text:
\begin{document}

\baselineskip=16pt

\title[Toric degenerations of weight varieties]{Toric degenerations of weight varieties and applications}

\author[P. Foth, Yi Hu]{Philip Foth and Yi Hu}

\address{Philip Foth  \newline
Department of Mathematics, University of Arizona, Tucson, AZ
85721-0089, USA}

\email{foth@math.arizona.edu}

\address{Yi Hu \newline
Department of Mathematics, University of Arizona, Tucson, AZ
85721-0089, USA \newline
 Center for Combinatorics, IPMC, Nankai
University, Tianjin 300071, China}

\email{yhu@math.arizona.edu}

\subjclass{}

\keywords{}

\date{May 01, 2004}

\begin{abstract} We show that a weight variety, which is
a quotient of a flag variety by the maximal torus, admits a flat
degeneration to a toric variety. In particular, we show that the
moduli spaces of spatial polygons degenerate to polarized toric
varieties with the moment polytopes defined by the lengths of their
diagonals. We extend these results to more general
Flaschka-Millson hamiltonians on the quotients of products of
projective spaces. We also study boundary toric divisors and
certain real loci.
\end{abstract}

\maketitle

\section{Introduction}

Let $G$ be a complex connected semisimple Lie group and let $X$ be
a flag variety of $G$, parameterizing parabolic subgroups of a
given type. Several authors studied flat degenerations of $X$ to a
toric variety, starting with the work of Gonciulea and Lakshmibai
\cite{GL}. Caldero \cite{Caldero} used Kashiwara-Lusztig's
canonical bases and their string parameterization in his
construction, which was later extended by Alexeev-Brion
\cite{AlexBrion} to the case of spherical varieties. In this paper
we use their methods to construct flat degenerations of weight
varieties, which are, by definition \cite{Knutson}, quotients of
the flag varieties by the action of the maximal torus. The
resulting polytopes are certain ``slices'' of the string
polytopes.

Interesting examples of weight varieties are the quotients of
complex grassmannians. For example, the quotients of ${\rm
Gr}_{\C}(2, n)$ can be identified \cite{HK} with the moduli spaces
of spatial $n$-gons studied by Klyachko \cite{Klyachko},
Kapovich-Millson \cite{KM}, and many other authors. There are
remarkable integrable systems on these spaces, where the action
variables are given by the lengths of diagonals emanating from a
fixed vertex and the angle variables define the so-called bending
flows, which have a transparent geometric meaning \cite{KM}. As an
application, we show that there exist flat degenerations of the
moduli spaces of polygons to polarized toric varieties, {\it toric
polygon spaces}, whose moment polytopes are defined by the action
variables. These action variables can be computed using the
Gelfand-Tsetlin functions and the Gelfand-MacPherson
correspondence \cite{HK}. Our results can also be generalized for
other grassmannians and the bending flows defined by the
Flaschka-Millson hamiltonians \cite{FlMill}.

We show that there are some real cycles of codimension 2 on the
moduli spaces of polygons which degenerate to the toric
subvarieties corresponding to the facets of the polytopes. We also
extend the work of Kamiyama-Yoshida \cite{Kamiyama} and construct
topological spaces with compact torus action, which conjecturally
are homeomorphic to the central fibers of the flat families.

In the last section of the paper we consider real loci of the
aforementioned spaces and show that they map surjectively to the
moment polytopes and compute the cardinality of the fibers of
these maps.

Let $\overline{M}_{0,n}$ be the moduli space of stable $n$-pointed
rational curves. It can be realized \cite{Kapranov} as the Chow
quotient of the Grassmannian ${\rm Gr}_{\C}(2,n)$. For an
alternative construction using stable polygons, see \cite{Hu99}.
Then $\overline{M}_{0,n}$ can be flatly degenerated to a toric
moduli space whose associated fan is the common refinement of the
fans of all the toric polygon spaces. This and related topics will
appear in a forthcoming paper.

\section{Toric degenerations of weight varieties.}

In this section we use toric degenerations of flag varieties
constructed by Caldero \cite{Caldero} to construct toric
degenerations of weight varieties \cite{Knutson}, which are
defined as GIT quotients of flag varieties by the action of the
maximal torus.

Let $G$ be a connected complex semisimple group, $B$ a Borel
subgroup, $U$ its unipotent radical, and $H$ a Cartan subgroup
such that $B=HU$. Let also $\Phi=\Phi(G, H)$ be the system of
roots, $\Phi^+=\Phi^+(B, H)$ the subset of positive roots, and $\{
\alpha_1, ...,\alpha_r\}$ the basis of simple roots, where $r$ is
the rank of $G$. Let $\Lambda$ be the weight lattice of $G$ and
$\Lambda^+$ the subset of dominant weights. For
$\lambda\in\Lambda^+$ we denote by $V(\lambda)$ the irreducible
$G$-module with highest weight $\lambda$. Let $P_\lambda\supset B$
be the parabolic subgroup of $G$ which stabilizes a highest weight
vector in $V(\lambda)$. Also denote by $L_\lambda = G
\times_{P_\lambda} \C$ the $G$-linearized line bundle on
$X_\lambda:=G/P_\lambda$ corresponding to the character $\lambda$
extended to $P_\lambda$.

Let $W$ be the Weyl group and $w_0\in W$ the longest element of
length $\ell$. Choose a reduced decomposition
$$
\underline{w_0}=s_{i_1}s_{i_2}\cdots s_{i_\ell}
$$
into a product of simple reflections. Recall that there is a
rational convex polyhedral cone ${\mathcal
C}_{\underline{w_0}}\subset \Lambda_\R\times \R^\ell$ such that
the integral points in the {\it string polytope} $Q(\lambda)$
correspond to the elements of the (dual) canonical basis (for the
chosen string parameterization) $(b_{\lambda,\phi})$ for any fixed
$\lambda\in\Lambda^+$. For a fixed $\lambda$, we may identify
$Q(\lambda)$ with its image in $\R^\ell$ via the projection
$\Lambda_\R \times \R^\ell \to \R^\ell$.  If we denote the string
parameterization by
$$
b_{\lambda,\phi}\mapsto (\lambda, t_1, ..., t_\ell)\in
\Lambda^+\times\N^\ell,
$$
then the projection
$$
\pi_\lambda:\ \R^\ell\to \Lambda_\R, \ \ (t_1, ..., t_\ell)\mapsto
-\lambda+t_1\alpha_{i_1}+\cdots +t_\ell\alpha_{i_\ell}
$$
maps the string polytope $Q(\lambda)$ onto the convex hull of the
Weyl group orbit of the dual weight $\lambda^*=-w_0\lambda$:
$$
\pi_\lambda(Q(\lambda))={\rm Conv}(W.\lambda^*)=-{\rm
Conv}(W.\lambda):=\Delta(\lambda)
$$

Caldero \cite{Caldero} (see also \cite{AlexBrion}) has constructed
a flat deformation of the polarized flag variety $(X_\lambda,
L_\lambda)$ to a polarized toric variety $(X_{\lambda; 0},
L_{\lambda; 0})$ such that the corresponding moment polytope is
exactly the string polytope $Q(\lambda)$. His construction is
based on the key multiplicative property of the (dual) canonical
basis:
$$
b_{\lambda_1, \phi_1}b_{\lambda_2, \phi_2}=b_{\lambda_1+\lambda_2,
\phi_1+\phi_2}+ \sum_{\phi < \phi_1+\phi_2} {\rm
coeff}.b_{\lambda_1+\lambda_2, \phi},
$$
which allows to put a filtration on the homogeneous coordinate
ring which is compatible with grading and the torus action such
that the associated graded ring is the algebra of
the semigroup of integral points in a rational convex polyhedral
cone, see \cite{Caldero}, \cite{AlexBrion} for details.

Let $\Phi_\lambda: X_{\lambda; 0} \rightarrow Q(\lambda) \subset
\R^\ell$ be the moment map for ${\mathbb T}$-action, where
${\mathbb T}$ is the compact part of $(\C^*)^\ell$. Then the
composition $\pi_\lambda \circ \Phi_\lambda: X_{\lambda; 0} \to
\Delta_\lambda \subset \Lambda_\R$ is the moment map for the torus
$H$.

Recall that a {\it weight variety} $M_{\lambda; \mu}$ is a GIT
quotient of $X$ by the action of $H$ associated with an integral
point $\mu \in \Delta(\lambda)$. More precisely, let $\mu \in
\Delta(\lambda)$ be a character of $H$, and $L_\lambda (-\mu)$ be
the linearized line bundle $L_\lambda$ twisted by $-\mu$. Then
$M_{\lambda; \mu} = X_\lambda^{ss}(L_\lambda (-\mu))/\!/G$. In
symplectic terms, the $H$- moment map for the polarized variety
$(X_\lambda,L_\lambda (-\mu))$ is $\pi_\lambda \circ \Phi_\lambda
- \mu$, and $M_{\lambda; \mu}$ is identified with the reduction of
$X_\lambda$ at the level $\mu$. The line bundle $L_\lambda (-\mu)$
descends to the quotient $M_{\lambda; \mu}$. Let us denote the
descended line bundle by $L_{\lambda;\mu}$.

\begin{theorem} There exists a flat degeneration of the weight variety
$M_{\lambda; \mu}$ to a projective toric variety $N_{\lambda;
\mu}$ with a polarization $L_{\lambda;\mu;0}$ such that the
deformation carries $L_{\lambda;\mu}^n$ to $L_{\lambda;\mu;0}^n$
for sufficiently large positive integer $n$. In particular, the
moment polytope of the polarized toric variety $(N_{\lambda;
\mu},L_{\lambda;\mu;0})$ is $Q_{\lambda;
\mu}=\pi_{\lambda}^{-1}(\mu) \cap Q(\lambda)$.
\label{th:flat_weight}
\end{theorem}

\proof  By the Borel-Weil Theorem, we have the following
identification of the section ring
$$R_\lambda = \bigoplus_{n=0}^\infty H^0(X, L_\lambda^n) = \bigoplus_{n=0}^\infty
V(n\lambda^*).$$ $H$ acts on $R_\lambda$ via the induced action
twisted by $-\mu$. Let $R_\lambda^{H(-\mu)}$ denote the set of
invariants of this twisted $H$-action. Using the canonical bases,
we have
$$R_\lambda  = \bigoplus_{n=0}^\infty \bigoplus_\phi \C .b_{n
\lambda, \phi}.$$ Since $b_{n \lambda, \phi}$ is an eigenvector of
eigenvalue $- n \lambda + \sum_k t_{i_k} \alpha_{i_k}$ with
respect to the left $H$-action,   we see that after twisting the
$H$ action on $\C .b_{n \lambda, \phi}$ by $-n\mu$, the eigenvalue
becomes zero if $\sum_k t_{i_k} \alpha_{i_k} = n (\lambda + \mu)$.
Hence,
$$R_\lambda^{H (-\mu)} =\bigoplus_{n=0}^\infty \bigoplus_{\sum t_{i_k} \alpha_{i_k}  = n
(\lambda + \mu)} \C .b_{\lambda, \phi}.$$  Observe that the
filtration on $R_\lambda$ induces a filtration on $R_\lambda^{H
(-\mu)}$. Let $\hbox{Gr}(R_\lambda^{H (-\mu)})$ be the associated
graded algebra of  $R_\lambda^{H (-\mu)}$. Then the weight variety
$M_{\lambda; \mu} = \hbox{Proj} (R_\lambda^H (-\mu))$ degenerates
flatly to $N_{\lambda; \mu} =\hbox{Proj} (\hbox{Gr} R_\lambda^{H
(-\mu)})$, and the degeneration carries the set of ample line
bundles $L_{\lambda;\mu}^n$ to the set of  ample line bundles
$L_{\lambda;\mu;0}^n$ on $N_{\lambda, \mu}$ for sufficiently large
$n$ (\cite{AlexBrion}). Since in general, the moment map of a
polarization $L$ is the moment map of $L^n$ divided by $n$,  it
follows from \cite{KSZ} that the moment polytope of the polarized
toric variety $(N_{\lambda; \mu}, L_{\lambda;\mu;0})$ is
$\pi_\lambda^{-1}(\mu) \cap Q(\lambda)$.
\endproof

\begin{definition}
We may call the central toric fiber $N_{\lambda, \mu}$ the toric
weight variety of type $(\lambda, \mu)$, and call the polytope
$Q_{\lambda; \mu} = \pi_\lambda^{-1}(\mu) \cap Q(\lambda)$ the
weight polytope of type $(\lambda, \mu)$. Note that the polytope
$Q_{\lambda; \mu}$ is rational but needs not to be integral in
general.
\end{definition}

\begin{rem}{\em
Unlike the isomorphic type of the quotient $M_{\lambda; \mu}$
which only depends on the GIT chamber of $\mu$, the degeneration
depends on the choice of individual $\mu$, in particular, the
topological type of the central fiber $N_{\lambda; \mu}$ can
change even when $\mu$ varies within its GIT chamber. See Example
\ref{M11111}.}
\end{rem}

\begin{question}
Is there a chamber structure on $\Delta_\lambda$ so that
$N_{\lambda; \mu}$,  as (unpolarized) toric variety, remain the
same within a chamber?
\end{question}

\begin{rem}{\em As noted in \cite[Section 5]{AlexBrion},
the Ehrhart polynomial of the polytope $Q_{\lambda;\mu}$ is given
by
$$ n\mapsto {\rm dim}V(n\lambda^*)_{n\mu}, $$ the multiplicity
of the weight $n\mu$ for the right $H$-action in the irreducible
$G$-module with highest weight $n\lambda^*$. See also discussion
on Pieri's formula in \cite{FlMill}.}
\end{rem}

\section{Degeneration of polygon spaces}

Let us recall the generalities about the polygon spaces. We will
start with spatial polygons in Eucledian 3-space. Given an
$n$-tuple of positive real numbers ${\bf r}=(r_1, ..., r_n)$ we
consider the moduli space $M_{\bf r}$ of $n$-gons with consecutive
side lengths $r_1, ..., r_n$. This space can be viewed as the
symplectic quotient at the zero level of the product of
two-dimensional spheres $ (S^2)^n$ by the diagonal action of the
group ${\rm SO}(3)$. The symplectic structure on the $j$-th
multiple is taken to be $r_j$ times the standard unit sphere
volume form. On the algebraic geometry side, the space $M_{\bf r}$
can be realized as the GIT quotient of the $n$-fold product of the
projective line $\CP^1$ by the diagonal action of the group ${\rm
SL}(2, \C)$. The choice of linearization is given by $r_i$'s
(which we assume to be integers here). These spaces were studied
by Klyachko \cite{Klyachko}, Kapovich-Millson \cite{KM},
Hausmann-Knutson \cite{HK}, and many others.

