifold consists of a smooth
action of a Lie group $G$ on $\mathcal{X}$ and a smooth moment map
$\Phi:\mathcal{X}\rightarrow\mathfrak{g}^{*}$. Theorem 3.4.6 in their paper
says that by assuming that the $G$ action on $\mathcal{X}$ is clean. By clean
they mean of Reeb type. According to them cleanness means that there should
exist an ideal of the Lie algebra of the groups, called the ’null ideal’,
which at every point of the manifold spans the tangent space of the
intersection of the group orbit with the leaf of $\mathcal{F}$. S, if the
action of clean then for every $\epsilon\in\mathfrak{g}$, $\Phi^{\epsilon}$ is
the corresponding Morse-Bott function. And the positive and negative bundles
of the critical set $\mathcal{X}^{[\epsilon]}$ with respect to the
$G$-invariant compatible Riemannian metric on $\mathcal{X}$ \- the symplectic
subbundles of $T\mathcal{X}$ are orthogonal to the subbundle. Section 4.5 of
that paper then introduces contact manifolds as one of the presymplectic
manifolds to which their results apply on which the action of $G$ leaves
$\alpha$ invariant. The Hamiltonian action is defined as
$\Phi^{\epsilon}=\iota(\epsilon_{\mathcal{X}})\alpha$. The action is of Reeb
type if there exists $\epsilon\in\mathfrak{g}$ with
$\epsilon_{\mathcal{X}}=v_{\alpha}$, where $v_{\alpha}$ is the Reeb vector
field introduced before. The Reeb type actions are leafwise transitive. The
Reeb vector field spans the null foliation $\mathcal{F}$ of $\omega$.

Lin and Sjamaar’s presymplectic convexity theorem applied to contact
structures, aptly named the ’Contact convexity theorem’ part I - theorem 4.5.1
is that if the action of $G$ on $\mathcal{X}$ is clean, i.e. of Reeb type,
then $\Phi(\mathcal{X})\cap C$ is a convex polytope denoted
$\triangle(\mathcal{X})$. There exist many compact contact Hamiltonian
$G$-manifolds $X$ where $\triangle(\mathcal{X})$ is not convex which implies
that the action is not clean.

They propose several methods for examples, for a manifold
$(\mathcal{X},\alpha)$, where $\triangle(\mathcal{X})$ is convex, if $\alpha$
is replaced with a conformally equivalent contact form $e^{f}\alpha$ where $f$
is a $G$-invariant smooth function i.e. $\Phi\times e^{f}$ then this destroys
the convexity of $\triangle(\mathcal{X})$. And another a method for a failure
of convexity: Zhenqi He gave the first example of a presymplectic Hamiltonian
torus action with nonconvex momentum map in his PhD thesis titled ’Odd
dimensional symplectic manifolds’ completed at MIT. Such actions are not
typically clean. Introducing a hypersurface $Y\subset\mathbb{R}^{d}$ that is
transverse to the orthant faces of $\mathbb{R}^{d}$ and the radial vector
field on $\mathbb{R}^{d}$. For a quadratic momentum map $\varphi$ this implies
that radial vector fields on $\mathbb{C}^{d}$ map to two times the radial
vector fields on $\mathbb{R}^{d}$. For $\varphi^{-1}Y$, it is transverse to
the Liouville radial vector field on $\mathbb{C}^{d}$ which implies that
$\varphi^{-1}Y$ is of contact type. Only if $Y$ is an affine hyperplane does
this imply that $\varphi^{-1}Y$ is a contact ellipsoid and
$\Phi(\varphi^{-1}Y)$ is a simplex. Therefore $Y\cap\mathbb{R}^{d}_{\geq 0}$
is not convex. Now for Lin and Sjamaar their results regard the symplectic
leaf space $\mathcal{X}/\mathcal{F}$ of $\mathcal{X}$. For Chiang and
Karshon’s improvement [2] of Lerman’s theorem in [3], their results regard the
symplectisation as described before but where $\omega_{S}=-d(t\alpha)=t\omega-
dt\wedge\alpha$ for $t\in(0,\infty)$. For trivial action of $G$, the
Hamiltonian $G$-action on $S$ with $\Phi_{S}(x,t)=t\Phi(x)$ and the
$\text{Im}(\Phi_{S})$ is the conical set $\Phi_{S}(S)=\bigcup_{t>0}t\Phi(M)$.
The subset is defined as $\text{Im}(\Phi_{S})\cap C$ plus the origin:
$\triangle(S)=\\{0\\}\cup(\Phi_{S}(S)\cap C)$ of $C$ meaning that
$\triangle(S)=\bigcup_{t\geq 0}t\triangle(M)$ i.e. it is the union of all
dilations of $\triangle(S)$. _Then_ if $G$ is a torus of dimension greater
than or equal to 2, this implies that $\triangle(S)$ is convex but
$\triangle(M)$ is not. Another difference between [1] and [2] is that
$\text{Im}(\Phi(M))$ is highly dependent on the choice of contact form
$\alpha$. But the symplectic cone $(S,\omega_{S})$ is intrinsic and invariant
of the contact hyperplane bundle $\text{ker}(\alpha)$ which implies that
$\triangle(S)$ depends on the conformal class of $\alpha$.

However both these papers are not wholly disconnected from each other. The
connection between theorem 4.5.1 of [1] and [2] is that the Reeb vector field
$v_{\alpha}$ satisfies
$\iota_{v_{\alpha}}\omega_{S}=-t\iota_{v_{\alpha}}d\alpha-\iota_{v_{\alpha}}(dt\wedge\alpha)=0+dt\wedge\iota_{v_{\alpha}}\alpha=dt$
i.e. it’s the Hamiltonian vector field of $t$ on cone $S$. As $v_{\alpha}$ is
$G$-invariant this implies that $S$ is a Hamiltonian $\hat{G}$-manifold for
$\hat{G}=G\times\mathbb{R}$ and
$\hat{\Phi}:S\rightarrow\mathfrak{g}^{*}\times\mathbb{R}$ where
$\hat{\Phi}(x,t)=(t\Phi(x),t)$. If we have
$\hat{\triangle}(S)=\\{0\\}\cup(\hat{\Phi}(S)\cap(C\times\mathbb{R}))\subset
C\times\mathbb{R}$ and $\hat{\triangle}(S)=\bigcup_{t\geq
0}(t\triangle(M)\times\\{t\\})$ and
$\triangle(M)=\hat{\triangle}(S)\cap\\{t=1\\}$ as it should be! As the
symplectic quotient of $S$ at the level 1 with respect to the
$\mathbb{R}$-action is leaf space $X/\mathcal{F}$. Therefore $\triangle(S)$ is
the restriction of $\hat{G}$-action to $G$ i.e. projecting
$\hat{\triangle}(S)$ along the $\mathbb{R}$-axis. Therefore leading to a
nonabelian extension of the Lerman-Chiang-Karshon theorem (theorem 4.5.3 of
[1]) which states that $\hat{\triangle}(S)$ and $\triangle(S)$ are convex
polyhedral cones for a clean $G$-action on $M$. This proof does not work for
cone $\triangle(S)$ rational nor does it prove convexity if the action is not
clean.

Looking more closely at [3] where the symplectic cone is the symplectisation
of the contact manifold,

$\langle\Phi(q,p),\text{A}\rangle=\langle p,\text{A${}_{m}$}(q)\rangle$

where Am is the vector field induced on $S$ by $\text{A}\in\mathfrak{g}$. The
momentum cone $C(\Upsilon)=\Upsilon(\xi^{\circ}_{+})\cup\\{0\\}$ with contact
distribution $\xi$ and $\xi^{\circ}\subset T^{*}S$. The $G$-action preserves
$\xi$ coorientation which implies that the two components of $\xi-0$ is
preserved. $\xi^{\circ}_{+}$ is one component.
$\Upsilon=\Phi|_{\xi^{\circ}_{+}}$ where
$\Upsilon:\xi^{\circ}_{+}\rightarrow\mathfrak{g}^{*}$. $\Upsilon_{\alpha}$ is
the momentum map defined by the contact form $\alpha$ and $\Upsilon$ is the
momentum map defined by the contact distribution $\xi$. The proof of lemma 1.7
in [3] uses the fact that $\text{dim}S+1=2\text{dim}G$ which implies that
$\Upsilon_{\alpha}(x)\neq 0$ for all $x\in S$. If we suppose not then for some
point $x\in S$ ($\Upsilon_{\alpha}(x)=0$) the orbit $G\cdot x$ is tangent to
the contact distribution. See remark 1.9 in that paper. It is a condition on a
contact distribution and not on a particular choice of a contact form
representing the distribution. Chiang and Karshon answer the question, is the
transversality condition necessary?: “Eugene Lerman gave analogous theorem in
equivariant contact geometry when torus orbits are transverse to the contact
distribution, and asked whether transversality…in this paper we answer
Lerman’s question and give the ’convexity package’”. The solution of which is
in remark 1.2 in [2]. This paper is a step by step of several proofs for
turning one thing into another - namely providing as they said a convexity
package for contact structures.

Boyer and Galicki in [4] mainly prove Delzant’s theorem: Every compact toric
contact manifold whose Reeb vector field corresponds to an element of the Lie
algebra of the torus (Reeb) can be procured by contact reduction from an odd
dimensional sphere. For toric contact geometry, if a torus action is regular
this means that the image is a sphere. And if the action is singular then the
image is a closed convex poytope. According to Banyanga and Molino in CITE the
polytope determines a contact toric structure up to isomorphism. But not that
every such manifold can be procured from reduction (this may not be true). So
if one makes the added assumption that the Reeb vector field corresponds to
the element of the Lie algebra of the torus then it does become true. Their
method, starts with the Atiyah-Guillemin-Sternberg theorem holding true for
compact contact manifolds of Reeb type. Then if the contact 1-form is Pfaffian
this shows that the convex polytope on certain hyperplane is the
characteristic hyperplane in the dual of the torus. If we change the Pfaffian
function, this changes the polytope by a scale. It is always possible to
choose a Pfaffian so the polytope is rational. Now, according to Lerman and
Tolman’s theorem in CITE The geometry of their labelled polytopes are directly
related to the geometry of the contact manifold with fixed Pfaffian with toris
action and characteristic foliation. According to section 2 of that paper, if
we fix the contact structure by an equivalence class this provides a Pfaffian
structure. The proposition therein proposes that contact manifold $(S,\xi)$ is
orientable if and only if $C(S)=S+\mathbb{R}^{+}$ is a symplectic cone. Fixed
$\xi$ has a unique characteristic/Reeb vector field (as we know) but if
$v_{\alpha}$ is nowhere vanishing then a foliation is induced that is defined
$\mathcal{F}^{v_{\alpha}}$ of $S$. It is called the characteristic foliation
($v_{\alpha}$ and $\mathcal{F}^{v_{\alpha}}$ depend on a choice of $\xi$).

In [4], $\mathfrak{C}(S,\mathcal{D})$ is the group of contact transformations
(where $\mathcal{D}$ denotes the contact distribution in this case), a subset
of $\text{Diff}(S)$ that leave $\mathcal{D}$ invariant. By fixing
$\mathcal{D}=\text{ker}\xi$ which implies that $\mathfrak{C}$ is a subgroup
$\varphi:M\rightarrow M$ of diffeomorphisms with $\varphi^{*}\xi=f\xi$ for
$f\neq 0$. What’s wanted is a $\mathfrak{C}(M,\xi)$ for $\varphi^{*}\xi=\xi$
with $\xi$ fixed. Which is called a strict contact transformation.

For $\mathfrak{C}(S,\mathcal{D})$ and $\mathfrak{C}(S,\xi)$ the Lie algebras
are $\mathfrak{c}(S,\mathcal{D})$ and $\mathfrak{c}(S,\xi)$ respectively. And
the respective Lie algebras of the group of symplectomorphisms are
$\mathfrak{s}(\mathfrak{c}(S,\omega))$ and
$\mathfrak{s}_{0}(\mathfrak{c}(S,\omega))$. And on $C(S)$,
$\mathfrak{G}(C(S),\omega)$ denote the group of symplectomorphisms of
$(C(S),\omega)$, also $\mathfrak{G}_{0}(C(S),\omega)$ denote the subgroup of
$\mathfrak{G}(C(S),\omega)$ that commutes with the homotheties meaning that it
is the automorphism group of the symplectic Liouville structure.

They proposed that $\Phi(\mathcal{X})=\xi(\mathcal{X})$ where
$\xi(\mathcal{X})$ the contact Hamiltonian function, and $\Phi\mathcal{X}$ the
Lie algebra isomorphism between $\mathfrak{c}(S,\mathcal{D})$ of infinitesimal
contact transformations. And
$\mathcal{X}\mapsto\mathcal{X}_{S}\mapsto\xi(\mathcal{X}_{S})$ implies that
$\mathfrak{s}(C(S),\omega)\approx\mathfrak{c}(S,\xi)\approx
C^{\infty}(S)^{v_{\alpha}}$ where $v_{\alpha}$ is the center of
$\mathfrak{c}(S,\xi)$.

The Atiyah-Guillemin-Sternberg theorem holds for compact contact manifolds of
Reeb type. $\mathfrak{G}$ is the Lie group acting on symplectic cone
$(C(S),\omega)$ leaves with invariant, commutes with homothety group.
$\mathfrak{G}\subset\mathfrak{G}_{0}$ and
$\tilde{\mu}:C(S)\rightarrow\mathfrak{g}^{*}$ where
$\tilde{\mu}^{\tau}=\tilde{\xi}(\mathcal{X}_{\tau})$. The homotheties are such
that $r\mapsto e^{t}r$, $\omega\mapsto e^{t}\omega$, $\tilde{\xi}\mapsto
e^{t}\tilde{\xi}$ and $\tilde{\mu}\mapsto e^{t}\tilde{\mu}$. Consider the
$n+1$-dimensional torus, $\mathbb{T}^{n+1}$, $\\{e_{i}\\}^{n}_{i=0}$ is the
standard basis for $\mathfrak{t}_{n+1}\simeq\mathbb{R}^{n+1}$ for $e_{i}$ have
$\mathcal{X}^{e_{i}}$ denote $H_{i}$. $v_{\alpha}$ corresponds to $\zeta$, the
characteristic vector, in $\mathfrak{t}_{n+1}$. So, the Reeb vector is almost
periodic. So the torus action is of Reeb type. The dual basis of
$\mathfrak{t}_{n+1}^{*}$ is $\\{e^{*}_{i}\\}^{n}_{i=0}$.
$\langle\mu,\zeta\rangle=\sum^{i}_{i=0}\mu_{i}a_{i}=1$ the characteristic
hyperplane of codimension 1 ($\zeta=\sum^{n}_{i=0}a_{i}e_{i}$,
$\mu=\sum^{n}_{i=0}\mu_{i}e^{*}_{i}$).

