\section{Preliminaries on toric varieties}\mylabel{sec:torvar} 
 
This section will mostly serve to  remind the reader of the 
definition and some properties of smooth toric varieties as well 
as to fix the notation.
Of course, everything is already well known, see for example 
(in alphabetic order) \cite{aud91,bat93,cox97,dan78,del88,ful93,oda88}.


\subsection{The algebro--geometric construction of toric varieties} 

 
For all what follows we will fix the following notation: Let $d>0$ be a positive 
integer. Let $N=\Z^d$ be the $d$--dimensional integral lattice, and $M=\Hom(N, \Z)$ 
be its dual space. Moreover, let $N_\R=N\otimes_\Z\R$ and $M_\R=M\otimes_Z\R$ 
be the $\R$--scalar extensions of $N$ and $M$, respectively. 
 
A convex subset $\sigma\subset N_\R$ is called 
a {\em regular $k$--dimensional cone} if there exists 
a $\Z$--basis $v_1, \ldots, v_k, \ldots, v_d$ of $N$ such that the cone $\sigma$ is 
generated by $v_1,\ldots, v_k$. The vectors $v_1,\ldots, v_k\in N$ are the 
{\em integral generators of $\sigma$}. The origin $0\in N_\R$ is the only 
{\em regular zero dimensional cone}. Its set of integral generators is empty.
A {\em face} of a regular cone $\sigma$ is a cone $\sigma'$ generated by a subset of 
the integral generators of $\sigma$. If $\sigma'$ is a (proper) face of $\sigma$, we will 
write $\sigma'\prec\sigma$. 
 
A finite system $\Sigma=\{\sigma_1, \ldots, \sigma_s\}$ of regular cones in 
$N_\R$ is called a regular {\em $d$--dimensional fan} of cones, 
if any face $\sigma'$ of a cone $\sigma\in\Sigma$ in the fan and any intersection
of two cones $\sigma_1, \sigma_2\in\Sigma$ are again in the fan.
A fan $\Sigma$ is called a {\em complete fan} if the (set theoretic) 
union of all cones $\sigma_i$ in $\Sigma$ is all of $N_\R$, \ie. 
$N_\R= \bigcup_i \sigma_i$.
The {\em $k$--skeleton} $\Sigma^{(k)}$ of the fan $\Sigma$ 
is the set of all $k$--dimensional cones in $\Sigma$. 
 
By abuse of language, we will also consider cones $\sigma$ as fans, meaning 
in fact the fan $\Sigma_\sigma$ of $\sigma$ and all its faces: 
$ \Sigma_\sigma = \{ \sigma' \, | \, \sigma'\preccurlyeq\sigma\}$.
 
A subset ${\primcol} \subset \Sigma^{(1)}$ of the $1$--skeleton of a fan $\Sigma$ is 
called a {\em primitive collection} of $\Sigma$ (see \cite{bat91}) 
if ${\primcol}$ is not the set of generators of a cone in $\Sigma$, 
while any proper subset of $\primcol$ is. We will denote the set of primitive 
collections of $\Sigma$ by $\primcolset$. 
 
Let $n=\module{\smash{\Sigma^{(1)}}}$ be the cardinality of the one--skeleton of 
$\Sigma$, and $v_1, \ldots, v_n$ its elements. Let $z_1, \ldots, z_n$ be a set of 
coordinates in $\C^n$ and let 
$\iota: \C^n \longrightarrow N\otimes_\Z\C$ be a linear map such that $\iota(z_i)=v_i$. 
For each primitive collection $\primcol\in\primcolset$, 
$\primcol=\{v_{i_1},\ldots, v_{i_p}\}$, 
we define an $(n-p)$--dimensional affine subspace in $\C^n$ by 
\[ \AA(P) := \{(z_1,\ldots, z_n)\in \C^n \, |\, z_{i_1}=\ldots=z_{i_p}=0\}.\] 
Moreover, we define the set $U(\Sigma)$ to be the open algebraic subset of $\C^n$ 
given by 
\[ U(\Sigma) =\C^n - \bigcup_{\primcol\in\primcolset} \AA(P).\] 
 