Let us now recall the Gelfand-MacPherson correspondence
\cite{GelMac} in the form of Theorem 2.4.7 of \cite{Kapranov}. It
states that for any $n$-tuple of positive integers ${\bf r}=(r_1,
..., r_n)$, there is an isomorphism of GIT quotients
$$
({\rm Gr}(k, n)/\!/H_1)_{{\mathcal O}(1), {\bf r}}\simeq
((\CP^{k-1})^n/\!/{\rm SL}(k, \C))_{{\mathcal O}({\bf r}), \zeta},
$$
where the line bundle ${\mathcal O}(1)$ on the grassmannian is
linearized to correspond to the action of $H_1=\{ (t_1, ...,
t_n)\in (\C^*)^n: \prod t_i=1\}$ on $\C^n$ given by
$$
(t_1, ..., t_n)\mapsto {\rm diag}(t^{\bf r}t_1, ..., t^{\bf
r}t_n),
$$
and $t^{\bf r}=t_1^{r_1}\cdots t_n^{r_n}$ is the character of
$H_1$ corresponding to ${\bf r}$. On the other side, the line
bundle ${\mathcal O}({\bf r})$ on $(\CP^{k-1})^n$ is the tensor
product of pull-backs of line bundles ${\mathcal O}(r_i)$ from the
$i$-th multiple. This bundle has exactly one ${\rm SL}(k,
\C)$-linearization, which is denoted by $\zeta$.

Now, let us apply this correspondence to $M_{\bf r}$ using the
language of hermitian matrices as was first done in \cite{HK}. Let
us start with the complex vector space $\C^{2n}$ with coordinates
$(z_1, ..., z_n, w_1, ...,w_n)$. For convenience, we arrange
$z_i$'s and $w_j$'s in a matrix form
$$
A=\left(
\begin{array}{cccc}
z_1 & z_2 & \cdots & z_n \\
w_1 & w_2 & \cdots & w_n
\end{array}
\right)
$$
If we let $P=\sum_{i=1}^n r_i/2$ stand for half the perimeter,
then the moduli space $M_{\bf r}$ can be realized as the quotient
of the grassmannian ${\rm Gr}(2, n)$ by the action of the maximal
torus. The proof goes via the so-called reduction in stages. The
grassmannian  ${\rm Gr}(2, n)$ can be realized as the symplectic
quotient of the space $\C^{2n}$ with the standard Darboux
symplectic form by the action of the group ${\rm U}(2)$ by left
multiplication on the matrix $A$. The corresponding moment map
takes values in $2\times 2$ hermitian matrices and is given by
$$
A{\overline{A}}^t=\left(
\begin{array}{cc}
\sum_{i=1}^n |z_i|^2 & \sum_{i=1}^n z_i{\bar w}_i \\ {} & {} \\
\sum_{i=1}^n w_i{\bar z}_i & \sum_{i=1}^n |w_i|^2
\end{array}
\right).
$$
And we reduce at the level ${\rm diag}(P, P)$. Alternatively, we
can reduce the space $\C^{2n}$ by the action of $(S^1)^n$ on the
columns of the matrix $A$ at the level $(r_1, ..., r_n)$ to get
the n-fold product of two-dimensional spheres with the
aforementioned symplectic structure.

Using the $2\times 2$ hermitian matrices, we see that the moduli
space of polygons can be identified with the space of solutions of
matrix equation
$$
A_1+\cdots + A_n=\left( \begin{array}{cc} P & 0 \\ 0 & P
\end{array}\right),
$$
where each matrix $A_j$ has the prescribed spectrum $(0, r_j)$ and
in the language of the above correspondence:
$$
A_j=\left( \begin{array}{cc} |z_j|^2 & z_j{\bar w}_j \\ {} & {} \\
{\bar z}_jw_j & |w_j|^2 \end{array}\right).
$$

Let $d_j$ stand for the length of the diagonal of the polygon
connecting vertices $1$ and $j$. There is a remarkable integrable
system \cite{KM} on $M_{\bf r}$ whose action variables are these
functions $d_j$ for $3\le j\le n-1$. The corresponding flows are
the so-called {\it bending} flows and have a simple geometric
description by rotating the last $n-j$ spheres about the axis
defined by the diagonal $d_j$. This integrable system was shown
\cite{HK} to be the reduction of the so-called Gelfand-Tsetlin
integrable system on ${\rm Gr}(2, n)$ \cite{GS}, which we recall
now briefly.

Consider the $n\times n$ hermitian matrix $A^*A$ of rank $2$. The
two non-trivial eigenvalues of the $k\times k$ upper-left
submatrix denote by $a_j$ and $b_j$, $a_j\le b_j$. (For
completeness, we let $a_1=0$.) The value $b_{n-1}=P$ is
predetermined by the fact that the two non-zero eigenvalues of
$A^*A$ are both equal to $P$. Thus we have $2(n-2)$ independent
commuting hamiltonians on ${\rm Gr}(2,n)$, which are also called
the Gelfand-Tsetlin variables and commonly written in a triangular
form:
$$
\begin{array}{ccccccccccc}
0 & \ & 0 & \ & \cdots & \ & 0 & \ & P & \ & P \\
\ & 0 & \ & 0 & \cdots & 0 & \ & a_{n-1} & \ & P & \ \\
\ & \ & \ & \cdots & \ & \cdots & \ & \cdots & \ & \ & \ \\
\ & \ & \ & 0 & \ & a_3 & \ & b_3 & \ & \ & \ \\
\ & \ & \ & \ & a_2 & \ & b_2 & \ &  \ & \ & \ \\
\ & \ & \ & \ & \ & b_1 & \ & \ & \ & \ & \
\end{array}
$$
which also satisfy the interlacing inequalities.

The reduction of this system corresponds to choosing the diagonal
values of the matrix $A^*A$ to ensure the compatibility with
chosen side length, which amounts to fixing the sums in rows of
the Gelfand-Tsetlin diagram: $b_1=r_1$, $a_2+b_2=r_1+r_2$, ...,
$a_j+b_j=\sum_{i=1}^jr_i$, ... , $a_{n-1}=P-r_n$. Note that the
lengths of the diagonals can also easily be expressed in terms of
the variables $(a_i, b_i)$, namely: $$d_{j+1}=b_j-a_j.$$ These
variable diagonal lengths $d_j$'s, $3\le j\le n-1$, which are the
action variables define a polytope $\Pi_{\bf r}$ in $\R^{n-3}$.
This polytope is cut by $3(n-2)$ hyperplanes corresponding to the
triangle inequalities, some of which are redundant, for the
triangulation of the polygon by all these diagonals emanating from
the first vertex. The $j$-th triangle has the side lengths $d_j,
d_{j+1}$, and $r_{j+1}$ so it contributes three inequalities:
$$d_j+d_{j+1}\le r_{j+1}, d_j+r_{j+1}\le d_{j+1}, d_{j+1}+r_{j+1}\le
d_j. \;\;\;\;\;\;\;\; (*)$$

\begin{definition}
The weight polytope $\Pi_{\bf r}$ is a polytope in $\R^{n-3}$
defined by all the triangle inequalities of ${\rm (*)}$.
\end{definition}

Kogan-Miller \cite{KogMil} and Alexeev-Brion \cite{AlexBrion} have
explicitly constructed degenerations of complex flag varieties $X$
of type $A_n$ to polarized toric varieties which have the
Gelfand-Tsetlin polytopes (denoted by GT$(\lambda)$, $\lambda\in
\Lambda^+$) as their moment polytopes. In the case of
Alexeev-Brion degeneration, they used the simplest reduced
decomposition of the longest Weyl group element in $W=S_{n+1}$
$$
\underline{w}_0^{\rm std}=s_1s_2s_1s_3s_2s_1\cdots
s_ns_{n-1}\cdots s_1.
$$
As before, we denote by $\pi_\lambda: {\rm GT}(\lambda)\to
\Delta(\lambda)\subset\Lambda_\R$ the projection of the
Gelfand-Tsetlin polytope onto the moment polytope of the flag
variety $X$ with respect to the hamiltonian ${\rm SU}(n)$-action
and the invariant symplectic form $\omega$ on $X=G/P_\lambda$ such
that $[\omega]=c_1(L_\lambda)$. From Theorem \ref{th:flat_weight}
we immediately conclude

\begin{theorem}
\label{thm:MtoN} There exists a flat degeneration of the moduli
space of polygons $M_{\bf r}$ with integral side length ${\bf
r}=(r_1, ..., r_n)$ together with a polarizing line bundle $L_r$
on it to a polarized toric variety $N_{\bf r}$ whose moment
polytope is the weight polytope $\Pi_{\bf r}$.
\end{theorem}

The toric variety $N_{\bf r}$, even its topological type,  depends
on the choice of ${\bf r}$. In fact, the combinatorial type of the
weight polytope $\Pi_{\bf r}$ may also change when ${\bf r}$
changes, as shown in the following  examples.

\begin{exmp}
\label{M11111}
 Consider the case of pentagons. Already here we will
see how the degeneration depends on the choice of the linearized
line bundle, which in terms of the polygons corresponds to
choosing not only side length, but also the order in which they
appear. Let us consider the following three quintuples of
integers: ${\bf r}^1=(3,3,3,3,3)$, ${\bf r}^2=(3,3,3,3,4)$, and
${\bf r}^3=(3,4,3,4,3)$. For all these choices, the moduli spaces
$M_{{\bf r}^1}$, $M_{{\bf r}^2}$, and $M_{{\bf r}^3}$ are
biholomorphically isomorphic to the same space
${\overline{M}}_{0,5}$, the canonically compactified moduli space
of projective lines marked at 5 distinct points. This surface is
also isomorphic to the Del Pezzo surface of degree 5 (blow-up of
$\CP^2$ at 4 generic points). The corresponding weight polytops
are  2-dimensional polygons. The coordinates $(x,y)$ correspond to
the lengths of the first and the second diagonals respectively.

In the case of ${\bf r}^1$ we have a pentagon with vertices at the
points $(0,3)$, $(3,0)$, $(6,3)$, $(6,6)$, and $(3,6)$. This
corresponds to a toric surface with two isolated singular points
(local quotient singularities of $\C^2$ by the action of $\Z/2$).

In the case of ${\bf r}^2$ we have a hexagon with vertices at the
points $(0,3)$, $(2,1)$, $(4,1)$, $(6,3)$, $(6,7)$, and $(4,7)$.
This corresponds to a toric surface with one isolated singular
point (over $(3,0)$).

In the case of ${\bf r}^3$ we get a heptagon with vertices at the
points $(1,4)$, $(1,2)$, $(2,1)$, $(4,1)$, $(7,4)$, $(7,7)$, and
$(4,7)$. The corresponding toric variety is smooth (and
diffeomorphic to $\overline{M}_{0,n}$). This toric variety is a
blow up of each of the previous ones at one or two singular
points. Notice that we would get a different polytope with five
vertices should the same side lengths be arranged as
$(3,3,3,4,4)$.
\end{exmp}

\begin{exmp} In the equilateral case when $n=4,5,6$, or $7$ and ${\bf r}=(1,1,...,1)$, the
polytopes $\Pi_{\bf r}$ were considered in \cite[Section
4]{Kamiyama} and their virtual Poincar\'e polynomials were
computed.
\end{exmp}

\begin{exmp} Now consider an example when ${\bf r}=(2,2,2,..., 2, 2n-3)$,
when there is one ``very long'' side. In this case ${\bf r}$ lies
in a {\it favorable} chamber \cite[Section 3]{Hu99}. One can see
that the polytope $\Pi_{\bf r}$ is defined by the following set of
inequalities:
$$
3\le d_3\le 4, \ \ 5\le d_4\le d_3+2, \ \ ..., \ \ 2i-3\le d_i\le
d_{i-1}+2, \ {\rm for} \ 4\le i\le n-1.
$$
This polytope is actually a simplex, and the corresponding
polarized toric variety is $(\CP^{n-3}$, ${\mathcal O}(1))$. This
is not surprising, because for any element ${\bf r}$ from the
interior of a favorable chamber, the moduli space $M_{\bf r}$ is
itself isomorphic to $\CP^{n-3}$, and thus our flat family is
trivial in this case.
\end{exmp}

\begin{exmp} Let us consider the case when we have three ``very long'' sides.
More precisely, let $r_1=r_2=\cdots=r_{n-3}=1$ and
$r_{n-2}=r_{n-1}=r_n=n$. So the perimeter is equal to $4n-3$ and
no two long sides can ever be parallel. We will show that the
moduli space $M_{\bf r}$ is isomorphic to $(\CP^1)^{n-3}$ in a
natural way. Let us treat the moduli space $M_{\bf r}$ as the
moduli space of weighted configurations of points on $\CP^1$.
Denote these points by $z_1$, ..., $z_n$ respectively, where we
coordinatize, as usual, by $\C$ the complement of a point (the
``North pole'' or $\infty$) in $\CP^1$. Consider the cross-ratios
$$
w_i=\frac{z_{n-2}-z_{n}}{z_{n}-z_{n-1}} \cdot
\frac{z_i-z_{n-1}}{z_n-z_i}, \ \ {\rm where} \ \ 1\le i\le n-3.
$$
Since the points $z_{n-2}$, $z_{n-1}$ and $z_n$ never collide on
the subset of semi-stable configurations, these cross-ratios
define a global map from the moduli space $M_{\bf r}$ to
$(\CP^1)^{n-3}$, which is an isomorphism. It is important to
notice that the bending flows are {\bf not} the standard circle
actions on this space.

This example also shows that the toric variety $N_{\bf r}$ and the
polytope $\Pi_r$ depends on the order of side-lengths. If we had
switched the first and the $(n-2)$-nd sides, so that ${\bf r}=(n,
1, 1, ..., 1, n, n)$, then none of the diagonals emanating from
the first vertex would ever vanish, because it would not be
allowed by the triangle inequalities, and the central fiber would
be isomorphic to $(\CP^1)^{n-3}$. But if we stick with the
original choice of order on the side-lengths, then the resulting
toric variety will be singular, in general. This can be seen
already for $n=5$, when the resulting toric variety has an
isolated quotient singularity.
\end{exmp}

In fact, since all the inequalities on the lengths of the
diagonals emanating from a given vertex come from the triangles
which these diagonals break the polygon into, we immediately have
the following result:

\begin{proposition}
\label{facets} The weight polytope $\Pi_{\bf r}$ is the compact
intersection of the half spaces defined by affine hyperplanes of
the following forms: $x_1 = |r_1\pm r_2|, x_{n-3}=|r_n\pm
r_{n-1}|$ or $|x_i \pm x_{i+1}|=r_{i+2}$. In particular, let
$\{e_i\}$ be the coordinate basis vectors, then any facet is
perpendicular to one of the following: $e_1, e_{n-3}$, i.e., the
first or the last coordinate line; $e_i \pm e_{i+1}$, i.e., the
diagonal or anti-diagonal line in the two dimensional coordinate
plane spanned by $\{e_i, e_{i+1}\}$.
\end{proposition}

In the above Proposition $x_k$ corresponds to $d_{k+2}$.