$C(S)$S$\mathfrak{t}^{*}_{n+1}$$\tilde{\mu}$$\mu$

where the inclusion $S$ into $C(S)$ as $S\times{1}$.

They defined the facet as a codimension one face. A dimension $n$ simple
convex polytope - $n$ facets meeting at each vertex. And a rational convex
polytope as
$\Delta=\cap_{i=1}^{N}\\{\alpha\in\mathfrak{t}^{*}|\langle\alpha,y_{i}\rangle\leq\lambda_{i}\\}$
where $l\subset\mathfrak{t}$ is the lattice of the circle subgroups of
$\mathbb{T}$, where $N$ is the number of faces.

By the above, polytope $\mu(S)\subset\mathfrak{t}^{*}$ is simple, of dim
$\mathfrak{t}-1$, and rational if and only if characteristic vector $\zeta$
lies in lattice $l\subset\mathfrak{t}$ of circle subgroups of $\mathbb{T}$.

Theorem 4.5 of [4] states for $x\in S$:
(I) the naming and lettering of facets and vectors. And the polytope is the
convex hull of vertices.
(II) $\mathfrak{l}_{x}\in\mathfrak{h}_{x}$ (Lie algebra of isotropy subgroup
$\mathbb{H}_{x}$ of $\mathbb{T}$ at $x$, is the linear span of vectors
$p_{i}\forall i$ such that $i^{\text{th}}$ open facet $f_{i}$ lies in $F(x)$
(set of open faces)). $\mathfrak{l}_{x}$ denotes lattice of circle subgroups
of $\mathbb{H}_{x}$ and $\hat{\mathfrak{l}}_{x}$ denotes sublattice of $l$
generated by $\\{m_{i}p_{i}\\}_{f_{i}\in F(x)}$ generates $\mathcal{l}_{x}$
$\forall x\in M$ (here $m_{i}=1$ $\forall i$).

Then they discuss varying $\xi^{\prime}=f\xi$ with a corresponding lemma.

$(n+1)$-torus $\mathbb{T}^{n+1}$ $\zeta$ in $\mathfrak{t}_{n+1}$
$\mu_{i}=\xi(H_{i})$. The polytope for $\xi$ and the polytope for
$\xi^{\prime}$, the same number of facets and vertices but size is different
(depends on Pfaffian). But in $R\mathbb{P}^{n+1}$ lines through origin in
$\mathfrak{t}_{n+1}$ intersect lattice $\mathcal{l}$ of circle subgroups are
dense. So, perturb Reeb vector field and contact form. Which means that the
foliation is quasi-regular and that the polytope is rational. After defining
the toric contact manifold of Reeb type $(S,\mathcal{D},\mathbb{T})$ they
explained the proposition that $(S,\mathcal{D},\mathbb{T})$ implies that
$\mathcal{D}$ has quasi-regular contact form i.e. rational polytope.

So, for a $(S,\xi)$ a compact toric contact manifold of Reeb types and fixed
quasi-regular contact form $\eta$, then $(S,\xi)\simeq$ reduction by a torus
of $S^{2n+1}$ sphere with $\xi_{a}$. The proof involves looking at the space
of leaves $\mathcal{Z}$ of characteristic foliation $\mathcal{F}$. For $M$
toric this implies that $\mathcal{Z}$ is toric which implies $\mathbb{T}^{n}$
preserving $\omega$ on $\mathcal{Z}$. The integral class in
$H^{2}_{orb}(\mathcal{Z},\mathbb{Z})$ ($S^{1}$ $V$-bundle
$S\xrightarrow{\pi}\mathcal{Z})$. Chern class $d\xi=\pi^{*}\omega$ implies
$H^{2}_{orb}(\mu^{-1}_{N-n}(n)/\mathbb{T}^{N-n},\mathbb{Z})$ ($S^{1}$
$V$-bundle $\pi:p\rightarrow\mu^{-1}_{N-n}(\lambda)-\mathbb{T}^{N-n})$. With
$\tilde{\pi}:P\rightarrow\mu^{-1}_{N-n}(\lambda)-\mathbb{T}^{N-n}$ and
$\varphi:S\rightarrow P$ and
$\tilde{\varphi}\mathcal{Z}\rightarrow\mu^{-1}_{N-n}(\lambda)-\mathbb{T}^{N-n}$
so $\varphi^{*}d\hat{\xi}=d\xi$ means we can choose
$\varphi^{*}\hat{\xi}=\xi$.
$\mathbb{T}^{N-n}=S^{1}_{\zeta}\times\mathbb{T}^{N-n-1}$
$P=\mu^{-1}_{N-n}(\lambda)/\mathbb{T}^{N-n-1}$ so $(S,\xi)$ becomes
$(\mu^{-1}_{N-n}(\lambda)-\mathbb{T}^{N-n-1},\hat{\xi})$.
$\sum_{a}=\\{\sum_{i}a_{i}|z_{i}|^{2}\\}\simeq S^{2n+1}$. Defining
$p:\mu^{-1}_{N-n}(\lambda)\rightarrow S$, so $p^{*}\xi=\iota^{*}\xi_{a}$. So
$(S,\xi)$ is obtained from $(\sum_{a},\xi_{a})$ by contact reduction.

Their main theorem is that for a compact toric contact manifold of Reeb type
with a fixed quasi-regular contact form, the manifold is isomorphic to the
reduction by a torus of a $2n-1$ dimensional sphere with its standard contact
structure and with fixed 1-form (of particular Reeb type). Where every such
manifold admits a compatible Sasakian structure. As every symplectic toric
orbifold is in fact Kähler.

Now not every contact vector field is the Reeb field of some contact form.
Indeed, $v_{\alpha}$ is the Reeb field of some contact form which defines
$\xi$ if and only if $v_{\alpha}$ is transverse to $\xi$, i.e.
$\alpha(v_{\alpha})\neq 0$ for any defining form $\alpha$.

According to [2] a convex cone is defined as
$C(\Psi):=\\{0\\}\cup\Psi(S\times\mathbb{R}_{>0})$ $S$ compact connected
cooriented contact manifold equipped with an effective action of a torus of
dimension $k>2$ and let $\Psi:S\times\mathbb{R}_{>0}\rightarrow\mathbb{R}^{k}$
is the momentum map on the symplectisation. Their symplectisation of
$(S,\alpha)$ is the symplectic manifold $(S\times\mathbb{R}_{>0},d(t\alpha))$
where $t$ is the coordinate on $\mathbb{R}_{>0}$ and $\alpha$ is the contact
one form on $S$ and its pullback to $S\times\mathbb{R}_{>0}$. The
nondegeneracy of $d(t\alpha)$ follows from the property
$\alpha\wedge(d\alpha)^{n}\neq 0$. The positive connected component of the
annihilator of $\xi$ on cotangent bundle $T^{*}M$ is (similar to before)
$\xi^{0}_{+}:=\\{(x,\beta)|x\in S,\ \beta\in T^{*}_{x}M,\
|\beta(\xi|_{x})=0,\beta\ \text{induces coorientation of}\ \xi|_{x}\\}$ and
map $S\times\mathbb{R}_{>0}\rightarrow\xi^{0}_{+}$ sends $(x,t)$ to
$t\alpha_{x}$.

In general the manifold $(Q\times\mathbb{R}_{>0},d(e^{\theta}\alpha))$ is
called a symplectisation of $S$, where $\theta$ denotes the coordinate on
$\mathbb{R}$. Every (transversally orientable) contact manifold $(Q,\xi)$ can
be embedded as a hypersurface in an exact symplectic manifold. Choose any
contact form $\alpha$. Then $M=Q\times\mathbb{R}$ is a symplectic manifold
with symplectic form $\omega=e^{\theta}(d\alpha-\alpha\wedge
d\theta)=d\lambda$, where $\lambda=e^{\theta}\alpha$.

Going back to [2] they introduce momentum map
$\Psi_{\alpha}:S\rightarrow\mathfrak{t}^{*}$ whose $\mathfrak{x}$ component
$\Psi_{\alpha}^{\mathfrak{x}}:S\rightarrow\mathbb{R}$ for
$\mathfrak{x}\in\mathfrak{t}$ is:
$\Psi_{\alpha}^{\mathfrak{x}}(s)=\alpha(\mathfrak{x}_{S}(s))$ $\forall s\in S$
satisfying $d\Psi^{\mathfrak{x}}_{\alpha}=-\iota(\mathfrak{x}_{S})d\alpha$ on
$S$. Using $\alpha$ to identify $\xi^{0}_{+}$ with $S\times\mathbb{R}_{>0}$,
induced torus action on $S\times\mathbb{R}_{>0}$ is the given action on $S$
component, which is trivial on the $\mathbb{R}_{>0}$ component.
$\Psi:S\times\mathbb{R}_{>0}\rightarrow\mathfrak{t}^{*}$ with $(s,t)\mapsto
t\Psi_{\alpha}(s)$ which is contact momentum map corresponding to $\alpha$. So
the momentum cone is
$C(\Psi)=\\{0\\}\cup\Psi(S\times\mathbb{R}_{>0})=\mathbb{R}_{\geq
0}\cdot\Psi_{\alpha}(S)$.

[1] state that the cone $C(\mathcal{X})$ is obtained by restricting the
$\hat{G}$-action (defined above) to $G$, i.e. by projecting $\hat{C}(S)$ along
the $\mathbb{R}$ axis. Where for $C(S)$ to be a convex polytope, $\hat{C}(S)$
is the convex polyhedral cone and its projection $C(S)$ onto
$\mathfrak{t}^{*}$is a convex polyhedral cone.
$C(\mathcal{X})=\Phi(\mathcal{X})\cap$positive Weyl chamber.
$\hat{C}(S)=\bigcup_{t\geq 0}(tC(\mathcal{X})\times\\{t\\})$ and
$C(S)=\bigcup_{t\geq 0}tC(\mathcal{X})$ and $\Phi_{S}(x,t)=t\Phi(x)$.

Now $C(\mathcal{X})$ is rational if and only if the null subgroup
$N(\mathcal{X})$ of $G$ is closed. $\langle\xi,\cdot\rangle\geq a$ is the half
space or character lattice. According to [3] a cone or polytope is rational if
annihilators of codimension one faces are spanned by integral lattice i.e.
vectors if and only if the limit of the kernel of exp$:\mathfrak{g}\rightarrow
G$.

Section 2.7 of [1] says that irrationality ker$(\omega)=0$ foliation is
trivial but weakly rational i.e. normal vectors are contained in quasi-lattice
$\mathfrak{X}_{*}(\mathbb{T})/\mathfrak{X}_{*}(\mathbb{T}\cap\mathfrak{n}(\mathcal{X}))$
in quotient space $\mathfrak{t}/\mathfrak{n}(\mathcal{X})$ where
$\bar{x}=\mathcal{F}(x)$ and $G_{\bar{x}}$ is the stabiliser of $\bar{x}$ with
corresponding Lie algebra $\mathfrak{g}_{\bar{x}}$ and
$\mathfrak{n}(U)=\bigcap_{x\in G\cdot U}\mathfrak{g}_{\bar{x}}$ for all open
$U\subseteq\mathcal{X}$. $\mathfrak{n}(\mathcal{X})\cap\mathfrak{t}$ of
$\mathfrak{t}$ is rational so subspace
$\mathfrak{n}^{*}(\mathcal{X})^{\circ}\cap\mathfrak{t}^{*}$ is rational. By
this, $C(\mathcal{X})$ is rational and
$C(\mathcal{X})=\bigcap_{x\in\mathcal{X}}C_{x}\cap\mathfrak{n}(\mathcal{X})^{\circ}$.
Remark 1.3 of Lerman’s paper proves rationality - it is the main result of his
paper, [3]. And Banyango and Molina in [5] prove rationality in Lemma 2.2 and
Remark 2.3 of their paper, stating that for $W\subset S$ is a rational
polyhedral if there exist vectors $v_{1},\ldots,v_{k}$ in integral lattice
$\mathbb{Z}_{G}=\text{ker}\\{exp:\mathfrak{g}\rightarrow G\\}$ such that
$W=\\{f\in S|\langle f,v_{i}\rangle\geq 0,\ 1\leq i\leq k\\}$.

There are other papers that also tussle with the image of the momentum map of
an action on a contact structure. More recently T. Ratiu and N. T. Zhung write
in their preprint titled ”Presymplectic convexity and (ir)rational polytope”
that: ”Battaglia and Prato and Katzarkov-Lupercio-Meerseman-Verjovsky have
worked on irrational analogues of symplectic toric manifolds. However we
wanted to have a simpler understanding of the geometric structure and so
developed our own approach, which uses presymplectic realisations”.

##  2\. Background

###  2.1. Toric Varieties

####  2.1.1. Introduction

Toric varieties are geometric objects defined by combinatorial information.

A lattice $N\simeq\mathbb{Z}^{n}$, and real vector space
$N_{\mathbb{R}}=N\otimes_{\mathbb{Z}}\mathbb{R}$. For a strongly convex
rational polyhedral cone $\sigma$ in $N_{\mathbb{R}}$, rationality means that
it is generated by a finite number of vectors and a convexity is strong if it
doesn’t contain a line through the origin. So $\sigma$ is defined as a cone in
$N_{\mathbb{R}}$ that’s generated by vectors in the lattice with apex at the
origin. A fan $\Delta$ in $N$ is a collection of $\sigma$ where every face of
a cone in $\Delta$ is also a cone in $\Delta$ and the intersection of two
cones in $\Delta$ is a face of each.

The dual cone denoted $\sigma^{\smallsmile}$ in $M_{\mathbb{R}}$, where
$M=\text{Hom}(N,\mathbb{Z})$ is the dual lattice (with dual pairing
$\langle,\rangle$), is the set of vectors that are nonnegative in $\sigma$. A
finitely generated commutative semigroup is therefore established as
$S_{\sigma}=\sigma^{\smallsmile}\cap M=\\{u\in M:\langle u,v\rangle\geq 0\
\text{for all}\ v\in\sigma\\}$ where $\mathbb{C}[S_{\sigma}]$ is its
corresponding group algebra - a finitely generated commutative
$\mathbb{C}$-algebra. $U_{\sigma}=$Spec$(\mathbb{C}[S_{\sigma}])$ is the
affine variety.