The map $\iota: \C^n \longrightarrow N_\C$ induces a map between tori 
$(\C^*)^n \longrightarrow (\C^*)^d$ that we will also call $\iota$. Here, $\C^*=\C-\{0\}$. 
Let $\DD(\Sigma) := \ker (\iota: (\C^*)^n \longrightarrow (\C^*)^d)$ be the kernel of
this map, an $(n-d)$--dimensional subtorus.
 
\begin{defi}\mylabel{def:vt} 
Let $\Sigma$ be a regular $d$--dimensional\footnote{A $d$--dimensional fan is a fan 
in $\Z^d$ containing a cone of dimension $d$.} fan of regular cones. 
The quotient 
$\vt := U(\Sigma) / \DD(\Sigma)$
is called the {\em toric manifold associated with $\Sigma$}.  
\end{defi} 
 
The following proposition provides us with an atlas of charts for toric manifolds.

\begin{prop}\mylabel{thm:homcoords} 
Let $\sigma\in\Sigma^{(k)}$, and let $\{v_{i_1},\ldots, v_{i_k}\}$ be its 
set of generators. Let $\{v_{i_1},\ldots, v_{i_d}$\} be a $\Z$--basis of $N=\Z^d$ completing the 
set of generators of $\sigma$, and let $u_1,\ldots, u_d$ be its dual basis of $M=\Hom(N,\Z)$. 
Define the open subset $V(\sigma)\subset\C^n$ by 
\[ V(\sigma)=\left\{(z_1,\ldots, z_n)\,|\,z_j\neq0 \text{ for $j\notin \{i_1,\ldots, i_k\}$}\right\}.\] 
 
These open sets $V(\sigma)$ satisfy the following properties: 
\begin{enumerate} 
\item $U(\Sigma)=\bigcup_{\sigma\in\Sigma^{(d)}} V(\sigma)$; 
\item if $\sigma'\prec\sigma$, then $V(\sigma')\subset V(\sigma)$; 
\item $V(\sigma)$ is isomorphic to $\C^k\times(\C^*)^{n-k}$, and 
the torus $\DD(\Sigma)$ acts freely on $V(\sigma)$. The quotient $U_\sigma:=V(\Sigma)/\DD(\Sigma)$ 
is the toric subvariety associated to the cone $\sigma\in\Sigma$, whose co--ordinate 
functions $x_1^\sigma,\ldots, x_d^\sigma$ are the following Laurent monomials 
in $z_1,\ldots, z_n$: 
\[ x_j^\sigma=z_1^{\langle v_1, u_j\rangle}\cdots z_n^{\langle v_n,u_j\rangle}.\] 
\end{enumerate} 
\end{prop} 
 
\begin{rema} 
Note that our notation is slightly different to Batyrev's in \cite{bat93}: he defines the open 
sets $U(\Sigma)$ just for (complete) fans, while he calls $U(\sigma)$ what we call $V(\sigma)$. 
\end{rema} 
 
From now on we will consider at complete regular fans $\Sigma$ of regular cones.
 

\subsection{Support functions of a fan and dual polyhedra} 

 
A continuous function $\varphi : N_\R \mapright \R$ is called 
{\em $\Sigma$--piecewise linear}, if $\varphi$ is linear 
on every cone of $\Sigma$. Let $PL(\Sigma)$ be the set of all 
$\Sigma$--piecewise linear functions. Note that $PL(\Sigma)\cong \R^n$ since
$\Sigma$--piecewise linear functions are given by their values on the 
$1$--skeleton of $\Sigma$.