Let $U_{\bf r}$ be the (non-moduli) space of polygons with side
length vector ${\bf r}$ such that the first vertex is at the
origin. Note that $M_{\bf r} =  U_{\bf r}/{\rm SO}(3)$. We construct a
new quotient space by defining a new equivalence relation on
$U_{\bf r}$. For each $3 \le i \le n-1$, let $D_i$ be the set of
polygons with $d_i =0$. Let $D^\circ$ be the set of polygons where
none of the diagonals vanish. Then $$U_{\bf r} = D^\circ \cup_I
D_I,$$ where $I \subset \{3, \ldots, n-1\}$ and $D_I$ is the set
of polygons with vanishing diagonals $d_i, i \in I$ but the rest
of the diagonals do not vanish. For any $I$, every polygon in
$D_I$ can be decomposed as polygons $P_1, \ldots, P_\ell$ mutually
joining at a point, where $\ell=|I|+1$. In this case, set $$S_I =
D_I/({\rm SO}(3))^\ell$$ where $({\rm SO}(3))^\ell$ acts on each factor
component-wise. Let $S^\circ = D^\circ /{\rm SO}(3)$. We then obtain a
new topological space
$$V_{\bf r} = S^\circ \cup_I S_I.$$
There is an obvious continuous collapsing map
$$f: M_{\bf r} \rightarrow V_{\bf r}.$$
There is also a continuous map $$\Phi_{\bf r} : V_{\bf r}
\rightarrow \Pi_{\bf r}.$$ The map $f$ passes the action-angle
coordinates on $M_{\bf r}$ to $V_{\bf r}$.

\medskip\noindent{\bf Conjecture.} The space $V_{\bf r}$ is homeomorphic
to $N_{\bf r}$. \medskip

We were able to verify this conjecture in some particular cases,
however the rigorous proof evades us, because the explicit nature
of the deformation family is evasive.

\begin{rem}{\em
Our construction of the space $V_{\bf r}$ follows Kamiyama-Yoshida
\cite{Kamiyama}, where a similar space a continuous map to the
polytope $\Pi_{\bf r}$ was constructed in the equal side length
case. Despite the fact that they called it a ``symplectic toric
manifold'', their space only had a structure of a topological
space stratified into a finite union of orbifolds.}
\end{rem}

\subsection{Boundary toric divisors}

Corresponding to each facet, there is a toric subvariety of
$N_{\bf r}$ of codimension 1. Corresponding to the set of all
facets perpendicular to a given vector is the union of
subvarieties whose isotropy subgroup is generated by a prime
vector in that direction. By the above proposition, the one
dimensional isotropy subgroups are listed as below:
\begin{enumerate}
\item $T_1= \{t \in (\C^*)^{n-3} | t_2 = \ldots = t_{n-3} = 1\}$.
\item $T_{n-3} = \{t \in (\C^*)^{n-3} | t_1 = \ldots = t_{n-4} =
1\}$. \item $T_i^+ = \{t \in (\C^*)^{n-3} | t_i = t_{i+1}, t_k = 1
\; \hbox{for} \; k \ne i, i+1 \}$. \item $T_i^- =\{t \in
(\C^*)^{n-3} | t_{i+1} = t_i^{-1}, t_k = 1 \; \hbox{for}\; k \ne
i, i+1\}$. .
\end{enumerate}

Corresponding to the list, we have toric divisors (in parentheses
the equations of the corresponding facets of $\Pi_{\bf r}$ are
given):
\begin{itemize}
\item (1) $N^1_2$ ($x_1=r_1+r_2$) and $N^2_2$ ($x_1=r_2-r_1$) or $N^3_2$ ($x_1=r_1-r_2$)
\item (2) $N^3_{n-1}$ ($x_{n-3}=r_{n-1}+r_n$) and $N^1_{n-1}$ ($x_{n-3}=r_n-r_{n-1}$)
or $N^2_{n-1}$ ($x_{n-3}=r_{n-1}-r_n$)
\item (3) $N^1_i$ ($x_{i-1}-x_{i-2}=r_i$) and $N^3_i$ ($x_{i-2}-x_{i-1}=r_i$)
\item (4) $N^2_i$ ($x_{i-1}+x_{i-2}=r_i$)
\end{itemize}
These are all the toric divisors, some of which are reducible. In
the next section, we will show that they are the deformation
images of certain subpaces of $M_{\bf r}$.

The Chow ring $A^*(M_{\bf r})$ is generated, as a ring, by degree
$1$ cycles corresponding to divisors $Z_{ij}$, $i< j$ defined by
the condition that the $i$-th side of the polygon points in the
same direction as the $j$-th side. We call such sides ``positively
parallel''  (or strongly parallel), which is a stronger condition
than just being parallel. This statement can be verified by
considering the cycles $D^S$ defined in \cite{Keel} and which
generate $A^*({\overline{M}}_{0,n})$ together with the proper
surjective birational morphism ${\overline{M}}_{0,n}\to M_{\bf
r}$, and its properties \cite{FothCMB}. It is then well-known that
the map $A^*(M_{\bf r})\to H^{2*}(M_{\bf r}, \Z)$ is an
isomorphism. There are some natural linear relations among the
cycles $Z_{ij}$, which again can be deduced from the linear
relations on the cycles $D^S$ studied in \cite{Keel} and
\cite{KonMan}. However, these cycles are not stable under the
bending flows, and thus will not degenerate to toric subvarieties.

Instead, let us consider subsets of the space $M_{\bf r}$ which
are defined by the condition that the $i$-th diagonal is parallel
to the $(i+1)$-st diagonal, and thus to the $i$-th side  (we also
think that a zero vector is parallel to any other vector). Let us
introduce a notation for these subsets.
\begin{itemize}
\item denote by $Y^1_i$ the subset where $d_i+r_i=d_{i+1}$
\item denote by $Y^2_i$ the subset where $d_i+d_{i+1}=r_i$
\item denote by $Y^3_i$ the subset where $d_i-d_{i+1}=r_i$
\item denote by $Y_i=Y^1_i\cup Y^2_i\cup Y^3_i$ their union
\end{itemize}

Here we have $2\le i\le n-1$ as we think of the first and last
sides as $2$-nd and $n$-th diagonals respectively. The number of
such we get this way is bounded from above by $3n-8$. However,
depending on the choice of ${\bf r}$ some of these subsets might
be empty.

One can easily see that $Y^1_i$ and $Y^3_i$ always have empty
intersection. However, if $d_i$ can attain zero, then $Y^1_i$ and
$Y^2_i$ do intersect and when the side lengths are generic, and
$3<i<n-1$, this intersection is homeomorphic to an $S^1$-bundle
over an $S^2$-bundle over the space $M_{\bf r^{(1)}}\times M_{\bf
r^{(2)}}$, where ${\bf r^{(1)}}=(r_1, ..., r_{i-1})$ and ${\bf
r^{(2)}}=(r_{i}, ..., r_n)$.

Similarly, if $d_{i+1}$ can attain the value of zero, then then
$Y^2_i$ and $Y^3_i$ do intersect and when side lengths are
generic, and $2<i<n-2$, this intersection is homeomorphic to an
$S^1$-bundle over an $S^2$-bundle over the space $M_{\bf
r^{(1)}}\times M_{\bf r^{(2)}}$, where ${\bf r^{(1)}}=(r_1, ...,
r_{i})$ and ${\bf r^{(2)}}=(r_{i+1}, ..., r_n)$.

Note that when $r_1=r_2$ then the subsets $Y^2_{2}$ and $Y^3_2$
coincide and are equivalently defined by $d_3=0$. Same goes for
the case $r_{n-1}=r_n$, subsets $Y^1_{n-1}$ and $Y^2_{n-1}$
coincide and are defined by $d_{n-1}=0$. Unless stated otherwise,
we would like to exclude these from consideration in the statement
of the following result.

We denote by $M_{\bf r}^0$ the ``toric'' open submanifold of
$M_{\bf r}$, where none of the diagonals vanishes. We say that a
subset $Q\subset M_{\bf r}$ {\it degenerates to} $Q_0\subset
N_{\bf r}$ if $Q\cap M_{\bf r}^0$ (resp $Q_0$) is preserved by the
bending flows (resp. compact torus action), and they have the same
image in $\Pi_{\bf r}$.

\begin{lemma}
\begin{enumerate}
\item If $M_{\bf r }$ is smooth, then $Y_{i}\cap M_{\bf r}^0$, for
$2\le i\le n-1$, is a smooth closed submanifold of $M_{\bf r}^0$
of real codimension $2$.
\item Each $Y_i$ degenerates to the union of toric
divisors $N^1_{i}$, $N^2_i$, and $N^3_i$ of $N_{\bf r}$. All toric
divisors  of $N_{\bf r}$ arise this way.
\end{enumerate}
\end{lemma}
\proof (1) The intersections $$Y_{i} \cap M_{\bf r}^0$$ are
clearly non-empty. Each connected components of these
intersections is also a connected component of the fixed point set
of some one-dimensional torus (see the remark after Proposition
\ref{facets}), so it is smooth.

(2) is clear.
\endproof

\begin{rem}{\em The intersection $N^1_i\cap N^2_i$
is a toric subvariety which is isomorphic to the product of the
toric degenerations of polygon spaces with side lengths $(r_1,
...., r_{i-1})$ and $(r_i, ..., r_n)$ respectively. Analogously,
the intersection $N^2_i$ and $N^3_i$, if non-empty, is a toric
subvariety which is isomorphic to the product of the toric
degenerations of polygon spaces with side lengths $(r_1, ....,
r_{i})$ and $(r_{i+1}, ..., r_n)$ respectively. It is clear that
these divisors $\{ N^j_i\}$ for $j=1,2$, and $3$ and $2\le i\le
n-1$ generate $A^*(N_{\bf r})$. Although the additive structure on
homology of $M_{\bf r}$ can readily be understood if we work with
divisors $\{ Z_{ij}\}$, there are too many generators and linear
relations between these cycles. Using the real cycles represented
by connected components of the $Y_{i}$'s, the homology of $M_{\bf
r}$ can be understood with more ease, as we can see explicitly the
toric subvarieties that these cycles degenerate to in $N_{\bf
r}$.}
\end{rem}

\begin{question} Do the cycles of the (connected components of)
subspaces $Y_{i}$ for $2\le i\le n-1$, generate $H_*(M_{\bf r},
\Z)$?
\end{question}

In some examples that we were able to compute, this indeed was the
case. However, in general due to the fact that some of the $Y_i$'s
degenerate to the sum of two or three irreducible toric divisors,
the answer to this question might not be entirely obvious.

As follows from \cite[Proposition 2.2(ii)]{AlexBrion}, our family
is isomorphic to the trivial family over $\C^*=\C\setminus\{ 0\}$.
In the future we hope to establish an explicit trivialization, so
the degeneration of the aforementioned cycles will be transparent.

\section{Degeneration of Flaschka-Millson integrable systems}

In \cite{FlMill} the authors considered moduli spaces of weighted
configurations on $\CP^m$. Given an $n$-tuple of positive integers
${\bf r}=(r_1, ..., r_n)$ their moduli space is, again, identified
with the GIT quotient of $(\CP^m)^n$ by the diagonal action of
${\rm SL}(m+1, \C)$, with the linearizing line bundle ${\mathcal
O}({\bf r})$ is given by the tensor product of pull-backs of the
bundles ${\mathcal O}(r_i)$ from the $i$-th factor. It has a
unique ${\rm SL}(m+1, \C)$ linearization. In order for this moduli space
to be nonempty, we need to impose, in addition to the obvious
condition $n
> m+1$, the so-called strong triangle inequalities, one for each
$1\le i\le n$:
$$
r_i\le P,\ \ \ \hbox{where}\ \ \ P:=\frac{1}{m+1}\sum_{j=1}^n r_j.
$$
Note that when $m=2$, these are just the usual triangle
inequalities. On the symplectic side, this moduli space is
identified with the moduli space of solution to the matrix
equation:
$$
A_1+\cdots + A_n=P.{\rm Id},
$$
where each $A_i$ is a rank one $(m+1)\times (m+1)$ hermitian
symmetric matrix, with spectrum $\{ r_i, 0, 0, ..., 0\}$, and we
mod out by the diagonal action of the group ${\rm U}(m+1)$. Then
the Flaschka-Millson bending hamiltonians, by definition, are the
non-trivial eigenvalues of the partial sums
$$
\sum_{i=1}^k A_i.
$$
Using the Gelfand-MacPherson correspondence \cite{Kapranov}, the
above moduli space $M_{\bf r}$ can be identified with a GIT
quotient of the grassmannian ${\rm Gr}(m+1, n)$ with respect to
the torus action. The choice of the linearized line bundle is
analogous to the case $m=1$.