For a face of $\sigma$, $\tau$ \- the commutative semigroup $S_{\tau}$
contains $S_{\sigma}$ and $\mathbb{C}[S_{\sigma}]$ is a subalgebra of the
associated group algebra $\mathbb{C}[S_{\tau}]$, and the affine variety
$U_{\tau}$ is a principal open subset of $U_{\sigma}$, for $u\in S_{\sigma}$\-
so $\tau=\sigma\cap u^{\perp}$ then $U_{\tau}=\\{x\in U_{\sigma}:u(x)\neq
0\\}$ giving the map $U_{\tau}\rightarrow U_{\sigma}$. These affine varieties
form the algebraic variety $X(\Delta)$. Any product of projective and affine
varieties is a toric variety. $T$ the torus of algebraic groups
$\mathbb{C}^{*}\times...\times\mathbb{C}^{*}$ \- a normal variety $X$ contains
$T$ as a dense open subset and $T\times X\rightarrow X$. The simplest compact
example is indeed the projective space
$(\mathbb{C}^{*})^{n}\subset\mathbb{C}^{n}\subset\mathbb{P}^{n}$.

Now smaller cones mean smaller open sets therefore the geometry in $N$ is
preferred to the geometry in $M$.

####  2.1.2. Convex polyhedral cones

Next we will cover how to find generators of the semigroups. If we denote
$N_{\mathbb{R}}$ as $V$ and its dual $V^{*}=M_{\mathbb{R}}$, then the convex
polyhedral cone is the set $\sigma=\\{r_{1}v_{1}+...+r_{s}v_{s}\in V:r_{i}\geq
0\\}$ generated by any finites set of vectors $v_{1},...,v_{s}$ in $V$. The
generators for the cone $\sigma$ are the vectors $v_{i}$ or positive multiples
of $v_{i}$ called rays. The dimension of the linear space
$\mathbb{R}\cdot\sigma=\sigma+(-\sigma)$ spanned by $\sigma$, gives the
dimension of $\sigma$ (dim$\sigma$).

For any set $\sigma$ there is a corresponding dual set $\sigma^{\smallsmile}$,
and it is the set of equations of supporting hyperplanes,
$\sigma^{\smallsmile}=\\{u\in V^{*}:\langle u,v\rangle\geq 0\ \text{for all}\
v\in\sigma\\}$. If $v_{0}\notin\sigma$ in the case that $\sigma$ is a convex
polyhedral cone, then there exists some $u_{0}\in\sigma^{\smallsmile}$ where
$\langle u_{0},v_{0}\rangle<0$. Therefore
$(\sigma^{\smallsmile})^{\smallsmile}=\sigma$ (duality theorem by direct
translation of the previous sentence) and other conditions pertaining to the
facets also follow.

If $\sigma\cap(-\sigma)=\\{0\\}$, $\sigma$ does not contain a nonzero linear
subspace, $\sigma^{\smallsmile}$ spans $V^{*}$, and a $u$ in
$\sigma^{\smallsmile}$ exists such that $\sigma\cap u^{\top}=\\{0\\}$, then
$\sigma$ is called a strongly convex cone.

####  2.1.3. Affine toric varieties

For $\sigma$ is a strongly convex rational polyhedral cone,
$S_{\sigma}=\sigma^{\smallsmile}\cap M$ is a finitely generated semigroup and
any additive semigroup $S$ determines a group ring, $\mathbb{C}[S]$, that is a
commutative $\mathbb{C}$-algebra.

As $u$ varies over $S$, the basis of this complex vector space, $\chi^{u}$,
has multiplication given by addition in $S$, i.e.
$\chi^{u}\cdot\chi^{u^{\prime}}=\chi^{u+u^{\prime}}$. Where $\chi^{0}$ is the
unit 1. The generators $\\{\chi^{u_{i}}\\}$ for the $\mathbb{C}$-algebra are
determined by the generators $\\{u_{i}\\}$ for the semigroup $S$.

If we use $A$ to denote any finitely generated commutative
$\mathbb{C}$-algebra, $A$ determines a complex affine variety Spec$(A)$. If
generators of $A$ are specified, then for $I$ an ideal, $A$ is presented as
$\mathbb{C}[X_{1},\dots,X_{m}]/I$; and the subvariety $V(I)$ of affine space
$\mathbb{C}^{m}$ of common zeros of the polynomials in $I$ identifies
Spec$(A)$. Therefore $A$ is the domain, Spec$(A)$ is an irreducible variety,
and even though Spec$(A)$ officially includes all prime ideals of $A$
(corresponding to subvarieties of $V(I)$), a point in Spec$(A)$ is taken as an
ordinary closed point corresponding to the maximal ideal, which is denoted
Specm$(A)$. Any homomorphism $A\rightarrow B$ determines a morphism of
varieties Spec$(B)\rightarrow$ Spec$(A)$, and $\mathbb{C}$-algebra
homomorphisms from $A$ to $\mathbb{C}$. When $X=\text{Spec}(A)$ for each
element $f\in A$ with $f\neq 0$, then according to the localisation
homomorphism $A\mapsto A_{f}$, $X_{f}$ is the principal open subset, defined
$X_{f}=\text{Spec}(A_{f})\subset X=\text{Spec}(A)$.

When $\mathbb{C}=\mathbb{C}^{*}\cup\\{0\\}$ is taken as an abelian group via
multiplication, and $A=\mathbb{C}[S]$ \- constructed from a semigroup, then
the points simply correspond to homorphisms of the semigroup from $S$ to
$\mathbb{C}$, i.e.
$\text{Specm}(\mathbb{C}[S])=\text{Hom}_{sg}(S,\mathbb{C})$. For a semigroup
homomorphism, the value of the corresponding function $\chi^{u}$ at the
specific point of $\text{Specm}(\mathbb{C}[S])$, where $x$ denotes the
semigroup homomorphism map from $S$ to $\mathbb{C}$ is the image of $u$ by the
map $x$, $\chi^{u}(x)=x(u)$. And $S=S_{\sigma}$ is apparent from a strongly
convex rational polyhedral cone. We get $A_{\sigma}=\mathbb{C}[S_{\sigma}]$
and affine toric variety
$U_{\sigma}=\text{Spec}(\mathbb{C}[S_{\sigma}])=\text{Spec}(A_{\sigma})$ when
$S=S_{\sigma}$ because the polyhedral cone is strongly convex and rational.

For a strongly convex rational polyhedral cone, $S=S_{\sigma}$, setting
$A_{\sigma}=\mathbb{C}[S_{\sigma}]$ and
$U_{\sigma}=\text{Spec}(\mathbb{C}[S_{\sigma}])=\text{Spec}(A_{\sigma})$ the
corresponding affine toric variety.

Of the group $M=S_{\\{0\\}}$ all of the above semigroups will be sub-
semigroups, so that if $e_{1},\dots,e_{n}$ is a basis for $N$ and
$e_{1}^{*},\dots,e_{n}^{*}$ is the dual basis of $M$, rename
$X_{i}=\chi^{e_{i}^{*}}\in\mathbb{C}[M]$. The generators of semigroup $M$ are
$\pm e_{1}^{*},\dots,\pm e_{n}^{*}$, and therefore,
$\mathbb{C}[M]=\mathbb{C}[X_{1},\dots,X_{n}]_{X_{1}\cdot\dots\cdot X_{n}}$
which is a ring of Laurent polynomials in $n$ variables.
$\text{Spec}(\mathbb{C}[M])=U_{\\{0\\}}\simeq\mathbb{C}^{*}\times\mathbb{C}^{*}=(\mathbb{C}^{*})^{n}$
is an affine algebraic torus.

All subgroups will be considered to be sub-semigroups of a lattice $M$,
therefore $\mathbb{C}[S]$ will be the subalgebra of $\mathbb{C}[M]$, and
specifically $\mathbb{C}[S]$ is the domain. Elements of $\mathbb{C}[S]$ are
written as Laurent polytnomials in variables $X_{i}$ when the basis of $M$ is
as above. In other words, monomials in the variables $X_{i}$ generate all
those algebras.

The torus $T_{N}$ that corresponds to $M$ or $N$, can be written
$T_{N}=\text{Spec}(\mathbb{C}[M])=\text{Hom}(M,\mathbb{C}^{*})=N\otimes_{\mathbb{Z}}\mathbb{C}^{*}$.

A homomorphism between semigroups $S$ and $S^{\prime}$ implies a morphism of
the affine varieties
$\text{Spec}(\mathbb{C}[S^{\prime}])\rightarrow\text{Spec}(\mathbb{C}[S])$. If
$\sigma$ contains $\tau$, taking $u\in S_{\sigma}=\sigma^{\smallsmile}\cap M$,
$\tau$ is a rational convex polyhedral cone that is equal to $\sigma\cap
u^{\perp}$, and therefore every face, $\tau$ of $\sigma$ can be written
$S_{\tau}=S_{\sigma}+\mathbb{Z}_{\geq 0\cdot(-u)}$. Therefore every basis
element of the algebra of $S_{\tau}$ can be written as
$\chi^{w-pu}=\chi^{w}/(\chi^{u})^{p}$, where $w\in S_{\sigma}$, and therefore
$A_{\tau}=(A_{\sigma})_{\chi^{u}}$.

It follows that $S_{\sigma}$ is a subgroup of $S_{\tau}$, implying the
morphism $U_{\tau}\rightarrow U_{\sigma}$ which embeds $U_{\tau}$ as a
principal open subset of $U_{\sigma}$.

A homomorphism $\varphi$ of lattices $N$ and $N^{\prime}$ so that
$\varphi_{\mathbb{R}}$ maps the cone $\sigma^{\prime}$ in $N^{\prime}$ into a
cone $\sigma$ in $N$.Its dual map $\varphi^{\prime}$ maps $S_{\sigma}$ to
$S_{\sigma}^{\prime}$ establishing homomorphism $A_{\sigma}\rightarrow
A_{\sigma^{\prime}}$ and so the morphism from $U_{\sigma^{\prime}}$ to
$U_{\sigma}$.

Therefore $S_{\sigma}$ is saturated and $M=S_{\sigma}+(-S_{\sigma})$. For
$S_{\sigma}$ generated by $u_{1},\ldots,u_{t}$, $I$ is generated by
polynomials $Y_{1}^{a_{1}}\cdot Y_{2}^{a_{2}}\cdot\ldots\cdot
Y_{t}^{a_{t}}-Y_{1}^{b_{1}}\cdot Y_{2}^{b_{2}}\cdot\ldots\cdot Y_{t}^{b_{t}}$
where $a_{i},b_{i}\in\mathbb{R}_{\geq 0}$ and $a_{1}u_{1}+\ldots
a_{t}u_{t}=b_{1}u_{1}+\ldots b_{t}u_{t}$ and so

$A_{\sigma}=\mathbb{C}[\chi^{u_{1}},\ldots,\chi^{u_{t}}]=\mathbb{C}[Y_{1},\ldots,Y_{t}]/I$

.

The torus $T_{N}$ acts on $U_{\sigma}$, $T_{N}\times U_{\sigma}\rightarrow
U_{\sigma}$ for some $\sigma$ in $N$. The product $t\cdot x$ is a map of
semigroups $S_{\sigma}\rightarrow\mathbb{C}$: $u\mapsto t(u)x(u)$. Where $t\in
T_{N}$ corresponds with the map $M\rightarrow\mathbb{C}^{*}$, and in turn
$x\in U_{\sigma}$ with map $S_{\sigma}\rightarrow\mathbb{C}$. In fact, for
$u\in S_{\sigma}$ the map $\chi^{u}$ to $\chi^{u}\otimes\chi^{u}$ gives the
dual map on algebras
$\mathbb{C}[S_{\sigma}]\rightarrow\mathbb{C}[S_{\sigma}]\otimes\mathbb{C}[M]$,
and is the usual product of groups when $\sigma=\\{0\\}$, is in agreement with
the inclusion of open subsets that are associated with faces of the cone, and
expends to the action of $T_{N}$ on itself.

####  2.1.4. Fans and toric varieties

Henceforth all cones will be rational strongly convex polyhedral cones. And
fans are finite. To assemble $X(\Delta)$ from fan $\Delta$, is to take the
disjoint union of affine toric varieties $U_{\sigma}$, one for each $\sigma$
in $\Delta$. A gluing aspect is also performed: $\sigma\cap\tau$ is a face for
each cone $\sigma$ and $\tau$. This implies that $U_{\sigma\cap\tau}$ is the
principal open subvariety of $U_{\sigma}$ and $U_{\tau}$. That
$U_{\sigma\cap\tau}\rightarrow U_{\sigma}\times U_{\tau}$ is a closed
embedding implies the map $A_{\sigma}\otimes A_{\tau}\rightarrow
A_{\sigma\cap\tau}$ is surjective since
$S_{\sigma\cap\tau}=S_{\sigma}+S_{\tau}$. Where, of course,
$U_{\sigma\cap\tau}=U_{\sigma}\cap U_{\tau}$.

Any product of affine and projective spaces can be realised as a toric
variety.

####  2.1.5. Toric varieties from polytopes

Once the vertices of the polytope are established, then the fan $\Delta$ and
then the toric variety. A very important method of assembly begins with a
rational polytope $P$ in $M_{\mathbb{R}}$ that is $n$-dimensional and doesn’t
necessarily contain the origin. The corresponding fan $\Delta_{P}$ is
assembled by taking, for each face $Q$ of $P$ its cone $\sigma_{Q}=\\{v\in
N_{\mathbb{R}}:\langle u,v\rangle\leq\langle u^{\prime},v\rangle\ \text{for
all}\ u\in Q\ \text{and}\ u^{\prime}\in P\\}$.

###  2.2. Contact Manifold

Recalling that Toric varieties are geometric objects defined by combinatorial
information: complex contact manifolds arise naturally in differential
geometry, algebraic geometry and exterior differential systems. The geometry
of such a manifold is governed by the contact lines contained in it. These are
related to the notion of a variety of minimal rational tangents.

