%\nopagenumbers \topskip 25pt
\magnification=\magstep1
\input amstex
%\nopagenumbers
%\def\Log{\text{\,\rm Log}\,}
\define\Mul{\operatorname{Mul}}
\define\ir{\text{\rom{ir}}}
\define\Log{\operatorname{Log}}
\NoBlackBoxes
\documentstyle{amsppt}

\topmatter
\title{ A Function $\Log$ on Space of $n$-Dimensional Cycles and
Parshin-Kato Theory\footnotemark"*"}
\endtitle
\rightheadtext{ A Function $\Log$  and Parshin-Kato Theory}
 \footnotetext"*"{The work was partly supported by Cnadian grant
OGP0156833. }
\author{ A. Khovanskiii}
\endauthor
 \endtopmatter
 \document

\head{1. One dimensional case}
\endhead

Consider a complex algebraic curve $X$ and two nonzero meromorphic
functions $f$, $g$ on $X$. With each point $a\in X$ one can
associate the so-called  the  Weil symbol $[f,g]_a$.

Let $u$ be a local parameter about the point $a$, $u(a)=0$, and
let $f=c_1u^{k_1}+\dots$ and $g=c_2u^{k_2}+\dots$ be the leading
terms of the Laurent expansion of the functions $f$ and $g$ at the
point $a$.

{\bf The Weil symbol $[f,g]_a$ is the num\-ber} $(-1)^{k_1k_2}
c_1^{k_2}c_2^{-k_1}$.

For all the points on the curve $X$  except for  zeros and poles
of f and g, the Weil symbol is equal to one.


\proclaim{Weil's theorem} {\sl $$ \prod \limits_{a\in X
}[f,g]_a=1. $$}
\endproclaim

\example{Example (Weil's theorem and Vieta formula)} Consider the
Riemann sphere as the curve $X$, the coordinate function $x$ as
the function $f$, and  a polynomial $P = a_nx^n+\dots +a_kx^k$ of
degree $n$ as the function $g$. According to Weil's theorem we get
$$ \prod
x(a)=[x,P]_0^{-1}[x,P]^{-1}_{\infty}=(-1)^{-k}a_k^{-1}(-1)^n a_n
=(-1)^{n-k}a_n/a_k $$ where the product is taken over all nonzero
roots $a$ of the polynomial $P$. This formula coincides with the
Vieta formula.
\endexample



 \subhead{\bf 1.1. Toric Surfaces and Weil's Theorem [5]}
\endsubhead
Consider a a compact (maybe singular) toric surface $M$. Let $D$
be a zero-dimensional positive divisor located on the union of
one-dimensional orbits. Does an algebraic curve on the surface~$M$
exist that does not pass through zero-dimensional orbits and
intersects one-dimensional orbits at the given  divisor D?


Let us fix the orientation in the plane of one-parameter subgroups
of $\Bbb C^{*2}$. Thereby we fix the parameterization $\pi_j :
\Bbb C^* \rightarrow M_j $ on each  one-dimensional orbit $M_j$ in
$M$. Consider a map $\pi:\bigcup_j M_j\rightarrow \Bbb C^*$ which
restriction on $M_j$ equals to $\pi_j^{-1}$.



\proclaim {Theorem} If $D=\sum k_i a_i$ is the divisor of
intersection of a curve not passing through zero-dimensional
orbits of $M$ with the union of one-dimensional orbits of $M$ then
$$\prod (-\pi(a))^{k_i} =1$$

\endproclaim

\demo{Sketch of the proof} Each curve in~$\Bbb (C^*)^2$ is given
by an equation $P=0$, where $P$ is a Laurent polynomial. Let
$\Delta$ be its Newton polygons. On each side $\bold n_j$ of
$\Delta$ a polynomial $P_{\bold n_j}$ in one variable is written.
According to Vieta formula the product of all nonzero roots of all
polynomials $P_{\bold n_j}$ is equal to 1.
\enddemo

The Weil's theorem follows from the previous theorem (see~[5]).

\midinsert \vspace{5cm} \botcaption{Fig 1. Newton polygon for a
laurant polynomial $P(x,y)$ such that $P(f,g)\equiv 0$}
\endcaption
\endinsert


\subhead{\bf 1.2. Topological Proof of the Weil's theorem}
\endsubhead
Consider a complex algebraic curve $X$ and  two nonzero
meromorphic functions $f$ and $g$ on $X$. Let $A$ be the union of
the supports of the principle divisors  $(f)$ and $(g)$ and let
$U$ be $X\setminus A$.