These hamiltonians also appear if we take the Gelfand-Tsetlin
integrable system on the grassmannian and consider its reduction.
More precisely, consider the Gelfand-Tsetlin triangle for this
case. The top line consists of $(n-m-1)$ zeroes and $(m+1)$ $P$'s.
(For the purposes of drawing the diagram we assumed that $m< n/2$,
but we don't hold this assumption in general):
$$
\begin{array}{ccccccccccccccccccc}
0 & & 0 & & \cdots & & 0 & & 0 & & P & & P & & \cdots & & P & & P \\
 & 0 & & 0 & \cdots & 0 & & 0 & & a_1 & & P & & P & \cdots & P & & P & \\
 & & 0 & & \cdots & & 0 & & a_2 & & b_2 & & P & & \cdots & & P & & \\
 & & & \cdots & & & & & \cdots & & & & & & \cdots & & & & \\
& & & \cdots & & 0 & & a_{n-2m} & & \cdots & & p_{n-2m} & & P & & & & & \\
& & & & & \cdots & & & & \cdots & & & & & & & & & \\
& & & & 0 & & a_{m+1} & & & \cdots & & q_{m+1} & & & & & & & \\
& & & & & \cdots & & & \cdots & & \cdots & & & & & & & & \\
& & & & & & o_3 & & p_3 & & q_3 & & & & & & & & \\
& & & & & & & p_2 & & q_2 & & & & & & & & & \\
& & & & & & & & q_1 & & & & & & & & & & \\
\end{array}
$$
There are a total of $(m+1)\times (n-m-1)$ indeterminates $a_1$,
..., $q_{m+1}$ in the diagram. The reduction at the level ${\bf
r}=(r_1, ..., r_n)$ means that there are the following condition
imposed, analogous to Section 3, where we considered the case
$m=1$:
$$
q_1=r_1, \ \ p_2+q_2=r_1+r_2,\ \ ..., \ \ a_{m+1}+\cdots +
q_{m+1}=r_1+\cdots + r_{m+1}, \ ...\ \hbox{etc.}
$$
meaning that the sum of all elements in row $k$ from the bottom
equals $\sum_{i=1}^k r_i$. The Flaschka-Millson hamiltonians in
this presentation are the differences of adjacent elements in each
row:
$$
b_2-a_2, \ \ b_3-a_3, \ \ c_3-b_3, \ \ , ...., \ \ q_2-p_2.\
$$
There are precisely
$$
(m+1)(n-m-1)-(n-1)=mn-2m-m^2
$$
of them, which is exactly the complex dimension of the moduli
space. The polygon defined by the natural inequalities on these
hamiltonians is, as before, denoted by $\Pi_{\bf r}$.

Proceeding in a completely analogous way to Section 3, we can see
that there exists a flat degeneration of the moduli space $M_{\bf
r}$ to a polarized toric variety with the moment polytope
$\Pi_{\bf r}$.

\begin{rem} {\em If we denote by $P-{\bf r}$ the $n$-tuple of weights $(P-r_1, P-r_2, ..., P-r_n)$, then
Howard-Millson \cite{HowMill} showed that the GIT quotients
$$((\CP^m)^n//{\rm SL}(m+1, \C))_{{\mathcal O}({\bf r})}\ \ \ {\rm and} \ \ \
((\CP^{n-m-2})^n//{\rm SL}(n-m-1, \C))_{{\mathcal O}(P-{\bf r})}$$
are isomorphic. Applying the Gelfand-MacPherson correspondence,
this duality translates to the isomorphism of GIT quotients
$$({\rm Gr}(m+1, n)//H_1)_{{\mathcal O}(1), {\bf r}} \ \ \ {\rm and} \ \ \
({\rm Gr}(n-m-1, n)//H_1)_{{\mathcal O}(1),{P-{\bf r}}}\ \ , $$ which
they established by using the complex Hodge * operator. This
duality clearly induces an isomorphism between polytopes
$\Pi_{\bf r}$ and $\Pi_{P-{\bf r}}$ and the toric varieties
$N_{\bf r}$ and $N_{P-{\bf r}}$.}
\end{rem}

\section{Real loci}

First, let us have a brief discussion on real loci in general
weight varieties. Let $\tau$ be an involution of $G$ (group
automorphism of order $2$, and we treat $G$ as a real Lie group
with the Iwasawa decomposition $G=KAU$) satisfying the following
conditions:
\begin{itemize}
\item $\tau$ commutes with the Cartan involution $\theta$
\item $\tau$ maps $H$ to $H$
\item $\tau$ is anti-holomorphic
\item if $G^\tau$, $K^\tau$, and $A^\tau$ stand for the subgroups of $G$, $K$, and $A$ respectively fixed by
$\tau$ and $U^\tau$ is the subset of elements in $U$ fixed by
$\tau$, then $G^\tau=K^\tau A^\tau U^\tau$. (Note that we do not
require that $\tau$ maps $U$ to $U$.)
\end{itemize}
In fact, given a conjugacy class of real forms of $G$, one can
always find an involution $\tau$ as above so this conjugacy class
is represented by $G^\tau$. Alternatively, given a Satake diagram
$\Sigma$ \cite{Helga}, we can always find an involution $\tau$
such that $\Sigma$ corresponds to the real form $G^\tau$.

Now let us take a subset of simple roots $J$ such that it contains
all the black simple roots in the Satake diagram $\Sigma$ and
perhaps some white roots. If there are white roots connected by an
arrow, then we either include both of them to $J$ or neither. Let
$P_J$ be the corresponding parabolic subgroup containing $B$ (the
minimal among all parabolics containing $B$ and all the root
vectors $E_{-\alpha}$, $\alpha\in J$). The following is
straightforward:

\begin{lemma}The parabolic subgroup $P_J$ is $\tau$-stable.
\end{lemma}

This immediately implies that there is an induced involution, also
denoted by $\tau$, on the flag manifold $X=G/P$, and its fixed
point set is the real flag manifold $X^\tau\simeq G^\tau/P^\tau$.
In fact there exists an integral symplectic structure
\cite{FothReal} $\omega$ (corresponding to a polarizing line
bundle $L_\lambda$) such that $\tau$ is anti-symplectic with
respect to $\omega$ and $X^\tau$ is a lagrangian submanifold in
$X$.

In what follows we denote by  $T\subset H$ the connected component
of the maximal torus containing the identity element. The
involution $\tau$ maps $T$ to $T$ and let us denote by $T^+$ the
connected subgroup of $T$ on which $\tau$ acts identically and by
$T^-$ - where it acts by inverting all elements, so that
$T=T^+\times T^-$. Let us also denote by $Q$ a (not necessarily
connected) subgroup of $T$ consisting of {\it all} elements on
which $\tau$ acts by $\tau(t)=t^{-1}$. If the dimension of $T^+$
is denoted by $a$, then
$$
Q\simeq (\Z/2)^a\times T^-,
$$
and has $2^a$ connected components. Let us also denote by $T^\tau$
the subgroup of $T$ fixed by $\tau$. If the dimension of $T^{-}$
is $b$, then
$$
T^\tau\simeq T^+\times (\Z/2)^b.
$$

Note that the involution $\tau$ on $X$ satisfies the following
property with respect to $K$-action:
$$
\tau(k.x)=\tau(k)\tau(x).
$$
We can choose the moment map $\Phi$ for the $T$-action in such a
way that $\Phi(\tau(x))=-\tau(\Phi(x))$, where in the left hand
side the involution on $\ft^*$ is the one induced from $\tau$ on
$T$, and also denoted by $\tau$.  According to a theorem of
O'Shea-Sjamaar \cite{OSS} which generalizes Duistermaat's
\cite{Duister}, the image of $X^\tau$ under the moment map $\Phi$
for the $T$-action is the same as the intersection of $X$ with the
annihilator $(\ft^-)^\perp\subset \ft^*$ of $\ft^-$. Let us choose
a regular value $\mu$ of $\Phi$ such that $\mu\in(\ft^-)^\perp$.
Then the fixed point set of $\tau$ in $\Phi^{-1}(\mu)$, which is
the same as $\Phi^{-1}(\mu)\cap X^\tau$ is non-empty and
$T^\tau$-stable. If we reduce at $\mu$, and denote the induced
involution on $X//T$ also by $\tau$, then the natural map
$$
\psi:\ X^\tau//T^\tau:=(\Phi^{-1}(\mu)\cap X^\tau)/T^\tau \to
(X//T)^\tau
$$
has the following properties. The map $\psi$ is surjective onto a
(finite number of) connected component of $(X//T)^\tau$. The map
$\psi$ is a finite map, and is injective if $T$ acts freely on
$\Phi^{-1}(\mu)$. Replacing the involution $\tau$ acting on $X$ by
$s\tau$, where $s\in Q$ we can get all other connected components
of $(X//T)^\tau$ be in the image of maps analogous to $\psi$. The
induced involution on $X//T$ will still be the same, yet if $s$
acts non-trivially on $X$ and belongs to a different connected
component of $Q$ than the identity, then the $T^\tau$-orbits on
$X^{s\tau}$ are actually disjoint from those on $X^\tau$ and if
$X^{s\tau}$ is non-empty, then $X^{s\tau}\cap
\Phi^{-1}(\mu)/T^\tau$ will map onto different connected
components of $(X//T)^\tau$. To get all the connected components
of $(X//T)^\tau$, it is enough to choose one $s$ from each
connected component of $Q$, see details in \cite{FothReal}.

When the Satake diagram $\Sigma$ does not contain any black roots
(when $G^\tau$ is so-called quasi-split), then $B$ itself is
$\tau$-stable and the action of $\tau$ descends to the full flag
manifold $G/B$. And when $G^\tau$ is actually a split real form
(no arrows in $\Sigma$), then each $P\supset B$ is $\tau$-stable
and, moreover, $T=Q=T^{-}$ and $T^\tau$ is finite and isomorphic
to $(\Z/2)^r$, where $r={\rm rank}(G)$.

We will postpone a more detailed treatment of real loci in general
weight varieties and their degenerations to a future paper, and
now will concentrate on the case when $G={\rm SL}(n, \C)$, $\tau$
is the standard complex conjugation and $X$ is a complex
grassmannian. First, let us consider the case of spatial polygons.
Then the real quotient $X^{\tau}//T^\tau$ is the moduli space of
planar polygons (more precisely, its quotient by the mirror
reflections), which we denote by $M^{(2)}_{\bf r}$. If we, in
addition, fix admissible non-zero values of diagonals, then a
polygon can have only finitely many shapes. Their number
generically equals $2^{n-3}$, because we can do a 180${}^o$
``bending'' about each of the diagonals. Thus we have shown that
the map $M^{(2)}_{\bf r}\to \Pi_{\bf r}$ is surjective, and
generically $2^{n-3}$-to-one. However, this map is not globally
finite, and only becomes such at the special fiber $N_{\bf r}$. It
is easy to see that the involution $\tau$ on general fiber $M_{\bf
r}$ extends to a complex conjugate involution on the special
fiber, where its fixed point set is a lagrangian locus
$N^{(2)}_{\bf r}$ which maps surjectively and finitely onto
$\Pi_{\bf r}$. This statement has a natural generalization to the
moduli spaces $M_{\bf r}=(\CP^m)^n//{\rm SL}(m+1, \C)$ considered
in \cite{FlMill}:

\begin{proposition}
Let ${\bf r}=(r_1, ..., r_n)$ be an admissible $n$-tuple of
positive numbers. Let $\tau$ be a complex conjugate involution,
with fixed point set ${\rm SL}(n, \R)$ in ${\rm SL}(n, \C)$ and
${\rm Gr}_{\R}(m+1, n)$ in ${\rm Gr}_{\C}(m+1, n)$. Then $\tau$
extends to the special fiber $N_{\bf r}$ of the flat family and
the fixed point set $N^{\tau}_{\bf r}$ maps finitely and
surjectively under the moment map to the polytope $\Pi_{\bf r}$,
and the generic fiber has cardinality $2^{mn-2m-m^2}$.
\end{proposition}

\section*{Acknowledgments}
We thank Ben Howard and John Millson for showing us a preliminary
version of \cite{HowMill} and useful correspondence. We also thank
Michel Brion for a useful comment. The second author was partially
supported by an NSF grant.


text:
\begin{document}

\title{On the growth exponent of c-holomorphic functions with algebraic graphs}

\author{Maciej P. Denkowski}\address{Jagiellonian University, Faculty of Mathematics and Computer Science, Institute of Mathematics, \L ojasiewicza 6, 30-348 Krak\'ow, Poland}\email{maciej.denkowski@uj.edu.pl}\date{February 8th 2014}
\keywords{Complex analytic and algebraic sets, c-holo\-morph\-ic
functions, Liouville type theorem, rational functions, regular functions.}
\subjclass{32B15, 32A17, 32A22}

\begin{abstract}
This paper is the first of a series dealing with c-holomorphic functions defined on algebraic sets and having algebraic graphs. These functions may be seen as the complex counterpart of the recently introduced \textit{regulous} functions. Herein we their growth exponent at infinity. A general result on injectivity on fibres of an analytic set together with a Theorem of Tworzewski and Winiarski gives a bound for the growth exponent of a c-holomorphic function with algebraic graph in terms of the projective degrees of the sets involved. We prove also that algebricity of the graph is equivalent to the function being the restriction of a rational function (a Serre-type theorem). Then we turn to considering proper c-holomorphic mappings with algebraic graphs and we prove a B\'ezout-type theorem.
\end{abstract}

\maketitle

\section{Introduction}

The main object we will be dealing with in this article are continuous functions defined on a given algebraic subset of a complex finite-dimensional vector space $M$ and having algebraic graphs. To make the notation clearer, we will consider $M={\C}^m$ (anyway, everything here is invariant under linear isomorphisms). Our general aim is to present several effective results concerning this class of functions and thus we study their growth exponent and --- in a forthcoming paper --- also their \L ojasiewicz's exponents at infinity, the Nullstellensatz in this class. Actually, the class of functions we are interested in coincides with the class of c-holomorphic functions with algebraic graphs. When dealing with such functions we are bound to use purely geometric methods.

It may be interesting to note that c-holomorphic functions with algebraic graphs can be treated as a kind of complex counterpart of the recently introduced \textit{regulous} functions \cite{FHMM} as well as in \cite{Kol}, in connection with the important results of \cite{K}.

For the convenience of the reader we recall the definition of a c-holo\-morph\-ic mapping. Let $A\subset\om$ be an analytic subset of an open set $\om\subset{\C}^m$.

\begin{dfn} (cf. \cite{L}, \cite{Wh}) A mapping $f\colon
A\to{\C}^n$ is called {\it c-ho\-lo\-morph\-ic} if it is
continuous and the restriction of $f$ to the subset of regular
points ${\rm Reg} A$ is holomorphic. We denote by $\ch (A,{\C}^n)$
the ring of c-holomorphic mappings (with pointwise multiplication), and by $\ch (A)$ the ring of
c-holomorphic functions.
\end{dfn}

It is a way (due to R. Remmert) of generalizing the notion of holomorphic mapping onto
sets having singularities and a more convenient one than the usual notion of {\it weakly holomorphic functions} (i.e. functions defined and holomorphic on $\mathrm{Reg} A$ and locally bounded on $A$). The following theorem is fundamental for all what we shall do (cf. \cite{Wh} 4.5Q):

\begin{thm} A mapping $f\colon A\to {\C}^n$
is c-holomorphic iff it is continuous and its graph
$\Gamma_f:=\{(x,f(x))\mid x\in A\}$ is an analytic subset of
$\om\times{\C}^n$.\end{thm}

For a more detailed list of basic properties of c-holomorphic mappings see \cite{Wh} and \cite{D}.