######  Definition 2.1.

A contact structure on a $(2n+1)$-dimensional differentiable manifold $M$ is a
smooth codimension 1 distribution (i.e. tangent hyperplane field) $\xi\in TM$
that is maximally non-integrable in the following sense: We require that $\xi$
be a locally defined as $\xi=\text{ker}\alpha$ with a differential 1-form
$\alpha$ that satisfies $\alpha\wedge(d\alpha)^{n}\neq 0$ on its whole domain
of definition, If such a 1-form exists globally on $M$, it is called a contact
form on $M$. A manifold with a contact structure is called a contact manifold.

######  Example 2.1.

The standard contact structure on $\mathbb{R}^{2n+1}$ is the kernel of
$\alpha=dz-\sum_{i=1}^{n}y_{i}dx_{i}$ for coordinates
$(x_{1},y_{1},\ldots,x_{n},y_{n},z)\in\mathbb{R}^{2n+1}$. If we look at the
standard contact structure in $\mathbb{R}^{3}$
$(\mathbb{R}^{3},\text{ker}(dz+xdy))$, the conditions would be that
$\alpha=dz+xdy$, so $d\alpha=dx\wedge dy$ and certainly $\alpha\wedge
d\alpha\neq 0$. Plotting this contact plane, the 1-form $\alpha$ is always
tangent to the vector $\frac{\partial}{\partial x}\in\text{ker}(\alpha)$; and
this is further improved by again looking at the kernel, giving us
$x\frac{\partial}{\partial z}-\frac{\partial}{\partial y}$, so it is spanned
by these two vectors $\frac{\partial}{\partial x}$ and
$x\frac{\partial}{\partial z}-\frac{\partial}{\partial y}$.

######  Theorem 2.1 (Darboux’s theorem).

Let $\alpha$ be a contact form on the $(2n+1)$-dimensional manifold $M$ and
$p$ a point in $M$. Then there are coordinates
$(x_{1},\ldots,x_{n},y_{1},\ldots,y_{n},z)$ on a neighbourhood $U\subset M$ of
$p$ such that

$\alpha|_{U}=dz+\sum_{j=1}^{n}x_{j}dy_{j}$

.

One can easily find the respective duals $v_{i}^{*}$ to $v_{1}$ and $v_{2}$
that construct the dual vector space $M$ with respect to the dual pairing
$\langle,\rangle$ (as an example one can find the duals corresponding to the
standard contact structure). And therefore one can create a variety of minimal
rational tangents.

######  Definition 2.2.

Let $X$ be a normal variety, $U=X-\text{Sing}X$ and $j:U\rightarrow X$ the
natural embedding; the sheaf of differential p-forms or of p-differentials (in
the sense of Zariski-Steenbrink) to be $\Omega^{p}_{X}=j_{*}(\Omega^{p}_{X})$.

Let $\mathfrak{U}$ denote the ideal generated by all monomials defined before.
The sheaf $\Omega_{X}^{p}$ on the toric variety $X$ corresponds to a certain
$\mathbb{C}[M]$-module. $\Omega_{A}^{p}$ is the $N$-graded $\mathbb{C}$ vector
space $\Omega^{p}_{A}$ as defined in [6] is isomorphic to $\mathfrak{U}$.

##  3\. proof of theorem

A _contact structure_ on a smooth algebraic variety is the date of a subbundle
$D\subset\mathcal{T}_{X}$ of rank $\text{dim}X-1$ such that the
$\mathcal{O}_{X}$-bilinear form on $D$ with value in the line bundle
$L=\mathcal{T}_{X}/D$ induced by the Lie bracket on $\mathcal{T}_{X}$ is non
degenerate in each points of $X$. This implies that $X$ has odd dimension
$2n+1$ and that the canonical bundle $K_{X}$\- is isomorphic to $L^{-1-n}$. We
can also define the contact structure by the data of an element $\theta\in
H^{0}(X,\Omega_{X}^{1}\otimes L)$, the _contact form_ , such that
$\theta\wedge(d\theta)^{n}$ is everywhere nonzero.

Complex projective spaces and varieties $\mathbb{P}_{Y}(\mathcal{T}_{Y})$,
with $Y$ a smooth variety, are examples of contact varieties.

In short, a _toric variety_ is an equivariant compactification of a torus [6],
[7], [8], [9].

The study of contact varieties happens to be difficult. In dimension 3 they
are classified by using Mori theory. Other authors study contact structures on
Fano varieties but the results are not complete yet.

######  Theorem 3.1.

[10] Let X be a smooth projective toric variety with dimension $2n+1$ $(n\geq
1)$ endowed with a contact structure. Then $X$ is isomorphic either to the
complex projective space $\mathbb{P}^{2n+1}$ or to the variety
$\mathbb{P}_{\mathbb{P}^{1}\times\cdots\times\mathbb{P}^{1}}(\mathcal{T}_{\mathbb{P}^{1}\times\cdots\times\mathbb{P}^{1}})$

###### Proof.

The intersection pairing between 1-cycles and divisors induces a duality
between the following two real vector spaces:

$N_{1}(X)=(\\{1-\text{cycles}\\}/\equiv)\otimes\mathbb{R}\ \text{and}\
N^{1}(X)=(\\{1-\text{divisors}\\}/\equiv)\otimes\mathbb{R},$

with $\equiv$ the numerical equivalence. As stated before, dimension of both
vector spaces is called the Picard number of $X$. If we consider the cone as
$NE(X)$, and let it be spanned by classes of effective 1-cycles, $NE(X)\subset
N_{1}(X)$. Then an extremal ray is an half-line $R$ in $\bar{NE}(X)$ (the
closure of $NE(X)$ in $N_{1}(X)$), such that $K_{X}.R^{*}<0$ and such that for
all $Z_{1},Z_{2}\in\bar{NE}(X)$ if $Z_{1}+Z_{2}\in R$ then $Z_{1},Z_{2}\in R$
[11]. An extremal rational curve is an irreducible rational curve such that
$\mathbb{R}^{+}[\mathbb{C}]$ is an extremal ray and
$-K_{X}.C\leq\text{dim}X+1$. The first result from Mori theory is that every
extremal ray is spanned by an extremal rational curve. The second fundamental
result is that any extremal ray admits a contraction, i.e. there exists a
normal projective variety $Y$ and a morphism $\varphi:X\rightarrow Y$,
surjective with connected fibers, contracting irreducible curves $C$ such that
$[C]\in R$ (Kawamata-Shokurov theorem).

For toric varieties, there is a more precise result [Reid:1983]. Let $X$ be a
smooth projective toric variety associated to the $\Delta$. We use notations
from [9]. There there exist elements $\tau_{1},\ldots,\tau_{s}\in\Delta(d-1)$
such that

$NE(X)=\bar{NE}(X)=\mathbb{R}^{+}[V(\tau_{1})]+\ldots+\mathbb{R}^{+}[V(\tau_{s})].$

The half lines $\mathbb{R}^{*}[V(\tau_{i})]$ are called generalised extremal
rays. Let $R\subset NE(X)$ a generalised extremal ray. Then there exists a
projective toric variety $Y$ and an equivariant morphism $\varphi:X\rightarrow
Y$ which is a contraction in Mori’s sense. Denote by $A\subset X$ the
exceptional locus $\varphi$ and $B=\varphi(X)$:

$X$$Y$$\cup$$\cup$$A$$B$$\varphi$$\varphi|_{A}$

Then $A$ and $B$ are two irreducible toric stratas and the morphism
$\varphi_{A}:A\rightarrow B$ is flat and its fibers are weighted projective
spaces. Finally, when the contraction is of fibered type, the variety $Y$ is
regular and the morphism $\varphi$ is smooth.

Finally we recall a general result from J. Wisniewski [12], [13] about the
exceptional locus of extremal contraction. Let $F$ be an irreducible component
of a non-trivial fiber of an elementary contraction associated to the extremal
ray $R$. We call the locus of $R$, the locus of curves with numerical
equivalence class in $R$. Then we have the inequality:

$\text{dim}F+\text{dim}(\text{locus of}\ R)\geq\text{dim}X+l(R)-1$

where $l(R)$ is the length of the extremal ray $R$:

$l(R)=\text{inf}\\{-K_{X}.C_{0}|\ \text{with}\ C_{0}\text{a rational curve
and}C_{0}\in R\\}.$

The rest of the proof of this theorem relies on the study of extremal
contractions of $X$. Denote by $T$ the torus acting on $X$. As the canonical
divisor $K_{X}$ of $X$ is not effective, there exists an extremal ray $R$ in
the sense of Mori spanned by $C$ a smooth, rational, $T$-invariant curve. As
$K_{X}=-(n+1)L$, we have $l(R)=n+1$ or $l(R)=2n+2$ with $l(R)$ the length of
the extremal ray $R$. For the last case, $X$ is Fano and $Pic(X)=\mathbb{Z}$
([12] Prop. 2.4). Then $X$ is isomorphic to the complex projective space
$\mathbb{P}^{2n+1}$ ([8] Thm. 7.1).

It remains to study the case $l(R)=n+1$. Denote by $\varphi:X\rightarrow Y$
the Mori contraction associated to the ray $R$, with $Y$ a projective toric
variety. Denote by $T^{\prime}$ the torus acting on $Y$. Denote by $A\subset
X$ the exceptional locus of $\varphi$ and $B=\varphi(X)$:

$X$$Y$$\cup$$\cup$$A$$B$$\varphi$$\varphi|_{A}$

As the curve $C$ is $T$-invariant, it is contracted to a fixed point $y\in B$
under $T^{\prime}$. The morphism $\varphi$ being equivariant, the fiber
$F=\varphi^{-1}(y)$ is $T$-invariant. Hence, as this fiber is irreducible, we
deduce that $F$ is a toric strata and that $G$ is smooth as $X$ is not. Then
the fiber $F$ is isomorphic to projective space $\mathbb{P}^{k}$ $(k\geq 1)$
and, as $C$ is a toric strata of the toric variety $\mathbb{P}^{k}$, the curve
$C$ is identified to a line in $\mathbb{P}^{k}$. We know that there exists a
rational curve $C_{1}\subset X$ such that $C_{1}\in R$ and $-K_{X}.C_{1}=n+1$.
But, as there exists a divisor $D$ on $X$ such that $D.C=1$ and as
$R=\mathbb{R}^{+}[C]$, we easily deduce that $C\equiv C_{1}$ hence $C.L=1$.
Therefore we have
$(F,L_{|F})=(\mathbb{P}^{k},\mathcal{O}_{\mathbb{P}^{k}}(1))$.

As $H^{0}(\mathbb{P}^{k},\Omega^{1}_{\mathbb{P}^{k}}(1))=(0)$, the restriction
of the contact form $\theta$ to $\mathbb{P}^{k}$ vanishes and by a computation
with local coordinates, we check that for all $x\in\mathbb{P}^{k}$, the vector
subspace $\mathcal{T}_{\mathbb{P}^{k}}(x)\subset D(x)$ is totally isotropic
for the alternate contact form which is non-degenerate by hypothesis. This
implies the inequality $k\leq n$ and the equality $k=n$ as $\text{dim}F\geq
l(R)-1=n$. Applying again Wisniewski inequality, we deduce that the
contraction is of fibered type and therefore $Y$ is regular,
$n+1$-kdimensional and $\varphi$ is smooth.

Consider the exact sequence:

$0\rightarrow\mathcal{T}_{X/Y}\rightarrow\mathcal{T}_{X}\rightarrow\varphi^{*}\mathcal{T}_{Y}\rightarrow
0$

The map $\mathcal{T}_{X/Y}\rightarrow L$ obtained by composition with the
projection $\mathcal{T}_{X}\rightarrow L$ being identically zero by Grauert
theorem and Lemma 3.1 and Lemma 3.2 in [10], there exists a surjective map
$\varphi^{*}\mathcal{T}_{Y}\rightarrow L\rightarrow 0$ and therefore a
morphism $X\rightarrow\mathbb{P}_{Y}(\mathcal{T}_{Y})$ over $Y$ which induces
an isomorphism on each fiber. Then this morphism is in fact an isomorphism.
Therefore the result follows from
(1) the fact that for $Y$ a smooth projective toric variety of dimension $n$
and $\mathcal{E}$ a vector bundle of rank $r+1$ over $Y$. And with the
assumption that $X=\mathbb{P}_{Y}(\mathcal{E})$ is a toric variety and the
natural morphism $\varphi:X\rightarrow Y$ is equivariant. Then the bundle
$\mathcal{E}$ is the direct sum of its isotypical components.
And (2) for $X$ a smooth projective toric variety of dimension $n\geq 1$ with
a totally reducible tangent bundle. Then $X$ is isomorphic to
$\mathbb{P}^{1}\times\cdots\times\mathbb{P}^{1}$. ∎

######  Theorem 3.2.

For $X$ a smooth projective toric variety of dimension $2n+1$ $(n\geq 1)$,
defined over the field $\mathbb{C}$ of complex numbers and endowed with a
contact structure. Then either $X$ is isomorphic to the complex projective
space $\mathbb{P}^{2n+1}$ or $X$ is isomorphic to the variety
$\mathbb{P}_{\mathbb{P}^{1}\times\cdots\times\mathbb{P}^{1}}(\mathcal{T}_{\mathbb{P}^{1}\times\cdots\times\mathbb{P}^{1}})$.
The momentum map is defined

$\mu(x)=\frac{1}{\sum|\chi^{u}(x)|}\sum\limits_{u\in P\cap M}|\chi^{u}(x)|u$

where $\chi^{u}$ for $u\in P\cap M$ are the sections of
$\varphi:X\rightarrow\mathbb{P}^{2n}$ which is convex.

###### Proof.

For X a smooth projective toric variety with dimension $2n+1$ $(n\geq 1)$
endowed with a contact structure. $X$ is then isomorphic either to the complex
projective space $\mathbb{P}^{2n+1}$ or to the variety
$\mathbb{P}_{\mathbb{P}^{1}\times\cdots\times\mathbb{P}^{1}}(\mathcal{T}_{\mathbb{P}^{1}\times\cdots\times\mathbb{P}^{1}})$.

Since the toric varieties have been established, what is left is to talk about
how a toric variety will be redefined in the event of singularities, and
according to compactness. This is before fully identifying the orbits,
topology and line bundles all required in order to formulate the momentum
polytope.