Such a function 
$\varphi\in PL(\Sigma)$ is called {\em upper convex} if for any $x,y\in N_\R$,
$\varphi(x)+\varphi(y)\ge\varphi(x+y)$. If moreover for any two different
$d$--dimensional cones $\sigma_1, \sigma_2\in\Sigma$, the restrictions
$\varphi_{|\sigma_1}$ and $\varphi_{|\sigma_2}$ are different 
linear functions, then $\varphi$ is called {\em strictly upper convex support
function} for $\Sigma$.

\begin{prop}\mylabel{prop:sucsp1} 
A $\Sigma$--piecewise linear function $\varphi$ is a strictly upper convex 
support function if and only if for all primitive collections $\primcol\in\primcolset$, 
$\primcol=\{v_{i_1},\ldots,v_{i_k}\}$, the following inequality holds: 
\[ \varphi(v_{i_1})+\ldots+\varphi(v_{i_k}) > \varphi(v_{i_1}+\ldots+ v_{i_k}).\] 
\end{prop} 
 
We will give another criterion in terms of convex polytopes that will 
be useful in particular for the construction via a moment map: 
 
\begin{prop}\mylabel{prop:sucsp2} 
Let $\Sigma$ be a complete, regular fan in $N=\Z^d$. 
Let $\varphi\in PL(\Sigma)$ be a $\Sigma$--piecewise linear function on $\Sigma$. 
Define a polytope $\Delta_\varphi\in M$  by 
\[ \Delta_\varphi = \{ m \in M_\R\, | \, \langle m,n\rangle \ge -\varphi(n), \,\,\forall n\in N\, \}.\] 
 
Then the function $\varphi$ is a strictly upper convex support function if and 
only if the integral convex polytope $\Delta_\varphi$ is $d$--dimensional 
and has exactly $\{l_\sigma \, |\, \sigma\in\Sigma^{(d)}\}$ as the set of 
its vertices. Here, the $l_\sigma\in M_\R=\Hom(N, \R)$ are given by 
$l_\sigma=\varphi_\sigma$.
\end{prop} 
 

\subsection{Divisors, cohomology and first Chern class} 


The cohomology of a toric manifold $X_\Sigma$ is generated by its $T^N$--invariant
divisors $D_1, \ldots, D_n$ that are given by $D_i=(\{z_i=0\}\cup U(\Sigma))/\DD(\Sigma)$.
To each $\Sigma$--piecewise 
linear functions $\varphi: N_\R \longrightarrow \R$ we can associate a divisor by setting 
$ D_\varphi = \sum_{i=1}^n \varphi(v_i) D_i$, yielding a canonical isomorphism 
$H^2(\vt,\R)\cong PL(\Sigma)/M_\R$.

The cohomology ring of $\vt$ is therefore the quotient of $\R[Z_1, \ldots, Z_n]$ 
by an ideal of relations. As we have seen above, the ideal of linear relations is
$\Lin(\Sigma):=\langle \sum_i u_1(v_i)Z_i, \ldots, \sum_i u_d(v_i) Z_i \rangle$, where
$u_1,\ldots ,u_d$ is some basis of $M=\Hom(N, \Z)$. 
The higher--degree relations in the cohomology ring are given by the so--called
Stanley--Reisner ideal 
$SR(\Sigma):= \left\langle\prod_{v_j\in\primcol} Z_j\right\rangle_{\primcol\in\primcolset}$.

\begin{prop}\mylabel{prop:homcomtor}
The cohomology ring of the compact toric manifold $\vt$ is canonically isomorphic
to the quotient of $\R[Z_1,\ldots, Z_n]$ by the ideal $\Lin(\Sigma)+SR(\Sigma)$:
\[ H^*(\vt, \R) \cong \R[Z_1, \ldots, Z_n]/(\Lin(\Sigma)+SR(\Sigma)).\]
Moreover, the first Chern class $c_1(\vt)$ of $\vt$ is represented by $Z_1+\cdots+Z_n$.
\end{prop} 

Dually, let $R(\Sigma)\subset \Z^n$ be the subgroup of $\Z^n$ defined by 
\[ R(\Sigma) = \{ (\lambda_1,\ldots,\lambda_n) \in \Z^n \,|\, 
\lambda_1 v_1+\ldots+\lambda_n v_n =0\} \cong \Z^{n-d}.\] 
Then the group $R(\Sigma)_\R=R(\Sigma)\otimes_\Z\R$ of $\R$--linear 
extensions of $R(\Sigma)$ is canonically isomorphic to $H_2(\vt,\R)$.
 