Finally, we recall some notions we will be using. For a polynomial $P\in\mathbb{C}[x_1,\dots, x_m]$ let $P^+$ denote its homogeneous part of maximal degree, i.e. $\mathrm{deg} P^+=\mathrm{deg} P$ and $\mathrm{deg} (P-P^+)<\mathrm{deg} P$. We denote by $\tilde{P}(t,z)=\sum_{|\alpha|\leq d}a_{\alpha} z^\alpha t^{d-|\alpha|}$ the homogenization of $P(z)=\sum_{|\alpha|\leq d} a_{\alpha} z^\alpha$ where $d=\mathrm{deg} P$. Then, $\tilde{P}(0,z)=P^+(z)$.

Recall also (see \cite{L} VII.\S 7 and \cite{Ch}) that if $\Gamma\subset{\C}^n$ is algebraic of pure dimension $k$, then its projective degree $\mathrm{deg} \Gamma=\#(L\cap \Gamma)$ for any $L\subset{\C}^n$ affine subspace of dimension $n-k$ transversal to $\Gamma$ and such that $L_\infty\cap\overline{\Gamma}=\varnothing$, where $\overline{\Gamma}=\overline{\PP(\{1\}\times \Gamma)}$, for $\PP\colon{\C}^{1+n}_*\to\PP_n$ the canonical projection, is the projective closure and $L_\infty$ denotes the points of $L$ at infinity (i.e. the intersection of $\overline{L}$ with the hyperplane at infinity in $\mathbb{P}_n$). The point is that the condition  $L_\infty\cap\overline{\Gamma}=\varnothing$ (equivalently: $L_\infty\cap{\Gamma}_\infty=\varnothing$) is equivalent to the inclusion
$$
\Gamma\subset\{u+v\in L'+L\mid |v|\leq\mathrm{const.} (1+|u|)\}
$$
where $L'$ is any $k$-dimensional affine subspace such that $L'+L={\C}^n$. Moreover, for any $(n-k)$-dimensional affine subspace $L$ cutting $A$ in a zero-dimensional set (with no additional hypotheses) there is $\#(L\cap\Gamma)\leq\mathrm{deg} \Gamma$.
Writing $G'_{n-k}({\C}^n)$ for the set of affine hyperplanes of dimension $n-k$, we have
$$
\mathrm{deg} \Gamma=\max\{\#(L\cap \Gamma)\mid L\in G'_{n-k}({\C}^n)\colon \dim (L\cap \Gamma)=0\}.
$$
The projective degree of $\Gamma$ is in fact equal to the local degree (Lelong number) at zero of the cone defined by $\Gamma$. Namely, if $C_\Gamma$ denotes the closure in ${\C}\times{\C}^n$ of the pointed cone $S_\Gamma:=\{{\C}_*\cdot (1,x)\mid x\in \Gamma\}$, then $\mathrm{deg} \Gamma=\mathrm{deg}_0 C_\Gamma$. Of course, $C_\Gamma=\PP^{-1}(\overline{\Gamma})\cup\{0\}$.

Let $\Gamma^*:=\{x\in{\C}^n\mid (0,x)\in C_\Gamma\}$ which corresponds to the points at infinity but seen in the affine space. In particular, $\{P^+=0\}=\{P=0\}^*$. Moreover, $\Gamma^*=\{v\in{\C}^n\mid \exists \Gamma\ni v_\nu\to \infty, \lambda_\nu\in{\C}\colon \lambda_\nu v_\nu\to v\}$ and this is a cone. Finally, observe that now $V_\infty\cap W_\infty=\varnothing$, where $V,W\subset{\C}^n$ are algebraic, is equivalent to $V^*\cap W^*=\{0\}$.

Note by the way that with this approach it is easy to check
the following:
\begin{prop}\label{stopien wykresu}
 Let $F\colon{\C}^m\to{\C}$ be a non-constant polynomial. Then $\mathrm{deg} F=\mathrm{deg}\Gamma_F.$
\end{prop}
\begin{proof}
Observe that $\Gamma_F$ is the zero set of the polynomial $G(x,t)=t-F(x)$ of degree $d:=\mathrm{deg} F$ (we may assume that $d>1$, otherwise there is nothing to prove). Clearly, $G^+=-F^+$ and so $\Gamma_F^*=\{x\colon F^+(x)=0\}\times{\C}$. Therefore, an affine line $L\in G_1'({\C}^{m+1})$ realizing $\mathrm{deg}\Gamma_F$ can be chosen of the form $\ell\times\{t_0\}$ with $\ell\in G_1'({\C}^m)$ such that $\ell^*\cap\{F^+=0\}=\varnothing$. But then $L\cap \Gamma_F$ corresponds to the degree of $F|_\ell$ which is $d$.
\end{proof}
\begin{rem}
 It is worth noting that such an equality is no longer true for polynomial mappings (where the degree is defined to be the greatest degree of the components, which is accounted for in the next section). Indeed, it suffices to consider a proper polynomial mapping $F=(F_1,\dots, F_m)\colon{\C}^m\to{\C}^m$ such that $\mathrm{deg} F_j>1$ and $\bigcap\{F^+_j=0\}=\{0\}$. Then by the B\'ezout Theorem, for the generic $w\in{\C}^m$ (\footnote{by `generic' we always mean `apart from a nowheredense algebraic set'.}) there is $\#F^{-1}(w)=\prod\mathrm{deg} F_j>\mathrm{deg} F$ and since $F^{-1}(w)=\Gamma_F\cap({\C}^m\times\{w\}^m)$, it follows that $\mathrm{deg}\Gamma_F\geq \#F^{-1}(w)$. For instance, $m=2$, $F(x,y)=(x^2,y^3)$ is a good example.

Nonetheless, if $F\colon {\C}_x^m\to{\C}_y^n$ is a proper polynomial mapping (hence $n\geq m$), then in view of the main result of \cite{TW1} applied to the natural projection $\pi(x,y)=x$ restricted to $\Gamma_F$ (it is obviously one-sheeted), we obtain
$$
\Gamma_F\subset\{(x,y)\in{\C}^m\times{\C}^n\mid |y|\leq C(1+|x|)^{\mathrm{deg}\Gamma_F}\}
$$
with some $C>0$ (here $|\cdot|$ denotes any norm). This means that $|F(x)|\leq C(1+|x|)^{\mathrm{deg}\Gamma_F}$ and so $\mathrm{deg} F\leq\mathrm{deg}\Gamma_F$ (cf. section 3).
\end{rem}

\section{Generic injectivity of c-holomorphic non-constant mappings on fibred analytic sets}

We consider the following general situation.
Let $A\subset D\times{\C}^n$ be a pure $k$-dimensional analytic set with proper projection $\pi(x,y)=x$ onto the domain $D\subset{\C}^k$. Let $d$ denote the multiplicity of $\pi|_A$ as a branched covering and $\sigma_\pi\subsetneq D$ its critical (or discriminant) set.

We assume throughout this section that for projections $\pi'$ close to $\pi$ we still have $\pi'|_A$ a $d$-sheeted branched covering. Close, in this case, means close in the space of all epimorphisms (projections, as it were) ${\C}^{k+n}\to{\C}^k$. Equivalently, that means that $\mathrm{Ker}\pi$ and $\mathrm{Ker}\pi'$ are close in the $n$-th Grassmannian $G_n({\C}^{k+n})$, i.e. in the sense of the Kuratowski convergence (cf. \cite{DP} where this is explained in details).  As a matter of fact, we are particularly interested in the following two situations where these assumptions are satisfied:

\smallskip
\begin{enumerate}                                                                                \item Suppose that $D={\C}^k$ and $A$ is algebraic. If $\pi$ is a projection realizing the projective degree $\deg A$, then $A\subset\{(x,y)\mid |y|\leq \mathrm{const.}(1+ |x|)\}$. Thence it is easy to see that for $n$-dimensional linear subspaces $L\in G_n({\C}^{k+n})$ close enough to $L_0:=\{0\}^k\times{\C}^n$ the projection $\pi^L$ along $L$ is still proper on $A$ and has multiplicity $\mathrm{deg} A$ (\footnote{Indeed, for $L$ close to $L_0$ we will still have $A\subset \{x+u\mid x\in {\C}^k, u\in L, |u|\leq\mathrm{const.} (1+|x|)\}$; this follows from the fact that this inclusion is equivalent to the intersection at infinity being void: $A_\infty\cap \overline{L_0}=\varnothing$.}).

\smallskip
\item Suppose that $A\subset{\C}$ is locally analytic, $0\in A$ and $d$ is the local degree (Lelong number) $\mathrm{deg}_0 A$. If $\pi|_A$ realizes $\mathrm{deg}_0 A$ (as its multiplicity), then for the tangent cone $C_0(A)$, keeping the notations introduced so far, we have $L_0\cap C_0(A)=\{0\}$ and this property is open in the Grassmannian (cf. \cite{Ch} or \cite{L} (\footnote{By \cite{L} B.6.8, for any $L_0\in G_n({\C}^{k+n})$, the sets $\{L\in G_n({\C}^{k+n})\mid L\subset\{u+v\in L_0+L_0^\perp\mid |v|\leq c|u|\}\}$ for varying $c>0$ form a basis of neighbourhoods of $L_0$.})). Since there is a bounded neighbourhood $W\subset{\C}^n$ of zero such that $(\{0\}^k\times \overline{W})\cap A=\{0\}$, by taking $A\cap (G\times W)$, for some bounded neigbourhood $G\subset{\C}^k$ of the origin, instead of $A$, we may assume that $L_0\cap A=\{0\}$. Note that $L_0\cap A=\{0\}$ implies $A\cap (U\times {\C}^n)=A\cap( U\times V)$ for some bounded neighbourhoods of the origin $U\subset{\C}^k$, $V\subset{\C}^n$. It is easy to see that we may choose the same $U$ for any vector complement $L$ of ${\C}^k\times\{0\}^n$ sufficiently close to $L_0$ (\footnote{This is a simple consequence of the observation that $L_0\cap C_0(A)=\{0\}$ is equivalent to saying that $A\cap (U\times V)\subset\{(x,y)\mid |y|\leq \mathrm{const.}|x|\}$ for some neighbourhoods of zero. In our situation we can take $V={\C}^n$, as already noted. The form of neighbourhoods of $L_0$ (see the previous footnote) account for the rest.}), i.e. in such a way that $\pi^L(A\cap (U+L))=U$. Since $U$ may be chosen connected, we put $D:=U$.

\end{enumerate}

Accordingly with the notations above, we will identify the projections $\pi^L$ with their kernels $L=\mathrm{Ker}\pi^L$. Write $L_0=\mathrm{Ker}\pi$ and $L'=\mathrm{Ker}\pi'$, if necessary.

\bigskip
Let $f\colon A\to {\C}$ be a c-holomorphic function which is \textit{non-constant on any irreducible component of} $A$.

 Let $\mathscr{P}\subset G_n({\C}^{k+n})$ denote the neighbourhood of $\pi$ for which the multiplicity of the projections onto $D$ is $d$.

\begin{thm}
Under the assumptions made above, there is a projection $\pi'\in\mathscr{P}$ arbitrarily close to $\pi$ and such that $f$ is injective on the generic fibre of $\pi'|_A$.
\end{thm}
\begin{proof}
We will show that in any neighbourhood of $L_0$ in $G_n({\C}^{k+n})$, there is an $L$ such that
$$
\forall x\in D\setminus\sigma_L,\> \#f(A\cap (x+L))=d,\leqno{(\#)}
$$
for some analytic set $\sigma_L\subsetneq D$. If we denote $\Gamma_L=(\pi^L\times \mathrm{id}_{\C})(\Gamma_f)$ and $p(x,t)=x$, then condition $(\#)$ means precisely that $p|_{\Gamma_L}$ has multiplicity $d$ (it could only be less) as a branched covering. From this we infer that in $(\#)$ we may take $\exists$ instead of $\forall$.

Let $\sigma_{\pi^L}$ denote the critical set of the branched covering $\pi^L|_A$. For a given $L$ near $L_0$ put $$Z_L=\{x\in D\mid \#f(A\cap (x+L))<d\}.$$ Of course, there is always $\sigma_{\pi^L}\subset Z_L$.

If for $L_0$ we have $Z_{L_0}=D$, then given a point $x_0\in D\setminus\sigma_{\pi^{L_0}}$ and its (simply) connected neighbourhood $V\subset D\setminus\sigma_{\pi^{L_0}}$ for which $(\pi^{L_0})^{-1}(V)\cap A=\bigcup_1^d \Gamma_j$ is a disjoint union of graphs $\Gamma_j$ of holomorphic functions $\gamma_j$, there are indices $i\neq j$ such that $$\forall z\in V,\ f(z,\gamma_i(z))=f(z,\gamma_j(z)).$$ This follows easily from the identity principle. Indeed, write $g_i(z):=f(z,\gamma_i(z))$, $z\in V$. These are holomorphic functions and so $V_{ij}:=\{z\in V\mid g_i(z)=g_j(z)\}$ are analytic. By assumptions $\bigcup_{1\leq i<j\leq d} V_{ij}=V$ and so at least one of the sets $V_{ij}$ must coincide with $V$.

Let us suppose hereafter that for all $L$ in a neighbourhood $\mathscr{P}'\subset\mathscr{P}$ of $\pi$ we have $Z_L=D$.

Of course, $V$ and $\Gamma_j$ depend \textit{a priori} on $L_0$. But observe that the convergence $L\to L_0$ implies the convergence (in the Hausdorff measure) $A\cap (x_0+L)\to A\cap (x_0+L_0)$ (use for instance \cite{TW2}). Moreover, since $L_0$ is a common complement to the tangent spaces $T_{(x_0,\gamma_j(x_0))}A$, then we easily see that for all $L$ close enough to $L_0$, we will be able to find a neighbourhood $V$ of $x$ (necessarily contained in $D\setminus\sigma_{\pi_L}$ since $A$ is a union of $d$ disjoint graphs over it), i.e. $V$ can be chosen independently of $L_0$ for all $L$ sufficiently close to $L_0$.

To be more precise, suppose that we have separated the $d$ points of $A\cap (x_0+L_0)$ by small, disjoint balls $B_\iota$ of radius $\varepsilon$. Of course, the tangent spaces $T_w\Gamma_\iota$ for $w\in\Gamma_\iota\cap B_\iota$, $\iota=1,\dots, d$, do not vary much, if we take a sufficiently small $\varepsilon$, which means that for some neighbourhood $\mathscr{P}_1\subset\mathscr{P}'$ of $L_0$, any $L\in\mathscr{P}_1$ is transversal to each such tangent space. On the other hand, by \cite{TW2} there is a neighbourhood $\mathscr{P}_2\subset\mathscr{P}_1$ of $L_0$ such that for any $L\in \mathscr{P}_2$, there are $d$ points in $A\cap (x_0+L)$, each belonging to a different ball $B_\iota$. From this we easily see that a common choice of $V$ is possible (\footnote{I.e. we possibly need to shrink the neighbourhood $V$ found for $L_0$. A formal proof is based on the description of the topology of the Grassmannians as introduced in \cite{L} and in the previous footnotes and would be rather tedious to write, whereas the geometric idea is very clear.}).