The considerations for singularities and compactness

For specific point $x_{\sigma}$ of its affine variety,
$S_{\sigma}=\sigma^{\smallsmile}\rightarrow(1,0)\subset\mathbb{C}^{*}\cup\\{0\\}=\mathbb{C}$
where $u$ maps to 1 if $u\in\sigma^{\perp}$ and 0 otherwise. The existence of
fixed points in $U_{\sigma}$ is contingent on $\sigma$ spanning
$N_{\mathbb{R}}$, and in turn $x_{\sigma}$ is then a unique fixed point of
$T_{N}$ acting on $U_{\sigma}$. For singular points of the toric varieties,
suppose $U_{\sigma}$ is nonsingular at point $x_{\sigma}$, which means that
$\sigma^{\smallsmile}$ can’t have more than $n$ edges. And eventually,
$U_{\sigma}$ is isomorphic to $\mathbb{C}^{n}$. If we let
$N_{\sigma}=N\cap\sigma+(-\sigma\cap N)$ be a sublattice of $N$, then
$N(\sigma)=N/N_{\sigma}$ is the quotient group that is also a lattice, and the
map $N_{\sigma}\rightarrow N\rightarrow N(\sigma)$ . And so a cone is
nonsingular if it is generated by part of a basis for the lattice. And if all
the cones of a fan are nonsingular, then in turn that fan is nonsingular.
Although a toric variety can be singular, every toric variety is normal.
$\sigma$ is generated by part of a basis of $N$ if affine toric variety
$U_{\sigma}\simeq\mathbb{C}^{k}\times(\mathbb{C}^{*})^{n-k}$ is nonsingular,
where $\text{dim}(\sigma)=k$. And each ring
$A_{\sigma}=\mathbb{C}[S_{\sigma}]$ is integrally closed. Therefore we have a
method for extrapolating a nonsingular case from a singular case.

For a lattice $N$ of any rank, a sublattice $N^{\prime}\subset N$ of finite
index, with corresponding duals $M\subset M^{\prime}$, then
$M^{\prime}/M\times N/N^{\prime}\rightarrow\mathbb{Q}/\mathbb{Z}$ by
$\langle,\rangle$ pairing, and
$\mathbb{Q}/\mathbb{Z}\hookrightarrow\mathbb{C}^{*}$ by
$q\mapsto\text{exp}(2\pi iq)$. Therefore $G=N/N^{\prime}$ acts on the algebra
$\mathbb{C}[M^{\prime}]$ by $v\cdot\chi^{u^{\prime}}=\text{exp}(2\pi i\langle
u^{\prime},v\rangle)\cdot\chi^{u^{\prime}}$ for $u^{\prime}\in M^{\prime}$ and
$v\in N$ and therefore $\mathbb{C}[M^{\prime}]^{G}=\mathbb{C}[M]$. And
$T_{N^{\prime}}/G=T_{N}$.

For $\sigma$ any simplex, meaning that it is generated by independent vectors,
for example if $\sigma$ is an $n$-simplex and therefore generated by $n$
independent vectors, therefore the minimal elements in $\sigma\cap N$ along
all the $n$ edges generate the sublattice $N^{\prime}$ giving a cone
$\sigma^{\prime}$, $\mathbb{C}^{n}=U_{\sigma^{\prime}}\rightarrow U_{\sigma}$.
And we have the result of action
$U_{\sigma^{\prime}}/G=U_{\sigma}=\mathbb{C}^{n}/G$. Now if $\sigma$ is any
simplex, and in particular if all the cones in $\Delta$ are simplicial, making
it in turn a simplicial fan, then the toric variety only has quotient
singularities (is an orbifold).

For non-affine toric varieties, starting with the same fan used in the
assembly of the projective space, meaning this fan’s cones are generated by
proper subsets of vectors $\\{v_{0},v_{1},\ldots,v_{n}\\}$ of which any $n$
linearly independent vectors exist and whose sum is zero. Whilst $N$ is
generated by vectors $(1/d_{i})\cdot v_{i}$, $0\leq i\leq n$. And so we have a
’twisted’ or ’weighted projective space’,
$\mathbb{P}(d_{0},\ldots,d_{n})=\mathbb{C}^{n+1}\backslash\\{0\\}/\mathbb{C}^{*}$,
where the $\mathbb{C}^{*}$ action is defined,
$\varsigma\cdot(x_{0},\ldots,x_{n})=(\varsigma^{d_{0}}x_{0},\ldots,\varsigma^{d_{n}}x_{n})$.
This therefore makes the group action compatible on affine open subvarieties.

Therefore one can find a sublattice for any cone in a simplicial fan, for
which the associated open subvariety of the toric variety can be expressed as
a quotient by any finite abelian group. Like for the weighted projective
spaces, there usually exists a sublattice for all the open sets, but not
always. There exist fans with particular edges for which every sublattice, the
variety is singular.

If we take $\mathbb{G}_{m}$ to be the multiplicative algebraic group for, say,
$\mathbb{C}^{*}$, then every one-parameter subgroup
$\lambda:\mathbb{G}\rightarrow T_{N}$ exists for a unique $v\in N$, denoted
$\lambda_{v}$. For $z\in\mathbb{C}^{*}$ $\lambda_{v}(z)\in T_{N}$ meaning it
is given by a group homomorphism from $M$ to $\mathbb{C}^{*}$. Where
explicitly for $u$ in $M$,
$\lambda_{v}(z)(u)=\chi^{u}(\lambda_{v}(z))=z^{\langle u,v\rangle}$ and
$\langle,\rangle$ is the dual pairing $M\otimes N\rightarrow\mathbb{Z}$. Where
the dual $M=\text{Hom}(N,\mathbb{Z})=\text{Hom}(T_{N},\mathbb{G}_{m})$, and
every character $\chi:T_{N}\rightarrow\mathbb{G}_{m}$ is given by a unique $u$
in $M$. The function $\chi^{u}$ in the coordinate ring
$\mathbb{C}[M]=\Gamma(T_{N},\mathcal{O}^{*})$ gives the character
corresponding to $u$. This gives how to ascertain $N$ from $T_{N}$.

Looking at the limits $\lim_{z\to 0}\lambda_{v}(z)$ for different values of
$v\in N$, then this way given a $\sigma$ we can recover $\sigma$ from the
$T_{N}\subset U_{\sigma}$ torus embedding.

If $|\Delta|$ contains $v$, and $\Delta$ has cone $\tau$ that contains $v$ in
its relative interior then $x_{\tau}=\text{lim}_{z\rightarrow
0}\lambda_{v}(z)$. And if $v$ is not in any cone of $\Delta$, then
$\text{lim}_{z\rightarrow 0}\lambda_{v}(z)$ does not exist in $X(\Delta)$.
There is no limit or converging subsequence for $v$s not in the union of cones
$\Delta$ (not in the support $|\Delta|$).

$X(\Delta)$ is compact if and only if $|\Delta|=N_{\mathbb{R}}$. For a
homomorphism $\varphi:N^{\prime}\rightarrow N$, which maps fan
$\Delta^{\prime}$ to $\Delta$, the morphism
$\varphi_{*}:X(\Delta^{\prime})\rightarrow X(\Delta)$ is proper if and only if
$|\Delta^{\prime}|=\varphi^{-1}(|\Delta|)$. An example of a proper map is
what’s called blowing up.

Two dimensional nonsingular complete toric varieties are given by sequence of
lattice points, $v_{0},v_{1},\ldots,v_{d-1},v_{d}=v_{0}$, in
$N=\mathbb{Z}^{2}$, so that successive pairs generate the lattice. Since
$v_{0}$ and $v_{1}$ are a basis for a lattice, it follows that $v_{1}$ and
$v_{2}$ are also a basis where $v_{2}=-v_{0}+a_{1}v_{1}$ for
$a_{1}\in\mathbb{Z}$ (i.e. $a_{i}v_{i}=v_{i-1}+v_{i-1}$ where $1\leq i\leq d$
for some integers $a_{i}$). Cones created (for example between $v_{i-1}$ and
$v_{i+1}$) cannot have a third vector (e.g. $v_{j}$) in the angle strictly
between them. This is an example of how the possible configurations are
topologically constrained. All of these surfaces can be classified. In fact,
all toric surfaces that are complete and nonsingular are procured from
$\mathbb{P}^{2}$ or $\mathbb{F}_{a}$ by a series of blow-ups at $T_{N}$-fixed
points.

A strongly convex cone is generated by $v_{j-1}$ and $v_{j+1}$ for some $j$
$1\leq j\leq d$ when $d\geq 5$. And we have that $v_{j}=v_{j-1}+v_{j+1}$.

Before where we fixed the cone to refine the lattice, in order to resolve
singularities, let us now look at fixing the lattice and subdividing the
cones. A refinement $\Delta^{\prime}$ can be created from $\Delta$, so each
cone of $\Delta$ is a union of cones in $\Delta^{\prime}$. We obtain a
birational and proper morphism $X(\Delta^{\prime})\rightarrow X(\Delta)$ from
the identity map of $N$. This map is an isomorphism on $T_{N}$ contained in
each. More details can be found in section §2.6 of [7], including an example
of creating new generators of the cone using the Hirzebruch-Jung continued
fraction of $m/k$ so that, for example, $v=e_{2}$ and
$v^{\prime}=me_{1}-ke_{2}$ generate $\sigma$ rather than $e_{1}$ and $e_{2}$
for a cone that is not generated by a basis for $N$. The indicated subdivision
provides a nonsingular $X(\Delta^{\prime})$ mapping birationally and properly
to $U_{\sigma}$. In similar such ways this construction can be used on
singular toric varieties to resolve singularities.

In more detail, given a fan $\Delta$ in any lattice $N$ and any lattice point
$v$ in $N$, then $\Delta$ can be created from subdividing $\Delta$. Meaning
that each cone that contains $v$ is exchanged with the sums of faces that have
a ray through $v$, and every cone that doesn’t contain $v$ is left unchanged.
The mapping from $X(\Delta^{\prime})$ to $X(\Delta)$ is proper and birational
since $\Delta$ and $\Delta^{\prime}$ have the same support. Such subdivisions
are taken until a nonsingular toric variety is reached.

With accordance to orbits, topology and line bundles

So we have established that $X=X(\Delta)=\bigcup_{disjoint}$ its orbits by
$T_{N}$-action. There exists an orbit $O_{\tau}$ for each cone $\tau$ in
$\Delta$ containing distinguished point $x_{\tau}$, and if
$\text{dim}(\tau)=k$ then $O_{\tau}\simeq(\mathbb{C}^{*})^{n-k}$; if
$x_{\tau}$ is $n$-dimensional then $O_{\tau}$ is $x_{\tau}$; and if
$\tau=\\{0\\}$, then $O_{\tau}=T_{N}$. Denote by $V(\tau)$ the open subvariety
of the closure of $O_{\tau}$. It is the closed subvariety of the toric variety
$X$ and $V(\tau)=\bigcup_{disjoint}O_{\gamma}$ where $\tau$ is contained in
$\gamma$ as a face.

In general $O_{\tau}=\text{Hom}(\tau^{\perp}\cap M,\mathbb{C}^{*})$. Let
$N_{\tau}$ denote the sublattice of $N$ generated by $\tau\cap N$ and define
the quotient lattice and its dual as $N(\tau)=N/N_{\tau}$, and
$M(\tau)=\tau^{\perp}\cap M$. Therefore
$T_{N(\tau)}=\text{Hom}(M(\tau),\mathbb{C}^{*})=O_{\tau}$ is the torus
corresponding to these lattices, of dimension $n=k$ where $k$ is the value of
the dimension of $\tau$. And we have that
$O_{\tau}=\text{Spec}(\mathbb{C}[M(\tau)])=N(\tau)\otimes_{\mathbb{Z}}\mathbb{C}^{*}$,
$T_{N}$ acts transitively via the projection $T_{N}\rightarrow T_{N(\tau)}$.

Next we will introduce something called the ’Star of the cone $\tau$’ which
can be loosely defined as the set of cones $\sigma$ in $\Delta$ that contain
$\tau$ as a face. The cones
$\bar{\sigma}=\sigma+(N_{\tau})_{\mathbb{R}}/(N_{\tau})_{\mathbb{R}}$ where
$\bar{\sigma}$ is such that $\tau<\sigma$, is a subset of
$N_{\mathbb{R}}/(N_{\tau})_{\mathbb{R}}=N(\tau)_{\mathbb{R}}$. Forming a fan
in $N(\tau)$, denoted $\text{Star}(\tau)$. Conceptualise $\text{Star}(\tau)$
as the cones that contain $\tau$ but register it as a fan in $N(\tau)$.

The $(n-k)$-dimensional toric variety $V(\tau)$ is therefore defined as
$X(\text{Star}(\tau))$.

As $\sigma$ varies over all the cones in $\Delta$, the corresponding toric
variety $V(\tau)$ has an affine open covering $U_{\sigma}(\tau)$, with $\tau$
as a face. This affine open covering is defined as
$\text{Spec}(\mathbb{C}[\sigma^{\smallsmile}\cap\tau^{\perp}\cap M])$ where
the corresponding dual face to $\tau$ is
$\sigma^{\smallsmile}\cap\tau^{\perp}$, a face of $\sigma^{\smallsmile}$.

$U_{\sigma}(\tau)$ is embedded in $U_{\sigma}$, this can be seen due to the
respective semigroup homomorphisms:
$\text{Hom}_{sg}(\sigma^{\smallsmile}\cap\tau^{\perp}\cap M,\mathbb{C})$ is
embedded in $\text{Hom}_{sg}(\sigma^{\smallsmile}\cap M,\mathbb{C})$. Since
the extension by zero of a semigroup homomorphism is a semigroup homomorphism
as $\sigma^{\smallsmile}\cap\tau^{\perp}$ is a face of $\sigma^{\smallsmile}$.
In fact if $u$ is in $\sigma^{\smallsmile}\cap\tau^{\perp}\cap M$, then
$\chi^{u}$ is projected to $\chi^{u}$, by the surjection of rings
$\mathbb{C}[\sigma^{\smallsmile}\cap
M]\twoheadrightarrow\mathbb{C}[\sigma^{\smallsmile}\cap\tau^{\perp}\cap M]$.
And we therefore have the closed embedding

$V(\tau)\hookrightarrow X(\Delta).$

And closed embeddings $V(\tau^{\prime})\hookrightarrow V(\tau)$ if $\tau$ is a
face of $\tau^{\prime}$, given by extension by zero from
$\text{Hom}_{sg}(\sigma^{\smallsmile}\cap\tau^{\prime\perp}\cap M,\mathbb{C})$
to $\text{Hom}_{sg}(\sigma^{\perp}\cap\tau^{\perp}\cap M,\mathbb{C})$, on
$U_{\sigma}$ for $\sigma\in\text{Star}(\tau^{\prime})$. Or also given by
considering $V(\tau)$ as a toric variety and exerting the construction
described above. Therefore establishing an order-reversing correspondence from
$\tau$ in $\Delta$ to closures $V(\tau)$ in $X(\Delta)$.