{\bf With the couple $\bold f,\bold g$  one can associate  a
certain cohomology class $\bold{[f,g]\in H^1(U,\bold C^*)}$} which
is:

1)  skew symmetric $[f,g]=[g,f]^{-1}$,

2) multiplicative  $[f_1 f_2, g]=[f_1, g][ f_2, g]$


3) its value $[f,g,\gamma]$  at a loop $\gamma$ running around a
point $a\in A$ is equal to the Weil symbol: $ [f,g,\gamma]=
[f,g]_a$.

We will define the class $[f,g,\gamma]$ using so-called {\bf
Beilinson Integral}.


Notations:

$\gamma : I\rightarrow U, I=[0,1]$ --- a loop in $U$,
$\gamma(0)=\gamma(1)$;

$\ln f$ --- a continuous branch of the function $\ln f(t)$ on the
interval $0<t<1$;

$\deg_{\gamma} (f)$ --- a mapping degree of a map $\dfrac
{f}{|f|}:\gamma \to S^1$.

{\bf  Beilinson Integral} against the  loop $\gamma:I\rightarrow
U$ for the couple of functions $f,g $  a an  element $I_{\gamma}
(f,g)$ of the group $ \Bbb C/ \Bbb Z$  defined by a formula
$$I_{\gamma} (f,g) =\left(\dfrac{1}{2\pi i} \right)^2
\int_{\gamma}(\ln f) \dfrac{dg}{g} -\dfrac{1}{2\pi i}
\deg_{\gamma} (f)\ln g(c).$$


Beilinson integral is:


1) invariant under orientation preserving reparameterization of
the loop $\gamma$,

2) skew symmetric, i.e., $I_{\gamma} (f,g)= -I_{\gamma} (g,f)$,

3) additive, i.e., $I_{\gamma} (f_1f_2,g)=I_{\gamma}
(f_1,g)+I_{\gamma} (f_2,g)$,

4) if $f=g$ then $I_{\gamma} (f,g) = \frac{1}{2} \deg_{\gamma}
(f)$.

One can defined Beilinson integral $I_{\gamma} (f,g)$ against a
linear combination  $\gamma= \sum k_i \gamma_i$  of loops
$\gamma_i$ with integer coefficients $k_i$.

\proclaim{Theorem} Beilinson integral $I_{\gamma}(f,g)$ depends
only on a homology class of the cycle $\gamma =\sum k_i \gamma_i$
in $U$ and defines an element in $H^1 (U,\Bbb C/\Bbb Z)$.
\endproclaim

\proclaim{Corollary} A  number $[f,g,\gamma]=\exp 2\pi i
I_{\gamma}(f,g)$ depends only on a homology class of the cycle
$\gamma =\sum k_i \gamma_i$ and defines an element in $H^1 (U,\Bbb
C ^*$.
\endproclaim

Cohomology class $[f,g,\gamma]$ is:

1) skew symmetric, i.e., $[f,g,\gamma ] = [f,g,\gamma ]^{-1}$,

2) multiplicative, i.e., $[f_1f_2,g, \gamma]= [f_1,g,\gamma]
[f_2,g, \gamma]$,

\proclaim {Theorem} Consider a small ball $B_a$ centered at a
point $a\in X$. Let $\gamma$ be its boundary $\gamma= \partial B$.
Then $[f,g,\gamma ]$ equals to the Weil symbol $[f,g]_a$.
\endproclaim

Topological proof of the Weil theorem: Let $A$ be the union of the
supports of the principle divisors  $(f)$ and $(g)$ and let $U$ be
$X\setminus A$. Let $B$ be the union of small balls $B_a$ centered
at the points $a\in A$ and let $\gamma=\sum_{a\in A}
\partial B_a$ be its boundary. Then $\gamma =0$ in $H_1(U,\Bbb Z)$
and $[f,g, \gamma]=\prod_{a\in A} [f,g]_a =1$.