Write $(\pi^L)^{-1}(V)\cap A=\bigcup_1^d \Gamma^L_j$, numbering the sheets according to the points in the fibre of $x_0$. This numbering can be made consistent thanks to the convergence $A\cap (x_0+L)\to A\cap (x_0+L_0)$. Since for each $L\in \mathscr{P}_2$ $f$ glues up two sheets, then by the same type of argument as above, it follows that for some neighbourhood $\mathscr{P}_3\subset\mathscr{P}_2$, $f$ glues up the same two sheets (i.e. with the same two indices) for all $L\in\mathscr{P}_3$.  Let $i,j$ be the two indices in question.

It remains to fix $a=(x_0,\gamma_i(x_0))\in\Gamma_i$, put $b_L=\pi^L(a)$ and $c_L=\Gamma_j^L\cap (c_L+L)$. The latter point is well defined, provided $L$ is close enough to $L_0$. We may assume $c_L$ is well defined for all $L\in \mathscr{P}_4\subset\mathscr{P}_3$ (i.e. $b_L\in V$ and we obtain this using the continuity of the fibres when $L\to L_0$). Now, $f(a)=f(c_L)$ and it is obvious that the points $c_L$ sweep up a whole open subset of $\Gamma_j$ and thus of $A$ (cf. $\Gamma_j\cap \Gamma_j^L\neq\varnothing$). But $\Gamma_j$ is contained in an irreducible component $X\subset A$. Therefore, by the identity principle for c-holomorphic functions (see \cite{D2}), $f\equiv f(a)$ on $X$ which is a contradiction.
\end{proof}
Let us state clearly what we will need later on:
\begin{cor}\label{injekcja}
Let $A\subset{\C}^m$ be an algebraic irreducible set of dimension $k$ and let $f\in\mathcal{O}_c^\mathrm{a}(A)$. If $f$ is non-constant, then for the generic choice of coordinates in ${\C}^m_z={\C}^k_x\times{\C}^{m-k}_y$ the projection $\pi(x,y)=x$ restricted to $A$ realizes $\mathrm{deg} A$ and $f|_{\pi^{-1}(x)\cap A}$ is injective for the generic $x$.
\end{cor}

In the purely local case we also have the following result that gives a kind of complement to the results of \cite{D}:

\begin{prop}\label{Rzuty}
Let $f\colon (A,0)\to
({\C}_w^k,0)$ be a non-constant c-holo\-morph\-ic germ on a pure
$k$-dimensional analytic germ $A\subset {\C}^m$. Then we can
choose coordinates in ${\C}_z^m={\C}_x^k\times {\C}_y^{m-k}$ in such
a way that for the projections $\pi(x,y)=x$, $\eta:=(\pi\times
\mathrm{id}_{{\C}_w^k})$, $p(x,y,w)=w$, $\varrho(x,w)=w$, $\zeta(x,w)=x$
and the set $\Gamma:=\eta(\Gamma_f)$, we have \begin{enumerate}
\item[{\rm (i)}] $\pi^{-1}(0)\cap C_0(A)=\{0\}$, i.e.
$\mu_0(\pi|_A)=\mathrm{deg}_0
A$;
\item[{\rm (ii)}] $\mu_0(p|_{\Gamma_f})=\mu_0(\varrho|_\Gamma)$, i.e.
$m_0(f)=\mu_0(\varrho|_\Gamma)$ and so $\mu_0(\eta|_{\Gamma_f})=1$;
\item[{\rm (iii)}] $\mu_0(\zeta|_\Gamma)=\mathrm{deg}_0 A$,\end{enumerate}
and this holds true for the generic choice of coordinates.
\end{prop}
Here $\mu_0(\pi|_A)$ denotes the {\it covering number} of the branched covering $\pi|_A$ with $0$ as the unique point in the fibre $\pi^{-1}(0)$ (see \cite{Ch}) while $m_0(f):=\mu_0(p|_{\Gamma_f})$ is the geometric multiplicity of $f$ at zero.

So as to prove this proposition we begin with a most easy lemma:
\begin{lem}\label{lemmacik}
If $E\subset {\C}^m$ is such that
$\# E=\mu>0$ and $k\leq m$, then for the generic
epimorphism $L\in \mathrm{L}({\C}^m,{\C}^k)$ one has
$\# L(E)=\mu$.
\end{lem}
\begin{proof}
It suffices to prove the assertion for
$k=m-1$. The set $\{\ell\in G_1({\C}^m)\mid \exists x,y\in E\colon
x\not= y, x\in \ell+y\}$ is finite. Thus for the generic $\ell\in
G_1({\C}^m)$ the set $\bigcup_{x\in E} x+\ell$ consists of $\mu$
distinct lines. The orthogonal projection $\pi^\ell$ along $\ell$ is
hence the sought after epimorphism.
\end{proof}

\begin{proof}[Proof of proposition \ref{Rzuty}.]
We know that for the
generic projection $\pi$ we have $\pi^{-1}(0)\cap C_0(A)=\{0\}$ (cf. [Ch]).
Let us take such a projection which in addition `separates' the
points in the maximal fibre of $f$, i.e. for $w$ such that
$f^{-1}(w)$ consists of $m_0(f)$ points, $\pi(f^{-1}(w))$ consists
also of $m_0(f)$ points (cf. the previous lemma). That means $\pi|_{f^{-1}(w)}$ is an injection.

Observe that $p=\varrho\circ\eta$ and $p|_{\Gamma_f}$,
$\eta|_{\Gamma_f}$, $\varrho|_{\Gamma}$ are proper. Moreover
$m_0(f)=\mu_0(p|_{\Gamma_f})$.
Now it remains to observe that
$$\varrho^{-1}(w)\cap\Gamma=\pi(f^{-1}(w))\times\{w\}$$
and so by the choice of $\pi$ we have
$\mu_0(\varrho|_\Gamma)=m_0(f)$. Thence
$\mu_0(\eta|_{\Gamma_f})=1$, which means that
$\eta\colon\Gamma_f\to\Gamma$ is one-to-one. Indeed, if $(x_0,w_0)\in{\C}^k\times{\C}^k$ is fixed,
\begin{align*}
\#\eta^{-1}(x_0,w_0)\cap \Gamma_f&=\#\{y\in{\C}^{m-k}\mid (x_0,y)\in A,\ f(x_0,y)=w_0\}=\\
&=\#\{z\in f^{-1}(w_0)\mid \pi(z)=x_0\}=\\
&=\#f^{-1}(w_0)\cap \pi^{-1}(x_0).
\end{align*}
The latter is equal to one iff $f|_{\pi^{-1}(x_0)\cap A}$ is injective which is equivalent to $\pi|_{f^{-1}(w_0)}$ being an injection. That we know to be true. Thus, in particular, for any
$x,w\in{\C}^k$ there exists exactly one $y\in{\C}^{m-k}$ such that
$f(x,y)=w$.

Therefore $\mu_0(\zeta|_\Gamma)=\mathrm{deg}_0 A=:d$. Indeed, if
we take $x_0\in{\C}^k$ near zero such that $\#
\pi^{-1}(x_0)\cap A=d$, then obviously $\#
f(\pi^{-1}(x_0)\cap A)\leq d$. On the other hand if there were
$y\not=y'$ such that $f(x_0,y)=f(x_0,y')=:w_0$, then the set
$\eta^{-1}(x_0,w_0)\cap\Gamma_f$ would include the two points
$(x_0,y,w_0)\not=(x_0,y',w_0)$. If this held true for $x_0$
arbitrarily close to zero, then this would contradict
$\mu_0(\eta|_{\Gamma_f})=1$.
\end{proof}

\smallskip
\noindent{\bf Note.} It may be useful, in reference to \cite{D}, to observe that in the situation from the proposition above it is easy to check that the {\sl \L ojasiewicz exponent} $\mathcal{L}(f;0)={1}/{q_0(\Gamma,\varrho)}$ (with the notations from theorem (2.6) in \cite{D}).

\section{C-holomorphic functions with algebraic graphs}

Let $|\cdot|$ denote any of the usual norms on ${\C}^m$ (we shall not distinguish in notation the norms for different $m$ as long as there is no real need for such a distinction). We begin with the following Liouville-type lemma concerning c-holo\-morph\-ic mappings whose
graphs are algebraic sets (it is a consequence of the Rudin-Sadullaev criterion):

\begin{lem}\label{2.1} Let $A\subset {\C}^m$ be a purely $k$-dimensional analytic set and let
$f\in{\ch}(A,{\C}^n)$. Then $\Gamma_f$ is algebraic if and only if $A$ is algebraic and there are
constants $M,s>0$ such that
$$|f(x)|\leq M(1+|x|^s),\quad \textit{for}\ x\in A.$$
\end{lem}

\begin{proof} For the `only if' part remark that by Chevalley's theorem $A$ is algebraic as the
proper projection of the graph. Besides, by \cite{L} VII.7.2 we get immediately
$$\Gamma_f\subset\{(z,w)\in{\C}^m\times{\C}^n\mid |w|\leq M(1+|z|^s)\}$$
for some $M,s>0$.

For simplicity sake we assume now that the considered norms are the $\ell_1$ norms i.e. sum of moduli of the coordinates.

To prove the `if' part we apply \cite{L} VII.7.4 to $A$:
$$A\subset\{(u,v)\in{\C}^k\times{\C}^{m-k}\mid |v|\leq M'(1+|u|)\}$$
(in well-chosen coordinates) for some $M'>0$. Take now $(u,v,w)\in\Gamma_f$, we have then
$|f(u,v)|\leq M(1+|(u,v)|^s)=M[1+(|u|+|v|)^s]$ and by an easy computation:
\begin{align*}
|f(u,v)|+|v|&\leq M[1+(|u|+|v|)^s]+M'(1+|u|)\leq\\

&\leq 3C(1+|u|)^{s'},
\end{align*}
for $s':=\max\{1,s\}$ and $C:=\max\{M,M(M'+1)^s,M'\}$. Now Rudin-Sadullaev criterion yields $\Gamma_f$
algebraic. \end{proof}

\begin{rem} The condition `$A$ is algebraic' in the
equivalence is not redundant since any polynomial restricted to e.g. $A=\{y=e^x\}$ satisfies the inequality but has a non algebraic graph (otherwise $A$ would be algebraic too, by the Chevalley-Remmert Theorem).

Note also that $|f(x)|\leq M(1+|x|^s)$ on $A$ iff $|f(x)|\leq M(1+|x|)^s$ on $A$.
\end{rem}

Hereafter we are interested in particular in c-holomorphic functions with algebraic graphs. For simplicity sake we will denote their ring by $\mathcal{O}_c^\mathrm{a}(A)$ when $A\subset{\C}^m$ is a fixed algebraic set. As a matter of fact, we will assume most of the time that $A\subset{\C}^m$ is a pure $k$-dimensional algebraic set of degree $d:=\mathrm{deg} A$ (meaning the degree of the projective completion of $A$). Obviously, we shall assume also $k\geq 1$ unless something else is stated.

Observe that $P\circ f\in\mathcal{O}_c^\mathrm{a}(A)$, if $f\in\mathcal{O}_c^\mathrm{a}(A)$ and $P$ is a polynomial.

In connection with Strzebo\'nski's paper \cite{S}, for any $f\in\mathcal{O}_c^\mathrm{a} (A)$ with any algebraic set $A$, we introduce its {\it growth exponent}
$$
\mathcal{B}(f):=\inf\{s\geq 0\mid |f(x)|\leq C(1+|x|)^s,\ \hbox{\it on}\ A\ \hbox{\it with some constant}\ C>0\}
$$
and the set of all possible growth exponents on $A$:
$$
\mathcal{B}_A:=\{\mathcal{B}(f)\mid f\in\mathcal{O}_c(A)\colon \Gamma_f\ \hbox{\it is algebraic}\}.
$$
Observe that there is in fact
$$
\mathcal{B}(f)=\inf\{s\geq 0\mid |f(x)|\leq \mathrm{const.}|x|^s,\ x\in A\colon |x|\geq M\ \hbox{\it with some}\ M\geq 1\}.
$$
The growth exponent replaces in the c-holomorphic setting the notion of the {\it degree} of a polynomial (if $A={\C}^m$, then obviously $\mathcal{O}_c^\mathrm{a}(A)={\C}[x_1,\dots, x_m]$ and so $\mathcal{B}=\mathrm{deg}$ is the usual degree). Note by the way that for a  mapping $f=(f_1,\dots, f_n)$ with algebraic graph, defined on $A\subset{\C}^m$, $\mathcal{B}(f):=\max\mathcal{B}(f_j)$ is once again the minimal exponent $s$ for which $|f(x)|\leq\mathrm{const.}|x|^s$ for all $x\in A$ large enough.

Recall the following important lemma from \cite{S}:

\begin{lem}[\cite{S} lemma 2.3]\label{Adam}
Let $P(x,t)=t^d+a_1(x)t^{d-1}+\ldots+a_d(x)$ be a polynomial with $a_j\in{\C}[x_1,\dots,x_k]$. Then $\delta(P):=\max_{j=1}^d ({\mathrm{deg} a_j}/{j})$ is the minimal exponent $s>0$ for which the inclusion
$$
P^{-1}(0)\subset \{(x,t)\in{\C}^k\times{\C}\mid |t|\leq C(1+|x|)^s \}
$$
holds with some $C>0$.
\end{lem}

Note that in view of our preceding observation it is merely an avatar of the following P\l oski's crucial lemma:
\begin{lem}[\cite{P} lemma (2.1)]\label{P} If $P(x,t)$ is as above, then $\delta(P)$ is the minimal exponent $q>0$ such that
$$
\{(x,t)\in{\C}^k\times{\C}\mid P(x,t)=0, |x|\geq R\}\subset\{(x,t)\in{\C}^k\times{\C}\mid |t|\leq C|x|^q\}
$$
for some $R, C>0$.
\end{lem}

From Lemma \ref{Adam} we easily obtain the c-holomorphic counterpart of Strze\-bo\'nski's result from \cite{S} by taking the image $\Gamma$ of the graph $\Gamma_f$ by the projection $\pi\times\mathrm{id}_{{\C}}$, where $\pi\colon{\C}^m\to{\C}^k$ is a projection realizing $\mathrm{deg} A$. Then one can apply Lemma \ref{Adam} to $\Gamma$: it remains to observe that $\mathcal{B}(f)=\delta(P)$, where $P$ is the minimal polynomial describing the hypersurface $\Gamma$. We obtain the following theorem:

\begin{thm}\label{AG2.3} For any algebraic set $A$ and any $f\in\mathcal{O}_c^\mathrm{a}(A)$, the number $\mathcal{B}(f)$ is rational and is a growth exponent of $f$. Moreover, one has $$\mathbb{Z}_+\subset\mathcal{B}_A\subset\{p/q\mid p,q\in\mathbb{N}\colon 1\leq q\leq d,\ p,q\ \hbox{\it relatively prime}\},$$
where $d$ is the maximum of degrees of all the irreducible components of $A$.\end{thm}

\smallskip
P. Tworzewski conjectures that in this case the second inclusion is in fact an equality (when $A$ is irreducible).