This all implies that $\oplus\mathbb{C}\cdot\chi^{u}$ is the ideal of
$V(\tau)\cap U_{\sigma}$ in $A_{\sigma}$. This ideal is the sum over all $u$
in $S_{\sigma}$ so that for $v$ in the relative interior of $\tau$, $\langle
u,v\rangle>0$.

The following are the presiding relations between affine open sets
$U_{\sigma}$, orbits $O_{\tau}$ and orbit closures $V(\tau)$:
$U_{\sigma}=\coprod_{\tau\prec\sigma}O_{\tau}$;
$V(\tau)=\coprod_{\gamma\succ\tau}O_{\gamma}$;
$O_{\tau}=V(\tau)-\bigcup_{\gamma\succneqq\tau}V(\gamma)$.

According to the third relation,
$T_{N}=X(\Delta)-\bigcup_{\gamma\neq{0}}V(\gamma)$ for $\tau=\\{0\\}$. And
therefore, $X(\Delta)$ is the disjoint union of the $O_{\tau}$ that are the
orbits of the $T_{N}$-action. And when $\sigma$ is a face of $\tau$, is
precisely when $O_{\tau}$ is contained within the closure of $O_{\sigma}$. And
the closed orbits are equal to $O_{\sigma}$ for which $\sigma$ is maximal cone
in $\Delta$. Removing some maximal cones $\sigma$ from the fan $\Delta$,
whilst maintaining their faces - denote the resulting fan, $\Delta^{\circ}$ \-
then $X(\Delta^{\circ})$, the toric variety obtained from $X(\Delta)$ with its
closed orbits $O_{\sigma}$ removed.

If we construct a fan from a polytope $P$, then for each face $Q$ of the
polytope has a cone $\sigma_{Q}$, and $V(\sigma_{Q})$, the invariant
subvariety for each face, can simply be denoted $V_{Q}$ (when $Q$ is a face of
$Q^{\prime}$, then $V_{Q^{\prime}}$ contains $V_{Q}$). And the real dimension
of $Q$ is equal to the complex dimension of $V_{Q}$.

Each $V(\tau)$ is nonsingular if $X(\Delta)$.

Let us introduce $\pi_{1}(X(\Delta))$ as the fundamental group of a toric
group. Complete toric varieties are simply connected (where $\Delta$ is a fan
that includes an $n$-dimensional cone.)(The base points can be considered as
the origins of the embedded tori, and so will be omitted). Due to the
inclusion $T_{N}\hookrightarrow X(\Delta)$ provides the surjection
$\pi_{1}(T_{N})\twoheadrightarrow\pi_{1}(X(\Delta))$.

The normality is crucial: the closed complement isn’t small enough. Therefore
the variety is not simply connected but still have that the complement of the
point in the irreducible variety is simply connected. Consider as an example,
the variety that is equal to 2 simply connected points.

If we take $v\in N$ to $S^{1}\subset\mathbb{C}^{*}\rightarrow T_{N}$ where
$\lambda_{v}$ is the $\mathbb{C}^{*}\rightarrow T_{N}$ map defined before, we
have the canonical isomorphism between $N$ and $\pi_{1}(T_{N})$, for any
$T_{N}$. The loop $S^{1}$ can be contracted to $U_{\sigma}$ if $v$ is in
$\sigma\cap N$ for some cone $\sigma$. As $\text{lim}_{z\rightarrow
0}\lambda_{v}(z)=x_{\sigma}$ exists in $U_{\sigma}$, $\lambda_{v}$ enhances as
a $\mathbb{C}\rightarrow U_{\sigma}$ map. For $z\in S^{1}$ this contraction is
defined as $\lambda_{v}(tz)$ and $0<t\leq 1$, and $x_{\sigma}$ for $t=0$. For
all $t$ the general all-encompassing one-parameter subgroup for $v$ is denoted
$\lambda_{v,t}(z)$.

It follows that if $\sigma^{\prime}$ is a cone in lattice $N_{\sigma}$
generated by $\sigma$ then a canonical isomorphism from $\pi_{1}(U_{\sigma})$
to $\pi_{1}(T_{N(\sigma)})$ which is identically $N(\sigma)$ according to the
fibration $U_{\sigma^{\prime}}\rightarrow U_{\sigma}\rightarrow
T_{N(\sigma)}$. Of course as
$\pi_{1}(\mathbb{C}^{*})=\pi_{1}(S^{1})=\mathbb{Z}$ then
$U_{\sigma}=U_{\sigma^{\prime}}\times(\mathbb{C}^{*})^{n-k}$, then
$\pi_{1}(U_{\sigma})\simeq\mathbb{Z}^{n-k}$ when $\sigma$ is a $k$-dimensional
cone. When $\sigma$ is $n$-dimensional, then $N$ is generated as a group by
$\sigma\cap N$. And if such loops are trivial in $U_{\sigma}$ then all loops
are trivial.

If $N^{\prime}\subset N$ given by $\sigma\cap N$ as $\sigma$ varies over
$\Delta$, and since for each $\sigma$ $\pi_{1}(U_{\sigma})=N/N_{\sigma}$, this
means that as $\sigma$ varies over $\Delta$,
$\pi_{1}(X(\Delta))=N/N^{\prime}$.

And indeed $U_{\sigma}$ is contractible if $\sigma$ is $n$-dimensional, and
$O_{\sigma}\subset U_{\sigma}$ is a deformation contract if
$\text{dim}(\sigma)=k$. And where $\sigma^{\perp}\cap M=M(\sigma)$ then if we
define a homotopy $H:U_{\sigma}\times[0,1]\rightarrow U_{\sigma}$ between the
identity map and the retraction $r:U_{\sigma}\rightarrow x_{\sigma}$, the
canonical isomorphism
$H^{i}(U_{\sigma};\mathbb{Z})\simeq\bigwedge^{i}(M(\sigma))$.

Some information about the cohomology of $X(\Delta)$ can be provided from
knowing the cohomology of basic open sets $U_{\sigma}$, the cohomology of $X$
can be derived as the C̆ech cohomology of the covering.

Looking at the spectral sequence

$E_{1}^{p,q}=\bigotimes_{i_{0}<\ldots<i_{p}}H^{q}(U_{i_{0}}\cap\ldots\cap
U_{i_{p}})\Rightarrow H^{p+q}(X).$

This is the spectral sequence of the double complex
$\bigoplus\mathcal{C}^{q}(U_{i_{0}}\cap\ldots\cap U_{i_{p}})$ if
$\mathcal{C}^{*}$ is a resolution that is injective of the sheaf. And this is
true with any sheaf or coefficient group. It is the spectral sequence with
vertical maps coming from $\mathcal{C}^{*}$ (complex) and the horizontal maps
the so-called C̆ech maps according to alternating sums of restrictions.

Where $\sigma_{i}$ are the maximal cones of fan $\Delta$, can apply the
spectral sequence to the covering according to open sets
$U_{i}=U_{\sigma_{i}}$, then,

$E_{1}^{p,q}=\bigotimes_{i_{0}<\ldots<i_{p}}\bigwedge^{q}M(\sigma_{i_{0}}\cap\ldots\cap\sigma_{i_{p}})\Rightarrow
H^{p+q}(X(\Delta))$

which gives us $\chi(X(\Delta))$, that is the topological Euler
characteristic. For every $\text{dim}\tau<n$, then the alternating sum
$\sum(-1)^{q}\text{rank}(\wedge^{q}M(\tau))=0$, otherwise when
$\text{dim}\tau=n$ then $\sum(-1)^{q}\text{rank}(\wedge^{q}M(\tau))=1$ and so,
the topological Euler characteristic is
$\chi(X(\Delta))=\sum(-1)^{q}\text{rank}E_{1}^{p,q}=\sharp n-\text{dimensional
cones in}\Delta$. If $\Delta$ is complete i.e. $\text{dim}\Delta=n$, and
$U_{\sigma_{i}}$ is contractible, $E_{1}^{0,q}=0$ for $q\geq 1$. Additionally,
$E^{\cdot,0}_{1}$ is the cochain complex of a simplex, i.e. a Koszul complex.
Meaning that for $p$ greater than $1$, $E_{2}^{p,0}=0$. Then according to the
spectral sequence, $\text{Ker}(E_{1}^{1,1}\rightarrow
E_{1}^{2,1})=E_{\infty}^{1,1}=E_{2}^{1,1}=H^{2}(X(\Delta))=\text{Ker}(\bigoplus_{i<j}M(\sigma_{i}\cap\sigma_{j})\rightarrow\bigoplus_{i<j<k}M(\sigma_{i}\cap\sigma_{j}\cap\sigma_{k}))$.
Where $\chi^{u}\neq 0$ on $U_{\sigma}$ given by any element $u$ of
$M(\sigma)=\sigma^{\perp}\cap M$. The line bundle on $X(\Delta)$ is given by
the cocycle defined by the above kernel, and every element of the first Chern
class of a line bundle is $H^{2}(X)$.

On any variety, a finite formal sum $\sum a_{i}V_{i}$ of irreducible closed
subvarieties $V$ of codimension one in $X$ is called a Weil divisor. Let $D$
denote the Cartier divisor, which is given by the covering data of $X$ by
$U_{\alpha}$ and $f_{\alpha}$ -the affine open sets and nonzero rational
functions respectively. $f_{\alpha}$ are referred to as local equations so
that the ratios $f_{\alpha}/f_{\beta}\neq 0$ everywhere are are regular
functions on $U_{\alpha}\cap U_{\beta}$. If we look at the ideal sheaf of $D$,
denoted $\mathcal{O}(-D)$, it is a subsheaf of the sheaf of rational functions
generated by $f_{\alpha}$ on $U_{\alpha}$; and its inverse, $\mathcal{O}(D)$
is the subsheaf of the sheaf generated by the inverse rational functions
$1/f_{\alpha}$ on $U_{\alpha}$. The transition functions of this line bundle
which take us from $U_{\alpha}$ to $U_{\beta}$ are $f_{\alpha}/f_{\beta}$.
Denote the Weil divisor $[D]$ and $\text{ord}_{V}(D)$ the order of the
vanishing of an equation for $D$ in the local ring along $V$. Define $[D]$
according to $D$, $[D]=\sum_{\text{cod}(V,X)=1}\text{ord}_{V}(D)\cdot V$. Of
course the concept of order here is rather limited as these rings that are
local are valuation rings which are discrete when $X$ is normal. The map
$D\mapsto[D]$ denotes as embedding, for normal varieties, of the group of
Cartier divisors into the group of Weil divisors. A principal divisor
$\text{div}(f)$ for a $f\neq 0$ and rational, the local equation is each open
set is $f$.

Divisors on the toric variety that map to themselves by $T$, the irreducible
subvarieties of codimension one of which are $T$-stable attribute to the rays
(or edges) of the fan. If we let $v_{i}$ be the first lattice point met along
edge $\tau_{i}$, amongst edges $\tau_{1},\ldots,\tau_{d}$, then
$D_{i}=V(\tau_{i})$, i.e. these divisors are orbit closures. And for
$a_{i}\in\mathbb{Z}$ the $T$-Weil divisors are the sums $\sum a_{i}D_{i}$.

Let’s define $T$-Cartier divisors as, the Cartier divisors that are
equivariant by $T$. If we have the affine case where $\text{dim}(\sigma)=n$
and $X=U_{\sigma}$, then for the fractional ideal
$I=\Gamma(X,\mathcal{O}(D))$, $D$ is the divisor that $T$ preserves. And for a
unique $u\in\sigma^{\smallsmile}\cap M$, $I$ is generated by function
$\chi^{u}$. In other words, $I$ is a direct sum of spaces
$\mathbb{C}\cdot\chi^{u}$ over some set of $u$ in $M$, meaning that $I$ is
graded by $M$ is implied from the divisor being $T$-invariant. For
$\mathfrak{M}=\sum_{u\neq 0}\mathbb{C}\cdot\chi^{u}$, then immediately
$\text{dim}I/\mathfrak{M}I=1$ is a must, because $I$ is the principal at
determining point $x_{\sigma}$. Therefore we have that
$I=A_{\sigma}\cdot\chi^{u}$ for unique $u$. And for some unique $u\in M$, a
general $T$-Cartier divisor on $U_{\sigma}$ is of the form
$\text{div}(\chi^{u})$.

$\text{ord}_{V(\tau)}(\text{div}(\chi^{u}))=\langle u,v\rangle$ when $u\in M$
and $v$ is the first lattice point along edge $\tau$. Therefore
$[\text{div}(\chi^{u})]=\sum_{i}\langle u,v_{i}\rangle D_{i}$. A $T$-Cartier
divisor on $U_{\sigma}$ is of the form $\text{div}(\chi^{u})$ for some $u\in
M$ when $\text{dim}(\sigma)<n$. However, since
$U_{\sigma}=U_{\sigma^{\prime}}\times T_{N(\sigma)}$ where $\sigma$ and
$\sigma^{\prime}$ share the same edge and therefore the same Weil divisors,
and therefore $T$-Cartier divisors on $U_{\sigma}$ are in agreement with
elements of $M/M(\sigma)$, then $u-u^{\prime}$ is in $\sigma^{\perp}\cap
M=M(\sigma)$ if and only if $\text{div}(\chi^{u})$ and
$\text{dim}(\chi^{u^{\prime}})$ are equal.

A $T$-Cartier divisor is defined, on a general toric variety $X(\Delta)$, by
choosing an element $u(\sigma)\in M/M(\sigma)$ for each $\sigma$ in the fan.
And specifying $\text{div}(\chi^{-u})$ (minus sign is taken to conform to the
literature) on $U_{\sigma}$. Another way of saying this is,
$\Gamma(U_{\sigma},\mathcal{O}(D))=A_{\sigma}\cdot\chi^{u(\sigma)}$ meaning
that $\chi^{u(\sigma)}$ generates the fractional ideal of $\mathcal{O}(D)$ on
$U_{\sigma}$. These are true based on a lack of discrepancy concerning
overlaps. Meaning that when $\tau$ is a face of $\sigma$, $u(\sigma)\mapsto
u(\tau)$ under the canonical map $M/M(\sigma)\mapsto M/M(\tau)$. Basically,
$\sigma_{i}$ are the maximal cones, and
$\text{Ker}(\bigoplus_{i}M/M(\sigma_{j})\rightarrow\oplus_{i<j}M/M(\sigma_{i}\cap\sigma_{j}))=\text{lim}_{\leftarrow}M/M(\sigma)=\\{\text{$T$-Cartier
divisors}\\}$.