The pairing $H^2(\vt,\R)\otimes H_2(\vt,\R)\longrightarrow \R$ lifts to 
$PL(\Sigma)\otimes R(\Sigma)_R$ and is given there by the degree map: 
\[ \deg_\varphi(\lambda) = \sum_{i=1}^n \lambda_i\varphi(v_i).\] 
 


\subsection{Toric manifolds as symplectic quotient}\mylabel{sec:vtsq} 

 
\begin{defi}\mylabel{def:kcone} 
As before let $\Sigma$ be a complete, regular cone in $N$. 
Denote by $K(\Sigma)$ the cone in 
$H^2(\vt, \R)\cong PL(\Sigma)/M_\R$ consisting of the classes of all 
upper convex support function $\varphi$ for $\Sigma$. We denote by 
$K^\circ(\Sigma)$ the interior of $K(\Sigma)$, \ie. the cone consisting 
of the classes of all strictly convex upper support functions in $PL(\Sigma)$. 
\end{defi} 
 
\begin{prop} 
The open cone $K^\circ(\Sigma)\subset H^2(\vt,\R)$ consists of 
classes of K\"ah\-ler $(1,1)$--forms on $\vt$, \ie. $K(\Sigma)$ is 
isomorphic to the closed K\"ahler cone of $\vt$. 
\end{prop} 

If the K\"ahler cone is non--empty, the toric manifold can be constructed
as a symplectic quotient as follows. 
The $n$--dimensional complex space $\C^n$ has a natural 
symplectic structure. 
Remember from above, that $\DD(\Sigma)$ is an algebraic subtorus of $(\C^*)^n$, 
thus acting on $\C^n$. Let $G\cong (S^1)^{n-d}$ be the maximal compact subgroup 
of $\DD(\Sigma)$. Since $\DD(\Sigma)\subset(\C^*)^n$ acts as a subtorus, so does 
$G \subset T^n$. The action of $G\subset T^n$ is naturally Hamiltonian, and we 
obtain its moment map $\mu$ by composing the moment map $\mu_{T^n}$ of the 
$n$--dimensional torus action on $\C^n$ with the restriction map 
$\beta^*: (\gott^n)^* \longrightarrow \gotg^* $: 
\[ \mu: \C^n \MRE{\mu_{T^n}} (\gott^n)^* \MRE{\beta^*} \gotg^*.\] 
 
For almost all $\xi\in \gotg^*$, the moment map is regular. Moreover,
the action of $G$ on the level set $\mu^{-1}(\xi)$ is effective if and only 
if $\mu^{-1}(\xi)\subset U(\Sigma)$, the open subset of $\C^n$ used for the 
algebro--geometric quotient. 
 
\begin{theo}[\cite{del88}] 
Let $\vt$ be a projective simplicial toric variety. Then there exists a regular 
value $\xi\in\gotg^*$ of the moment function $\mu:M\longrightarrow \gotg^*$ such 
that the level set $\mu^{-1}(\xi) \subset U(\Sigma)$ is in the effective 
subset of the action $G$, and there is a diffeomorphism 
\[ \mu^{-1}(\xi)/G \mapright \vt \] 
preserving the cohomology class of the symplectic form. \proofend
\end{theo} 

\begin{prop} 
Let $\varphi$ be the strictly upper convex support function associated
with the symplectic form $\omega_\xi$ of the quotient $\mu^{-1}(\xi)/G$.
Then the polytope $\Delta_\varphi$ is the moment polytope of the induced 
Hamiltonian $T^N$--action on $\vt$. \proofend
\end{prop}
{"config": "arxiv", "file": "math0006156/torvarn.tex"}