\smallskip
In the second part of this paper we shall need some more information about $\mathcal{B}(f)$. It is easy to see from the definition that for any $h_1, h_2\in\mathcal{O}_c^\mathrm{a}(A)$ there is $\mathcal{B}(h_1h_2)\leq \mathcal{B}(h_1)+\mathcal{B}(h_2)$ and  $\mathcal{B}(h_1+h_2)\leq\max\{\mathcal{B}(h_1),\mathcal{B}(h_2)\}$. But what will turn out to be most important is that for any positive integer $n$ there is $\mathcal{B}(f^n)=n\mathcal{B}(f)$.

\smallskip
Using the results of \cite{TW1} we are able to give an estimate of $\mathcal{B}(f)$. Indeed, by applying \cite{TW1} theorem 3 we get in lemma \ref{Adam} above the estimate $\delta(P)\leq\mathrm{deg} P^{-1}(0)-d+1$. It remains to specify what actually $\mathrm{deg} P^{-1}(0)$ and $d$ are.

\smallskip

\begin{thm}
Let $f\in\mathcal{O}_c^\mathrm{a}(A)$ with $A\subset{\C}^m$ of dimension $k\geq 0$. Then
$$\mathcal{B}(f)\leq \mathrm{deg}\Gamma_f-\mathrm{deg} A+1.$$
\end{thm}
\begin{proof}
If $f$ is constant or $k=0$, then $\mathcal{B}(f)=0$ and the estimate holds. We may assume thus $f$ non-constant and $k>0$.

Suppose first that $A$ is irreducible.

Let $\pi\colon {\C}^m\to{\C}^k$ be a projection realizing the degree $\mathrm{deg} A$ and let $\Gamma:=(\pi\times\mathrm{id}_{\C})(\Gamma_f)$. The latter is clearly an algebraic set due to Remmert-Chevalley Theorem. A straightforward application of the main result of \cite{TW1} gives now $\mathcal{B}(f)\leq\mathrm{deg} \Gamma-d+1$, where $d$ is the covering number of the branched covering $\zeta\colon {\C}^k\times{\C}\to{\C}^k$ on $\Gamma$.

It is easy to see that $\mathrm{deg}\Gamma\leq\mathrm{deg}\Gamma_f$. Indeed, if $\ell\subset{\C}^{k+1}$ is an affine complex line such that $\mathrm{deg}\Gamma=\#(\ell\cap \Gamma)$, then the set $L:=\{z\in{\C}^m\mid (\pi\times\mathrm{id}_{\C})(z)\in \ell\}$ is an affine space of dimension $m+1-k$ intersecting $\Gamma_f$ in a zero-dimensional set. This follows from the properness of $\pi\times\mathrm{id}_{\C}$ on $\Gamma_f$. Therefore, we have $\#(L\cap\Gamma_f)\leq\mathrm{deg}\Gamma_f$ (cf. \cite{L} VII.11). Clearly  $\#(\ell\cap \Gamma)\leq\#(L\cap\Gamma_f)$.

The point is how to choose $\pi$ so as to have $d=\mathrm{deg} A$.

We are able to do this thanks to Corollary \ref{injekcja} asserting that for the generic $\pi$ we have $\#f(\pi^{-1}(x)\cap A)=\mathrm{deg} A$ for the generic $x\in{\C}^k$.

Now, if $A$ is reducible, then we apply the preceding argument to each irreducible component $S\subset A$ and $f|_S$ getting
$$
\mathcal{B}(f|_S)\leq\mathrm{deg} \Gamma_{f|_S}-\mathrm{deg} S+1.
$$ Observe that $\mathcal{B}(f)=\max\{\mathcal{B}(f|_S)\mid S\subset A\ \hbox{\it an irreducible component}\}$ (cf. \cite{S}) and $\mathrm{deg}\Gamma_f=\sum_S\mathrm{deg}\Gamma_{f|_S}$, since $\Gamma_{f|_S}$ is irreducible iff $S\subset A$ is irreducible (thus $\Gamma_{f|_S}$ are the irreducible components of the graph). Therefore
$$
\mathcal{B}(f)\leq\max_{S\subset A}(\mathrm{deg} \Gamma_{f|_S}-\mathrm{deg} S)+1,
$$
but since for each irreducible component $S'$ there is $$\mathrm{deg} \Gamma_{f|_{S'}}-\mathrm{deg} S'\leq\sum_{S\subset A} (\mathrm{deg} \Gamma_{f|_S}-\mathrm{deg} S),$$ we finally obtain the required inequality.
\end{proof}
The following example shows that the estimate is far from being the best one. Nevertheless, it is of some interest in view of the proposition following the example.
\begin{ex}\label{przyklad}  Let $A$ be the algebraic curve
$\{(x,y)\in {\C}^2\mid x^3=y^2\}$ and consider the c-holomorphic function
$$
f(x,y)=
\begin{cases}
\displaystyle
\frac{y}{x},\> \textit{\ for}\> (x,y)\in A\setminus (0,0)\\
0, \>\textit{\ for}\> x=y=0
\end{cases}
$$
It is easy to see (cf. Lemma \ref{2.1}) that $\Gamma_f$ is algebraic and $\mathcal{B}(f)=1/3$. Actually the algebricity of $\Gamma_f$ is not really
surprising because $f$ is the restriction to $\mathrm{Reg} A$ of a rational function. We will
show (next section) that in fact the algebricity of the graph is equivalent in this case to the fact that
the function has a rational `extension'.
\end{ex}

\begin{prop}
 Let $f\in\mathcal{O}^\mathrm{a}_c(A)$ with $A$ pure $k$-dimensional. Then the following conditions are equivalent:
\begin{enumerate}
 \item $\mathcal{B}(f)\leq 1$;
\item $\mathrm{deg} A=\mathrm{deg}\Gamma_f$;
\item $\Gamma_f^*\cap (\{0\}\times{\C})=\{0\}$;
\item $\{P=0\}^*\cap (\{0\}\times{\C})=\{0\}$,
\end{enumerate}
where $P$ is the minimal polynomial describing $(\pi\times\mathrm{id}_{\C})(\Gamma_f)$ with $\pi\colon {\C}^m\to {\C}^k$ a projection realizing $\mathrm{deg} A$.
\end{prop}
\begin{proof}

The implication from $(2)$ to $(1)$ follows from the preceding theorem. To prove the converse, suppose that coordinates in ${\C}^m={\C}^k_x\times{\C}_y^{m-k}$ are chosen in such a way that the projection onto the first $k$ coordinates realizes $\mathrm{deg} A$. Then $A\subset\{|y|\leq M(1+|x|)\}$. On the other hand, since $\mathcal{B}(f)\leq 1$, there is $|f(x,y)|\leq C(1+|x|+|y|)$ for $(x,y)\in A$. Combining the two facts, we obtain
$$
\Gamma\subset\{(x,y,t)\in{\C}^k\times{\C}^{m-k}\times{\C}\mid |y|+|t|\leq [M+C(1+M)](1+|x|)\}
$$
which implies that the projection $(x,y,t)\mapsto x$ realizes $\mathrm{deg}\Gamma_f$. Clearly, by the univalence of the graph, the multiplicity of this projection is equal to $\mathrm{deg} A$.

Now, we turn to proving the equivalence of the remaining conditions.
Choose affine coordinates so as to have ${\C}^m_z={\C}^k_x\times{\C}^{m-k}_y$ and $\pi(z)=x$. Write $P(x,t)=t^d+a_1(x)t^{d-1}+\ldots+a_d(x)$ and observe that $\mathrm{deg} P=\max\{d,\mathrm{deg} a_1+d-1,\dots, \mathrm{deg} a_d\}$. Therefore, in view of the fact that $\mathcal{B}(f)=\delta(P)$, $(1)$ is equivalent to $\mathrm{deg} P=d$. This in turn is equivalent to $(4)$, since $\{P=0\}^*=\{P^+=0\}$.

Now, clearly $\Gamma_f^*\supset \{0\}\times{\C}$ implies $\{P=0\}^*\supset\{0\}\times{\C}$ and thus $(4)$ implies $(3)$. On the other hand, by the choice of $\pi$,
$A\subset\{|y|\leq c(1+|x|)\}$ and so for all $(x,y)\in A$ large enough, $|y|\leq c'|x|$. Take a point $(0,t)\in\{P=0\}^*$. There are sequences $\{P=0\}\ni (x_\nu, t_\nu)\to\infty$ and $\lambda_\nu\in{\C}$ such that $\lambda_\nu(x_\nu, t_\nu)\to (0,t)$. Of course, there is a sequence $y_\nu\in{\C}^{m-k}$ such that $(x_\nu,y_\nu)\in A$ and $t_\nu=f(x_\nu,y_\nu)$. For all $\nu$ large enough, $|\lambda_\nu y_\nu|\leq c'|\lambda_\nu x_\nu|\to 0$, whence $(0,t)\in \Gamma_f^*$. Therefore, $(3)$ implies $(4)$.
\end{proof}

\section{Algebraic Graph Theorem}

Using Oka's theorem about universal denominators (cf. \cite{TsY} and \cite{Wh}) one can show that any c-holomorphic
function admits locally a universal denominator. We will detail this a little more
in the proof of the following theorem. For the convienience of the reader let us start with one useful construction of a universal denominator.

\begin{prop}\label{3.3} Let $A\subset U\times{\C}_t\times{\C}_y^{m-k}$ be a pure $k$-dimensional analytic set, where $U\subset{\C}_x^{k}$ is open and connected, such that $0\in A$ and the natural projection $\pi(x,t,y)=x$ is proper on $A$ and realizing $\mathrm{deg}_0 A=:d$. Then after a change of coordinates in ${\C}\times{\C}^{m-k}$ there exists a monic polynomial $P\in\mathcal{O}(U)[t]$ of degree $d$ such that $Q(x,t,y):=\frac{\partial P}{\partial t}(x,t)$ is a universal denominator at each point $a\in A$.
\end{prop}

\begin{proof} Let $\rho(x,t,y)=(x,t)$ and $\xi(x,t)=x$ be the natural projections. For any point $x\in U$ not critical for $\pi|_A$ we have exactly $d$ distinct points $(t_1,y^1),\dots, (t_d, y^d)$ over it in $A$. If we fix $x$, then taking if necessary a rotation in ${\C}\times{\C}^{m-k}$, we may assume that all the points $t_1, \dots, t_d$ are distinct. Thus $\xi$ on $\rho(A)$ has multiplicity $d$ as a branched covering. Note that by the Remmert theorem $\rho(A)\subset U\times{\C}$ is an analytic hypersurface. Thus there exist a reduced Weierstrass polynomial $P\in\mathcal{O}[t]$ such that $P^{-1}(0)=\rho(A)$. Its degree is obviously $d$.

Now for fixed $x$ in a simply connected neighbourhood $V$ not intersecting the critical set of $P$ we have $(t_1(x),y^1(x)),\dots, (t_d(x),y^d(x))$, exactly $d$ distinct points. Put
$$
h(x,t):=\sum_{j=1}^d f(x,t_j(x),y^j(x))\prod_{\iota\neq j} (t-t_\iota(x)),\quad (x,t)\in V\times {\C}.
$$
Observe that $h(x,t_j(x))=f(x,t_j(x),y^j(x))Q(x,t_j(x))$. Clearly the function $h(x,t)$ is locally holomorphic apart from the critical set of $f$ (because the functions $t_j(x)$ and $y^j(x)$ are locally holomorphic) and locally bounded near the critical points of $P$, and so by the Riemann theorem we obtain a holomorphic function $h\in\mathcal{O}(U\times {\C}\times{\C}^{m-k})$ being an extension of $Qf$. \end{proof}

\begin{thm}\label{3.4} Let $A\subset{\C}^m$ be a purely $k$-dimensional algebraic
set and let $f\in{\ch}(A)$. Then $\Gamma_f$ is algebraic if and only if there exists a rational
function $R\in{\C}(x_1,\dots,x_m)$ equal to $f$ on $A$ (in particular $R|_A$ is continuous). More precisely, there exists a polynomial $Q\in{\C}[x_1,\dots,x_m]$ with $\mathrm{deg} Q<\mathrm{deg} A$ such that $f=P/Q$ on $A$ for some polynomial $P\in{\C}[x_1,\dots, x_m]$.\end{thm}

\begin{proof} If $m=1$, then either $A$ is the whole ${\C}$, and then by the identity principle
$\Gamma_f$ is algebraic if and only if $f$ is a polynomial (cf. Serre's theorem on the algebraic graph),
or $A$ is a finite set and we apply a Lagrange interpolation. In both cases $Q\equiv 1$. Hence we may confine us to the case
$m\geq 2$.

If $k=0$, then $\#A<\infty$.
We follow lemma \ref{lemmacik}. The set $$\{\ell\in G_1({\C}^m)\mid \exists x,y\in A,\ x\neq y\colon y\in x+\ell\}$$ is finite (even algebraic). Take thus a line $\ell\in G_1({\C}^m)$ such that for all $x\in A$, $(x+\ell)\cap A=\{x\}$. If we denote by $\pi^\ell$ the natural projection along $\ell$ onto its orthogonal complement $\ell^\bot$, then $\#\pi^\ell(A)=\# A$. Continuing this procedure we find a one-dimensional subspace $L\subset{\C}^m$ such that $\#\pi(A)=\#A$, where $\pi$ is the orthogonal projection onto $L$. Now the Lagrange interpolation for $\pi(A)$ and the values $f(a), \pi(a)\in\pi(A)$ yields a polynomial $P\in{\C}[t]$. Then $\widetilde{P}(x):=P(\pi(x))$ is the polynomial interpolating $f$ on $A$. The `only if' part is clear.

\smallskip
Assume now that $k\geq 1$.