For any irreducible variety $X$, a homomorphism is given from the group of
Cartier divisors on $X$ onto the group of all line bundles $\text{Pic}(X)$,
where the group of principal divisors are the kernel. Let’s define the group
of all Weil divisors modulo $[\text{div}(f)]$, the subgroup of divisors of
rational functions, $A_{n-1}(X)$. The homomorphism $\text{Pic}(X)$ to
$A_{n-1}(X)$ is determined by the map $D\mapsto[D]$, which when $X$ is normal
is an embedding $\text{Pic}(X)\hookrightarrow A_{n-1}(X)$.

A homomorphism is given from $M$ to the group of $T$-Cartier divisors
$\text{Div}_{T}X$ by a principal Cartier divisor determined by any $u\in M$
for a $X$. The computation of $\text{Pic}(X)$ only requires $T$-Cartier
divisors and functions. This is similarly true for $A_{n-1}(X)$ with $T$-Weil
divisors. Both for complete $X$.

For a fan not contained within a proper subspace of $N_{\mathbb{R}}$ then we
have the exact sequences $0\rightarrow
M\rightarrow\text{Div}_{T}X\rightarrow\text{Pic}(X)\rightarrow 0$ and
$0\rightarrow M\rightarrow\bigoplus^{d}_{i=1}\mathbb{Z}\cdot D_{i}\rightarrow
A_{n-1}(X)\rightarrow 0$ with embeddings
$\text{Div}_{T}X\hookrightarrow\bigoplus^{d}_{i=1}\mathbb{Z}\cdot D_{i}$ and
$\text{Pic}(X)\hookrightarrow A_{n-1}(X)$ and the $M$s of each exact sequence
are equal. More specifically, for $d$ the number of edges of the fan,
$\text{rank}(\text{Pic}(X))\leq\text{rank}(A_{n-1}(X))=d-n$. And in
particular, $\text{Pic}(X)$ is free abelian.

Now $\text{Pic}(X(\Delta))\simeq H^{2}(X(\Delta),\mathbb{Z})$ if all maximal
cones of $\Delta$ are $n$-dimensional.

$u(\sigma)\in M/M(\sigma)$ which provides the data for $D$, gives us the
continuous piecewise linear function $\Psi_{D}$ on the support $|\Delta|$.
Restricting $\Psi_{D}$ to $\sigma$ is defined as the linear function
$u(\sigma)$: $\Psi_{D}(V)=\langle u(\sigma),v\rangle$ for $v\in\sigma$. This
function is well defined and continuous thanks to the compatibility of the
data. Any continuous function given by an element of the lattice $M$, that is
on the support $|\Delta|$ which is both integral and linear on each cone, is
derived from a unique $T$-Cartier divisor.

$[D]=\sum-\Psi_{D}(v_{i})D_{i}$, or otherwise one may say that because
$\Psi_{D}(v_{i})=-a_{i}$, this determines $\Psi_{D}$ (only when $D=\sum
a_{i}D_{i}$). $\Psi_{D+E}=\Psi_{D}+\Psi_{E}$ and $\Psi_{mD}=m\Psi_{D}$. The
linear function $-u$ is $\Psi_{\text{div}(\chi^{u})}$. And $\Psi_{D}$ and
$\Psi_{E}$ are different only up to the linear function $u\in M$ if $D$ and
$E$ are linearly equivalent divisors.

A rational convex polyhedron $P_{D}$ in $M_{\mathbb{R}}$ is defined as the
$u\in M_{\mathbb{R}}$ for which $\langle u,v_{i}\rangle\geq a_{i}$ for all
$i$, or equivalently, the $u\in M_{\mathbb{R}}$ for which $u\geq\Psi_{D}$ on
the support $|\Delta|$, for $T$-Cartier divisor $D=\sum a_{i}D_{i}$ on
$X(\Delta)$.

$\Gamma(X,\mathcal{O}(D))=\bigoplus_{u\in P_{D}\cap M}\mathbb{C}\cdot\chi^{u}$
represent the global sections of the line bundle $\mathcal{O}(D)$. It follows
that in correspondence with the direct sum over the intersection of the
$P_{D}(\sigma)\cap M$,
$\bigcap\Gamma(U_{\sigma},\mathcal{O}(D))=\Gamma(X,\mathcal{O}(D))$ therefore
the above works for restrictions to open sets that are smaller.

When the support is equal to $N_{\mathbb{R}}$, then the toric variety is
complete, in turn on any complete variety, the cohomology groups of a suitable
sheaf will be of finite dimension. Specifically, $\Gamma(X,\mathcal{O}(D))$ is
of finite dimension and $P_{D}\cap M$ is also finite when the cones $\Delta$
span $N_{\mathbb{R}}$.

When is a line bundle generated by its sections? Reconsider the real valued
function $\Psi$ on a vector space, it is upper convex if $\Psi(t\cdot
v+(1-t)\cdot w)\geq t\psi(v)+(1-t)\psi(w)$ for all $0\leq t\leq 1$ and for all
vectors $v$ and $w$. The convex function $\Psi$ is strictly convex if for all
$n$-dimensional cones, $\text{graph}(\Psi)$ on the complement of $\sigma$ lies
strictly beneath $\text{graph}(u(\sigma))$.

If all maximal cones in $\Delta$ are $n$-dimensional, and we have that $D$ is
a $T$-Cartier divisor on the toric variety, then $\Psi_{D}$ is convex if and
only if $\mathcal{O}(D)$ is generated by its sections. We can therefore
construct $\Psi_{D}$ from $P_{D}$:

$\Psi_{D}(v)=\text{min}_{u\in P_{D}\cap M}\langle u,v\rangle=\text{min}\langle
u,v\rangle$

where the vertices of polytope $P_{D}$ are given by $u_{i}$.

We have a corresponding mapping
$\varphi=\varphi_{D}:X(\Delta)\rightarrow\mathbb{P}^{r-1}$ with
$x\mapsto(\chi^{u_{1}}(x):\ldots:\chi^{u_{r}}(x))$ (where
$r=\text{Card}(P_{D}\cap M)$ and $\mathbb{P}^{r-1}$ is the projective space)
for a basis $\chi^{u}$ for the sections that generate $\mathcal{O}(D)$, where
$u\in P_{D}\cap M$. If $N_{\mathbb{R}}=|\Delta|$ then $\varphi_{D}$ is an
embedding (meaning that $D$ is very ample) if and only if $\Psi_{D}$ is
strictly convex and for every $n$-dimensional cone $\sigma$, the semigroup
$S_{\sigma}$ is generated by $\\{u-u_{\sigma}:u\in P_{D}\cap M\\}$. And in
fact on a complete toric variety, $D$ is ample, meaning that any positive
multiple of $D$ is very ample. This is true if and only if $\Psi_{D}$ is
strictly convex.

For a polytope that is convex and $n$-dimensional, with vertices in $M$ has a
fan $\Delta_{P}$ with complete toric variety $X_{P}=X(\Delta_{P})$, and
corresponding Cartier divisor $D_{P}$, where
$\Psi(v)=\Psi_{D}(v)=\text{min}_{u\in P}\langle u,v\rangle=\text{min}_{u\in
P\cap M}\langle u,v\rangle=\text{min}\langle u_{i},v\rangle$, where $u_{i}$
are the vertices of $P$ and $\chi^{u_{i}}$ generates $\mathcal{O}(D)$ on
$U_{\sigma_{i}}$, if $\sigma_{i}$ is the maximal cone for $u_{i}$.

A fan $\Delta$ is called ’compatible with $P$ \- the convex hull of any finite
set in $M$’, if $\Psi_{P}(v)=\text{min}_{u\in P}\langle u,v\rangle$ is linear
on each $\sigma$ in this complete fan. The line bundle of the determined
$D_{P}$ that is generated by its sections which are linear combinations of
$\chi^{u}$ as $u$ varies over $P\cap M$.

For $u\in M_{\mathbb{R}}$ where $u\geq\Psi$ on $|\Delta|$,
$H^{0}(X,\mathcal{O}(D))_{u}=\mathbb{C}\cdot\chi^{u}$ if $u\in P_{D}\cap M$
and $H^{0}(X,\mathcal{O}(D))_{u}=0$ otherwise. And the sections of
$\mathcal{O}(D)$ are the graded module: $H^{0}(X,\mathcal{O}(D))=\bigoplus
H^{0}(X,\mathcal{O}(D))_{u}$. This can be put more simply by introducing
$Z(u)$ to represent the closed conical subset of the support for each $u\in M$
and define it as $Z(u)=\\{v\in|\Delta|:\langle u,v\rangle\geq\Psi(v)\\}$. Then
when $Z(u)$ is exactly equal to the support, then $u$ perfectly belongs to
$P_{D}$. This can be put another way, if we introduce $H^{0}(|\Delta|-Z(u))$
as the $H^{0}$ that denotes the $0^{\text{th}}$ ordinary of sheaf cohomology
of the topological space with complex coefficients, then when that vanishes.

As previously defined, $H^{0}(X,\mathcal{O}(D))=\bigoplus
H^{0}(X,\mathcal{O}(D))_{u}$ where
$H^{0}(X,\mathcal{O}(D))_{u}=H^{0}_{Z(u)}(|\Delta|)$, is the $0^{\text{th}}$
local cohomology group, in other words the relative group of the pair
consisting of the support and the complement of $Z(u)$, and is defined,
$H^{0}_{Z(u)}(|\Delta|)=H^{0}(|\Delta|,|\Delta|-Z(u))=\text{Ker}(H^{0}(|\Delta|)\rightarrow
H^{0}(|\Delta|-Z(u)))$. And this is generally true for higher local cohomology
groups $H^{p}_{Z(u)}=H^{p}(|\Delta|,|\Delta|-Z(u);\mathbb{C})$ and higher
sheaf cohomology groups $H^{p}(X,\mathcal{O}(D))$. And in fact,
$H^{p}(X,\mathcal{O}(D))\simeq\bigoplus H^{p}(X,\mathcal{O}(D))_{u}$ are the
canonical isomorphisms, where
$H^{p}(X,\mathcal{O}(D))_{u}=H^{p}_{Z(u)}(|\Delta|)$ for all $p\geq 0$.

If the support $|\Delta|$ and $|\Delta|-Z(u)$ are both convex then all the
higher cohomology groups vanish, and this is usually the case for general
affine varieties, and where $X$ is affine, the support is a cone and $\Psi$ is
linear. This goes further to say that all higher cohomology groups of an ample
line bundle on a complete toric variety vanish. And even further, if the
support is convex and $\mathcal{O}(D)$ is generated by its sections then
$H^{p}(X,\mathcal{O}(D))$ vanishes for all $p>0$.

If also $X$ is complete, then the Cardinal, $\text{Card}(P_{D}\cap
M)=\text{dim}H^{0}(X,\mathcal{O}(D))=\sum(-1)^{p}\text{dim}H^{p}(X,\mathcal{O}(D))=\chi(X,\mathcal{O}(D))$\-
with arithmetic genus given
$\text{dim}H^{0}(X,\mathcal{O}_{X})=1=\chi(X,\mathcal{O}_{X})$.

The $p^{\text{th}}$ cohomology of, $C^{*}$, the C̆ech complex, is
$H^{p}(\mathcal{O}(D))$, so that

$C^{p}=\bigoplus_{\sigma_{0},\ldots,\sigma_{p}}H^{0}(U_{\sigma_{0}}\cap\ldots\cap
U_{\sigma_{p}},\mathcal{O}(D)),$

the sum over all cones $\sigma_{i}$ in the fan, which is equal to

$\bigoplus_{u\in
M}\bigoplus_{\sigma_{0},\ldots,\sigma_{p}}H^{0}_{Z(u)\cap\sigma_{0}\ldots\sigma_{p}}(\sigma_{0}\cap\ldots\sigma_{p}).$

Again we are aware that for all cones $\tau$ and for all $u$ for all $i>0$
that $H^{i}_{Z(u)\cap|\tau|}(|\tau|)=0$.

If $Y$ is the union of a finite number of closed subspaces $Y_{j}$ and $Z$ is
taken to be an arbitrary closed subspace of $Y$, and
$Y^{\prime}=Y_{j_{0}}\cap\ldots Y_{j_{P}}$ The sheaf $\mathcal{F}$ is on $Y$
so that $H^{i}_{Z\cap Y^{\prime}}=(Y^{\prime},\mathcal{F})$ vanishes for every
$i$ greater than $0$ and all $Y^{\prime}=Y_{j_{0}}\cap\ldots Y_{j_{p}}$. Then
for the complex $C^{*}(\\{Y_{j}\\},\mathcal{F})$ whose $p^{\text{th}}$ term is
$C^{p}(\\{Y_{j}\\},\mathcal{F})=\oplus_{j_{0},\ldots,j_{p}}\Gamma_{Z\cap
Y_{j_{0}}\cap\ldots\cap Y_{j_{p}}}(Y_{j_{0}}\cap\ldots\cap
Y_{j_{p}}.\mathcal{F})$ where the rows are the resolutions of the complex
$\Gamma_{Z}(Y,\mathcal{I}^{*})$ for injective resolution $\mathcal{I}^{*}$ of
$\mathcal{F}$ with double complex $C^{*}(\\{Y_{j}\\},\mathcal{I}^{*})$.

And finally for $\Delta^{\prime}$ a refinement of $\Delta$, we have birational
proper map $f:X^{\prime}=X(\Delta^{\prime})\rightarrow X=X(\Delta)$ and
therefore $f_{*}(\mathcal{O}_{X^{\prime}})=\mathcal{O}_{X}$ and
$R^{i}f_{*}(\mathcal{O}_{X^{\prime}})=0$ for all $i>0$. So this means that if
we take $X^{\prime}$ to be the resolution of singularities then every $X$ has
rational singularities.

Briefly we can say that for $X=U_{\sigma}$,
$\Gamma(X^{\prime},\mathcal{O}_{X^{\prime}})=\Gamma(X,\mathcal{O}_{X})=A_{\sigma}$
and since $|\Delta^{\prime}|=|\sigma|$, then
$H^{i}(X^{\prime},\mathcal{O}_{X^{\prime}})=0$ for all $i>0$.

To create the momentum polytopes from the toric varieties

Since toric varieties are naturally defined over integers, we are therefore
not restricted to complex toric varieties, as we have been looking at thus
far. We can rewrite the algebras with $\mathbb{Z}$ instead of $\mathbb{C}$:
$U_{\sigma}=\text{Spec}(\mathbb{Z}[\sigma^{\smallsmile}\cap M])$. In fact for
any field $K$, and $K$ is the multiplicative semigroup $K^{*}\cup\\{0\\}$ (so
that for $K=\mathbb{R}\subset\mathbb{C}$, the real points of the toric
variety) the semigroup homomorphisms $\text{Hom}_{sg}(\sigma^{\smallsmile}\cap
M,K)$ describe the $K$-valued points of $U_{\sigma}$.

This is also true for $K$ just a sub-semigroup of the complex field. For
$\mathbb{R}_{\geq}$ then we have the retraction $z\mapsto|z|$. In fact the
topological subspace, with retraction $U_{\sigma}\rightarrow(U\sigma)_{\geq}$
is determined $(U_{\sigma})_{\geq}=\text{Hom}_{sg}(\sigma^{\smallsmile}\cap
M,\mathbb{R}_{\geq})\subset
U_{\sigma}=\text{Hom}_{sg}(\sigma^{\smallsmile}\cap M,\mathbb{C})$ just as
$\mathbb{R}_{\geq}\subset\mathbb{C}\rightarrow\mathbb{R}_{\geq}$.

For the varieties we establish a corresponding retraction,
$X(\Delta)_{\geq}\subset X(\Delta)\rightarrow X(\Delta)_{\geq}$.

An example, for $N$ with partial basis $e_{1},\ldots,e_{k}$ that generate
$\sigma$, $(U_{\sigma})_{\geq}$ is isomorphic to $n-k$ copies of $\mathbb{R}$
and $k$ copies of $\mathbb{R}_{\geq}$. So considering the nonsingular case,
$X_{\geq}$ is therefore a manifold with corners. And when $X$ is singular,
looking at the retraction $X_{\geq}$ can expose worse singularities.

The retraction $X(\Delta)\rightarrow X(\Delta)_{\geq}$, the result of which is
simply the quotient space of $X(\Delta)$ that results from the action of the
compact torus $S_{N}$.

Let’s look at when the toric variety is the $n$-dimensional projective space,
then $\mathbb{P}^{n}_{\geq}=\mathbb{R}_{\geq}^{n+1}-\\{0\\}/\mathbb{R}^{+}$.
If we consider the usual covering by affine open sets $U_{i}=U_{\sigma_{i}}$
then $(U_{i})_{\geq}$ is made up of the points $(t_{0}:\ldots:1:\ldots:t_{n})$
where $t\geq 0$. And therefore
$\mathbb{P}^{n}_{\geq}=\\{(t_{0},\ldots,t_{n})\in\mathbb{R}^{n+1}\ \text{such
that}\ t_{i}\geq 0\ \text{and}\ t_{0}+\ldots t_{n}=1\\}$. This is the standard
simplex and so

$(x_{0}:\ldots:x_{n})\mapsto\frac{1}{\sum|x_{i}|}(|x_{0}|,\ldots,|x_{n}|)$

defines the retraction $\mathbb{P}^{n}\rightarrow\mathbb{P}^{n}_{\geq}$. The
compact torus whose dimension is one less than $\text{Card}(i:t_{i}\neq 0)$ is
exactly the fiber over the point $(t_{0},\ldots,t_{n})$.

The unit circle in $\mathbb{C}^{*}$ is denoted $S^{1}$ which is equal to
$U(1)$. The compact torus $S_{N}$ which is $\text{Hom}(M,S^{1})$ is clearly a
subset of $\text{Hom}(M,\mathbb{C}^{*})$ which is the algebraic torus. And so
$T_{N}=S_{N}\times\text{Hom}(M,\mathbb{R}^{+})=S_{N}\times\text{Hom}(M,\mathbb{R})=S_{N}\times
N_{\mathbb{R}}$.

Therefore The retraction $(O_{\tau})_{\geq}=X(\Delta)_{\geq}\cap
O_{\tau}=\text{Hom}(\tau^{\perp}\cap M,\mathbb{R}^{+})$, and as
$mathbb{R}^{+}$ is isomorphic with $\mathbb{R}$, this equals
$\text{Hom}(\tau^{\perp}\cap M,\mathbb{R})=N(\tau)_{\mathbb{R}}$. Therefore
$(O_{\tau})_{\geq}=N(\tau)_{\mathbb{R}}$ is the quotient space according to
the faithful action of $S_{N(\tau)}$ on $O_{\tau}=T_{N(\tau)}$. It is
according to the projection $S_{N}$ to $S_{N(\tau)}$ that its action on
$O_{\tau}$ is defined. This $S_{N(\tau)}$ (with
$\text{dim}S_{N(\tau)}=n-\text{dim}(\tau))$ is exactly the fiber of the
retraction $X\rightarrow X_{\geq}$ over $(O_{\tau})_{\geq}$. Just as how the
orbits $O_{\tau}$ fit together in $X$, so do the retracted spaces
$(O_{\tau})_{\geq}$ fit in the retracted variety $X_{\geq}$.

Consider the construction. A manifold whose corners are $X_{\geq}$ creates, if
$X$ is complete, a sort of dual polyhedron to the original fan, where each
cone of $n$-dimension has a vertex for each $X_{\geq}$. If cones share a
$(n-1)$ face then two vertices join at an edge. And $X_{\geq}$ is homeomorphic
to $P$ if $\Delta=\Delta_{P}$ comes from a convex polytope in
$M_{\mathbb{R}}$.

Another construction. For $r\in\mathbb{R}^{+}$ then the mapping $t\mapsto
t^{r}$ is an automorphism of $\mathbb{R}_{\geq}$. Therefore determining the
automorphism $(U_{\sigma})_{\geq}=\text{Hom}_{sg}(\sigma^{\smallsmile}\cap
M,\mathbb{R}_{\geq})$ that fit together to define the homeomorphism of
$X_{\geq}$ to itself. For $r\in\mathbb{Z}^{+}$, with map $z\mapsto z^{r}$, it
is an endomorphism of $\mathbb{C}$ with similar auto-endomorphism
$X_{\geq}\subset X\rightarrow X_{\geq}$.

Another construction. In accordance with the action of $T_{N}$ on $X$, we can
consider the action of the quotient $T_{N}/S_{N}=N_{\mathbb{R}}$ on
$X/S_{N}=X_{\geq}$. And the inclusion $X_{\geq}\subset X$ that is equivariant
since $N_{\mathbb{R}}=\text{Hom}(M,\mathbb{R}^{+})\subset
T_{N}=\text{Hom}(M,\mathbb{C}^{*})$.

Another construction. X is the quotient space that arises from $S_{N}\times
X_{\geq}\rightarrow X$.

Another construction. An example of a deformation retract, for any $\tau$
would be the inclusion $(O_{\tau})_{\geq}\subset(U\tau)_{\geq}$.

A momentum map can be described as the action of a Lie group on a varieties.
It is possible to construct maps for toric varieties and then draw the
connection to momentum maps.

A convex polytope $P$ in $M_{\mathbb{R}}$ with vertices in $M$, gives a toric
variety $X(\Delta_{P})$ and morphism $\varphi$ from $X$ to the projective
space of dimension $r-1$, ($\varphi:X\rightarrow\mathbb{P}^{r-1}$).

For Cartier divisor $D$ preserved by $T$, $\chi(u)$ is a function for unique
$\sigma^{\smallsmile}\cap M$ that generates the corresponding fractional
ideal.

The momentum map $\mu:X\rightarrow M_{\mathbb{R}}$ is defined by

$\mu(x)=\frac{1}{\sum|\chi^{u}(x)|}\sum\limits_{u\in P\cap M}|\chi^{u}(x)|u$

where $\chi^{u}$ for $u\in P\cap M$ are the sections of $\varphi$ and
specifically, $|\chi^{u}(x)|=|\chi^{u}(t\cdot
x)|=|\chi^{u}(t)|\cdot|\chi^{u}(x)|=|\chi^{u}(t\cdot x)|$, for $t\in S_{N}$
and $x\in X$ and $\mu$ $S_{N}$-invariant.

In other words,

$\tilde{\mu}:X_{\geq}\rightarrow M_{\mathbb{R}},$

is the map prompted on the quotient space $X/S_{N}=X_{\geq}$ by $\mu$.

Now for $Q$ a face of polytope $P$, with cone of the fan $\sigma$, the above
defined $\tilde{\mu}$ does in fact map
$(O_{\sigma})_{\geq}\xrightarrow{\simeq}\text{Int}(Q)$ bijectively, where
$\text{Int}(Q)$ is the interior of $Q$. If we introduce
$\sum|\chi^{u^{\prime}}(x)|$ as the sum over all $u^{\prime}$ in $P\cap M$ or
in a subset that contains vertices of $P$, then let
$\rho_{u}(x)=|\chi^{u}(x)|/\sum|\chi^{u^{\prime}}(x)|$ therefore
$\mu(x)=\sum\rho_{u}(x)u$ where of course $0\leq\rho_{u}(x)\leq 1$. For $x$ in
$(O_{\sigma})_{\geq}$ is also in the inclusion created by extending otside of
$\sigma^{\perp}$ by zero. $\text{Hom}(\sigma^{\perp}\cap
M,\mathbb{R}^{+}\subset\text{Hom}_{sg}(\sigma^{\perp}\cap
M),\mathbb{R}_{\geq})$. For that same $x$, if $u\in Q$ then $\rho_{u}(x)>0$
and for $u\notin Q$ then $\rho_{u}(x)$ is exactly zero.

For the finite set of vectors $u_{1},\ldots,u_{r}$ in the dual space $V^{*}$
of the finite dimensional real vector space $V$, if $K$ is the convex hull of
these vectors, and is not contained in a hyperplane. Then for
$\varepsilon_{1},\ldots,\varepsilon_{r}$ that represent any positive numbers,
then the map $\rho_{i}$ that takes $V$ to $\mathbb{R}$ is defined as

$\rho_{(}x)=\varepsilon_{i}e^{u_{i}(x)}/(\varepsilon_{1}e^{u_{1}(x)}+\ldots+\varepsilon_{r}e^{u_{r}(x)})$

and therefore the map $\mu:V\rightarrow V^{*}$ is the real analytic
isomorphism of $V$ onto $\text{Int}K$ and is defined,
$\mu(x)=\rho_{1}(x)u_{1}+\ldots+\rho_{r}(x)u_{r}$.

In fact due to the actions of $T_{N}$ on $X_{P}$ and $(\mathbb{C}^{*})^{r}$ on
$\mathbb{P}^{r-1}$ then we have a map from $X_{P}$ to $\mathbb{P}^{r-1}$,
where
$T_{N}=\text{Hom}(M,\mathbb{C}^{*})\rightarrow\text{Hom}(\mathbb{Z}^{r},\mathbb{C}^{*})=T$
by the map $\mathbb{Z}^{r}\rightarrow M$ which takes basic vectors to points
of $P\cap M$. And therefore we have a momentum map
$\mathfrak{M}:\mathbb{P}^{r-1}\rightarrow\text{Lie}S^{*}=\mathbb{R}^{*}=\mathbb{R}$
for the action of $(S^{1})^{r}$ on $\mathbb{P}^{r-1}$. Specifically we have

$\mathfrak{M}(x)=\frac{1}{\sum|x_{i}|^{2}}\sum^{r}_{i=1}|x_{i}|^{2}e_{i}^{*}$

up to a scalar factor, for $x\in\mathbb{P}^{r-1}$ and
$v=(x_{1},\ldots,x_{r})\in\mathbb{C}^{r}$.

And in fact,

$X_{P}\xrightarrow{\varphi}\mathbb{P}^{r-1}\xrightarrow{\mathfrak{P}}(\mathbb{R}^{r})^{*}\rightarrow
M_{\mathbb{R}}$

.

∎

## Acknowledgements

The author is funded by the Marie-Curie individual fellowship and would like
to thank the European Commission for fully funding project 898145 —
ROBOTTOPES. The author would also like to thank Professor Manuel de Leon for
bringing working on contact manifolds into consideration and reading drafts.
The author would also like to thank Yael Karshon for some consultations.

## References

  * [1] Y. Lin and R. Sjamaar.  Convexity properties of presymplectic moment maps.  J. of Sympl. Geom., 17:1159 – 1200, 2019.
  * [2] R. Chiang and Y. Karshon.  Convexity package for momentum maps on contact manifolds.  J. of Algebr. Geom. Topol., 10:925 – 977, 2010.
  * [3] E. Lerman.  A convexity theorem for torus actions on contact manifolds.  J. Math., 46:171 – 184, 2002.
  * [4] C. P. Boyer and K. Galicki.  A note on toric contact geometry.  J. of Geom. and Phys., 35:288–298, 2000.
  * [5] Molino P. Banyanga, A.  Complete integrability in contact geometry.  Nankai Tracts in Mathematics, 15:107 – 158, 2016.
  * [6] I. V. Danilov.  The geometry of toric varieties.  Russian Math. Surveys, 33:97 – 154, 1978.
  * [7] W. Fulton.  Introduction of toric varieties.  Princeton University Press, Princeton, New Jersey, 1993.
  * [8] T. Oda.  Lectures on torus embeddings and applications.  Tata Inst. of Fund. Research, 58:Springer, 1978.
  * [9] T. Oda.  Convex Bodies and Algebraic Geometry.  15, Springer-Verlag, 1988.
  * [10] S. Druel.  Structures de contact sur les variétés toriques.  Math. Ann., 3:429 – 435, 1999.
  * [11] S. Mori.  Threefolds whose canonical bundles are not numerically effective.  Ann. Math., 116:113 – 176, 1982.
  * [12] J. Wisniewski.  Length of extremal rays and generalised adjunction.  Math. Z., 200:409 – 427, 1989.
  * [13] J. Wisniewski.  On contractions of extremal rays on fano manifolds.  J. Riene U. Angew Math., 417:141 – 157, 1991.