Since $A$ is
algebraic of pure dimension $k$, there are coordinates in ${\C}^m$ such that the projection
$\pi$ onto the first $k$ coordinates is proper on $A$ (so it is a branched covering) and it realizes $\mathrm{deg} A$.

Now if we take $\rho(x_1,\dots, x_m)=(x_1,\dots, x_{k+1})$, then we are able to apply proposition \ref{3.3} getting a polynomial (since by Chevalley's theorem $\rho(A)$ is an algebraic hypersurface) $Q$ being a local universal denominator for $A$. Since ${\C}^m$ is a domain of holomorphy, $Q$ is in fact a global universal denominator for $A$ (actually this follows directly from the proof of \ref{3.3}).

That means that there exists $h\in\mathcal{O}({\C}^m)$ such that
$$
f=\frac{h}{Q}\quad \hbox{on}\ \mathrm{Reg} A.
$$
Note that for points $a\in\mathrm{Sng} A$, if we take any sequence $\mathrm{Reg} A\ni a_\nu\to a$, then by continuity we obtain $f(a)Q(a)=h(a)$. Therefore either $a$ is a point in which $h/Q$ is well defined, or it is a point of indeterminacy of $h/Q$. In the latter case, the function $h/Q$ has a finite and well defined limit along $A$, namely $f(a)$. Thus $h/Q$ is continous on $A$.

\smallskip
As a matter of fact $h$ is not uniquely determined. The proof of our theorem consists now in showing
\begin{enumerate}
\item[(a)] If $h$ is a polynomial, then $\Gamma_f$ is algebraic;

\item[(b)] If $\Gamma_f$ is algebraic, then we may choose $h\in{\C}[x_1,\dots,x_m]$.
\end{enumerate}
\noindent Ad (a):

Let $X:=(A\times{\C})\cap\{(x,t)\in{\C}^m\times{\C}\mid h(x)=Q(x)t\}$. It is an algebraic set of dimension at least $k$. Over points $x\in A\setminus Q^{-1}(0)$ this is exactly the graph of $f$. Thus for each such point $x$ and the only one  $t$ for which $(x,t)\in X$, we have $\mathrm{dim}_{(x,t)} X=k$. On the other hand, since $Q$ does not vanish on any irreducible component of $A$, the set $A\cap Q^{-1}(0)$ has pure dimension $k-1$ (see \cite{D2}). For each point $x\in A\cap Q^{-1}(0)$ we have a whole line $\{x\}\times{\C}\subset X$. Thus the set $X$ has pure dimension $k$.

Set $\Gamma:=\Gamma_f\setminus(Q^{-1}(0)\times{\C})=\Gamma_f\setminus [(A\cap Q^{-1}(0))\times{\C}]$. Then we have $\Gamma\subset X$ and so for closures $\overline{\Gamma}\subset\overline{X}=X$. But by continuity $\overline{\Gamma}=\Gamma_f$, and since $\Gamma_f$ has pure dimension $k$ it must be the union of some irreducible components of $X$. Since $X$ is algebraic, so is $\Gamma_f$.

\smallskip
\noindent Ad (b): This follows from Serre's algebraic graph theorem (for regular functions, see \cite{L}). Indeed, $fQ$ is a holomorphic function in ${\C}^m$ with algebraic graph over the algebraic set $A$ (to see this apply lemma \ref{2.1}; one can remark by the way that $\mathcal{B}(fQ)\leq\mathcal{B}(f)+\mathcal{B}(Q|_A)$). Thus it is on $A$ a regular function which means that it is in fact the restriction to $A$ of a polynomial $P$. \end{proof}

\begin{rem}
It is easy to check that in the theorem above we obtain $$\mathcal{B}(f)\geq\mathcal{B}(P|_A)-\mathcal{B}(Q|_A).$$
\end{rem}

\section{Proper c-holomorphic mappings with algebraic graphs}\label{proper}
C-holomorphic functions with algebraic graphs are a promising generalization of polynomials onto algebraic sets. Most of the theorems known for instance for polynomial dominating mappings should have their analogues at least for c-holomorphic proper mappings with algebraic graphs. Note, however, that in this setting we are naturally obliged to make do more with the geometric structure than the algebraic one (that is a hindrance when trying to extend the results of \cite{D2} to the c-holomorphic algebraic case).

We consider now the following situation:\\ Let $A\subset{\C}^m$ be algebraic of pure dimension $k>0$ and suppose $f\in\mathcal{O}_c(A,{\C}^k)$ is a proper mapping with algebraic graph. It is clear then that for each component $f_j$ of $f$ has an algebraic graph.

Since $\Gamma_f$ is algebraic with proper projection onto ${\C}^k$, then $\# f^{-1}(w)$ is constant for the generic $w\in{\C}^k$. We call this number, denoted by $\mathrm{d}(f)$, the {\it geometric degree} of $f$ just as in the polynomial case. We call {\it critical} for $f$ any point $w\in{\C}^k$ for which $\#f^{-1}(w)\neq\mathrm{d}(f)$. In that case one has actually $\#f^{-1}(w)<\mathrm{d}(f)$ (cf. e.g. \cite{Ch}, the projection onto ${\C}^k$ restricted to $\Gamma_f$ is a $\mathrm{d}(f)$-sheeted branched covering). Obviously $\mathrm{d}(f)\leq\mathrm{deg}\Gamma_f$ (cf. \cite{L}).

Of course, we could define this degree in a more general setting. To see this consider the set $W=\{x\in{\C}^k\mid \dim\pi^{-1}(x_0)\cap V=0\}$, where $V\subset{\C}^m\times{\C}^k$ is a pure $k$-dimensional algebraic set and $\pi$ the projection onto ${\C}^k$. Clearly, $W$ is open and $Z:={\C}^k\setminus W$ is algebraic (as it is algebraically constructible and closed). Moreover, $\mathrm{int} Z=\varnothing$, since otherwise we would have $\dim Z=k$ (the dimension being computed along \L ojasiewicz) whence $\dim A\cap (Z\times{\C}^{m-k})>k$ which is contrary to $\dim A=k$. Now we may conlude that $D:={\C}^k\setminus Z$ is connected and since $\pi$ is proper on $A\cap (D\times{\C}^{m-k})$, we have a well-defined multiplicity, denote it by $\mathrm{d}(\pi|_A)$, of this projection. We call this number the \textit{generic multiplicity}.

Now, if $f\colon \mathcal{O}_c^\mathrm{a}(A,{\C}^k)$ is non-constant on at least one irreducible component of the pure $k$-dimensional algebraic set $A\subset{\C}^m$ (\footnote{It is not equivalent to saying that $f$ is non-constant on $A$, think of $f$ defined as 0 on ${\C}\times \{0\}$ and 1 on ${\C}\times \{1\}$.}), then it follows easily from \cite{D2} that there exists a point $w_0$ for which $\dim f^{-1}(w_0)=0$. If $\pi$ denotes the projection onto the target space, then we call $\mathrm{d}(f):=\mathrm{d}(\pi|_{\Gamma_f})$ the {\it generic multiplicity} of $f$.

Of course, we can argue differently: in this case $\pi$ is a regular dominating mapping on $\Gamma_f\to{\C}^k$ in this case and so it has a generic multiplicity.

Anyway, we stick to the case of proper mappings. Similarly to the polynomial case, we have the following B\'ezout-type theorem:
\begin{prop}\label{Bezout}
Let $f\colon A\to{\C}^k$ be a c-holomorphic proper mapping with algebraic graph. Then
$$
\mathrm{d}(f)\leq\mathrm{deg} A\prod_{j=1}^k\mathcal{B}(f_j).
$$
\end{prop}
\begin{proof}
Let $q_j$ be any positive integers such that $q_j\mathcal{B}(f_j)\in \mathbb{N}$ for $j=1,\dots,k$. Then set $F:=(f_1^{q_1},\dots,f_k^{q_k})$. We still have $F\in\mathcal{O}_c^\mathrm{a}(A)$ and $F$ is proper with $\mathrm{d}(F)=\mathrm{d}(f)\prod_j q_j$. Besides, $\mathcal{B}(F_j)=q_j\mathcal{B}(f_j)$.

The idea now is to follow the idea used in the proof of proposition (4.6) from \cite{D} inspired by the methods of P\l oski and Tworzewski. To that aim consider the algebraic set
$$
\Gamma:=\{(z,w)\in A\times{\C}^k\mid w_j^{\mathcal{B}(F_j)}=F_j(z),\ j=1,\dots, k\}.
$$
Clearly, for any $a\in \Gamma$, there is $\mathrm{dim}_a\Gamma\geq k$ and since $\Gamma$ has proper projection $p(z,w)=z$ onto $A$, the converse inequality holds too and so $\Gamma$ is pure $k$-dimensional.

Take now any affine subspace $\ell\subset{\C}^m$ of dimension $k$ such that $\#(\ell\cap A)=\mathrm{deg} A$ and
$$
A\subset\{x+y\in\ell^\bot+\ell\mid |y|\leq C(1+|x|)\},
$$
where $\ell^\bot$ is an orthogonal complementary to $\ell$, $x+y=z$ and $C>0$ a constant. Then by construction $L:=\ell+{\C}^k$ (seen in ${\C}^{m+k}$) is transversal to $\Gamma$ and we have $\#(L\cap \Gamma)=\mathrm{deg} A\prod_j\mathcal{B}(F_j)$. We may assume that the norm in consideration is the sum of moduli. Now observe that for $(z,w)\in\Gamma$,
$$
|w_j|^{\mathcal{B}(F_j)}=|F_j(z)|\leq c_j|z|^{\mathcal{B}(F_j)}\ \hbox{when}\ |z|\geq R_j,
$$
for some $c_j,R_j>0$. Then $|w|\leq(\max_j c_j)|z|$ when $|z|\geq \max_j R_j$. Therefore, there exists a constant $K>0$ such that
$$
\Gamma\subset\{(x,y,w)\in\ell^\bot+\ell+{\C}^k\mid |y|+|w|\leq K(1+|x|)\}
$$
and so $\mathrm{deg}\Gamma=\mathrm{deg} A\prod_j\mathcal{B}(F_j)$.

Finally, it suffices to remark that one has $\mathrm{d}(F)\leq\mathrm{deg}\Gamma$ since there is  $\mathrm{d}(F)=\#(({\C}^m\times\{w\}^k)\cap\Gamma)$ for a well chosen $w$.
\end{proof}

\begin{ex}
Let $A$ and $f$ be as in Example \ref{przyklad}.
Since $f$ is injective, $\mathrm{d}(f)=1$. Clearly $\mathrm{deg} A=3$ and $\mathcal{B}(f)=1/3$. Thus $\mathrm{d}(f)=\mathrm{deg} A\cdot\mathcal{B}(f)$.
\end{ex}

This example hints at a more general observation:
\begin{prop}\label{krzywa}
Let $\Gamma\subset{\C}^m$ be an algebraic curve normalized by the polynomial mapping $\gamma=(\gamma_1,\dots,\gamma_m)\colon {\C}\to{\C}^m$ with $\mathrm{deg}\Gamma=\max_j\mathrm{deg}\gamma_j$ and let $f\in\mathcal{O}_c^\mathrm{a}(\Gamma)$ be non-constant. Then
$$
{\mathrm{d}(f)}={\mathrm{deg} \Gamma}\cdot \mathcal{B}(f).
$$
Moreover, $\mathcal{B}(f)$ is the greatest exponent (the \L ojasiewicz exponent) $q>0$ for which $|f(x)|\geq \mathrm{const}\cdot|x|^q$ when $x\in \Gamma$ is big enough.
\end{prop}
\begin{proof}
The idea of the proof is similar to that of theorem (3.2) from \cite{D}. We may assume that $|\cdot|$ is the maximum norm. Let $d:=\mathrm{deg}\Gamma$ and observe that since
$$
\lim_{|t|\to+\infty} \frac{|\gamma(t)|}{|t|^d}=\mathrm{const.}>0,
$$
the inequalities $c_1|x|^q\leq|f(x)|\leq c_2\cdot |x|^{\mathcal{B}(f)}$ for $x\in \Gamma$ with $|x|\gg 1$ are equivalent to
$$
c_1'|t|^{dq}\leq|f(\gamma(t))|\leq c_2'\cdot |t|^{d\mathcal{B}(f)},\quad |t|\gg 1
$$
with some other constants $c'_1,c_2'>0$.
Observe now that $f\circ \gamma$ is a polynomial by Serre's Graph Theorem and so there are two positive constants $c_1,c_2$ such that
$$
c''_1 |t|^{\mathrm{deg} (f\circ\gamma)}\leq |f(\gamma(t))|\leq c''_2\cdot |t|^{\mathrm{deg} (f\circ\gamma)},\quad |t|\gg 1.\leqno{(\circ)}
$$
But ${\mathrm{deg} (f\circ\gamma)}=\mathrm{d}(f)$ because $\gamma$ being generically injective, we obviously have $\#(f\circ\gamma)^{-1}(w)=\#f^{-1}(w)$ for the generic $w\in{\C}$. We have now on the one hand
$$\lim_{|t|\to+\infty}|t|^{d\mathcal{B}-\mathrm{d}(f)}\geq c''_1/c_2'\quad\hbox{and}\quad \lim_{|t|\to+\infty}|t|^{\mathrm{d}(f)-d\mathcal{B}(f)}\leq c'_2/c_1'',$$
while on the other $\lim_{|t|\to+\infty}|t|^{d(\mathcal{B}(f)-q)}\geq c'_1/c_2'$.
Therefore $\mathrm{d}(f)= d\mathcal{B}(f)$ and $q\leq\mathcal{B}(f)$. Moreover, the first inequality from $(\circ)$ is equivalent to $\mathrm{const}\cdot |x|^{\mathrm{d}(f)/d}\leq|f(x)|$ for $x\in\Gamma$ large enough. That ends the proof.
\end{proof}
\begin{rem}
The statement of the Proposition above remains true for $f\in{\oa}(\Gamma,{\C}^n)$ with $n>1$ as can be easily seen from the proof. In that case $\mathrm{d}(f)$ is defined in a slightly different way. Namely,
$$
\mathrm{d}(f):=\max_{j=1}^n\mathrm{deg} (f_j\circ\gamma),
$$
i.e. $\mathrm{d}(f):=\mathrm{deg} (f\circ\gamma)$ ($f\circ\gamma$ is a polynomial mapping).
\end{rem}

\section{Acknowledgements}
The author would like to thank Piotr Tworzewski for introducing him into the subject of c-holomorphic functions.

During the preparation of this note, the author was partially supported by Polish Ministry of Science and Higher Education grant  IP2011 009571.

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