\head{2. Product of roots of a system of equations with general
Newton polyhedra [4]}
\endhead

Consider  a system of equations $$ P_1=\dots=P_n=0\tag 1 $$ in
$(\Bbb C^*)^n$ where $P_1,\dots,P_n$ are Laurent polynomials. Let
$\Delta_1,\dots,\Delta_n$ be Newton polyhedra of $P_1,\dots,P_n$.

{\bf Problem}: Compute the product in the group $(\Bbb C^*)^n$ of
roots of the system (1) assuming that the collection of Newton
polyhedra $\Delta _1,\dots, \Delta _n$ is generic enough.

\subhead{2.1. Developed sets of polyhedra}
\endsubhead
Let $\Delta _1,\dots,\Delta _n$ be convex polyhedra in  $\Bbb R^n$
and let $\Delta $ be their Minkowski sum. Each face $\Gamma $ of
the polyhedron $\Delta $ can be uniquely represented as a sum $$
 \Gamma =\Gamma _1+\dots+\Gamma _n
$$ A collection of $n$ polyhedra $\Delta _1,\dots, \Delta _n$  is
{\bf developed} if for each face $\Gamma$ of the polyhedron
$\Delta $  among the components of its decomposition there is at
least one vertex.

For a developed collection of polyhedra $\Delta_1,\dots,\Delta_n$
a map $$ f:\partial \Delta\rightarrow  \partial \Bbb R^n_+ $$ of
the boundary of $\Delta=\sum \Delta_i $ into the boundary of the
positive octant is called {\bf characteristic} if the component
$f_i$ of the map $f=(f_1,\dots,f_n)$ vanishes on and only on the
faces $\Gamma $  for which the $i$-th component of the
decomposition is a point (a vertex of the polyhedron $\Delta _i$).
The preimage of the origin under the characteristic map is
precisely the set of all the vertices of the polyhedron $\Delta $.

\definition{Definition}
The {\bf combinatorial coefficient} $C_A$ of a  vertex $A$ of the
polyhedron $\Delta $ is the local degree of the germ of the map $$
f\:(\partial\Delta ,A)\to (\partial \Bbb R^n_+,0) $$ of the
restriction of a characteristic map on the boundary of the
polyhedron $\Delta$.
\enddefinition

\midinsert \vspace{6cm} \botcaption{Fig 2. Combinatorial
coefficient in two dimensional case equals to $+1, 0$ or $-1$. }
\endcaption
\endinsert


\subhead{2.2. Parshin Symbols}
\endsubhead
Consider $n+1$ monomials $c_1\bold x^{\bold k_1},\dots,
c_{n+1}\bold x^{\bold k_{n+1}}$ in $n$ complex variables, where
$c_i\in \Bbb C^*$, $\bold x =x_1,\dots, x_n$, $\bold k_i\in (\Bbb
Z)^n, \bold k_i=k_{i,1},\dots,k_{i,n}$, $c_i\bold x^{\bold
k_i}=c_i x_1^ {k_1}\cdot\dots \cdot x_n^ {k_n}$.

{\bf Parshin Symbol} of the sequens of monomials by definition is
$$[c_1\bold x^{\bold k_1},\dots, c_{n+1}\bold x^{\bold
k_{n+1}}]=$$ $$=(-1)^{D(\bold k_1,\dots,\bold
k_{n+1})}c_1^{-\det(\bold k_2,\dots, \bold k_{n+1})} \dots
c_{n+1}^{(-1)^{n+1}\det(\bold k_1,\dots, \bold k_{n})}=$$

$$ = (-1)^{D(\bold k_1,\dots,\bold k_{n+1})}\exp \left(-\det
\left(\matrix \ln c_1&k_{1,1}&\dots&k_{1,n}\\
\vdots&\vdots&&\vdots\\ \ln
c_{n+1}&k_{n+1,1}&\dots&k_{n+1,n}\endmatrix\right)\right), $$
where $D(\bold k_1,\dots,\bold k_{n+1})\in \Bbb Z/2\Bbb Z$  is a
special function. The function $D$ depends only on  $\pi (\bold
k_1),$$\dots,$ $ \pi( \bold k_{n+1}) \in (\Bbb Z/2\Bbb Z)^n $
where $\pi:(\Bbb Z)^n\rightarrow (\Bbb Z/2\Bbb Z)^n $ is a natural
projection. By definition $D$ is the only nonzero  multilinear
function of $n+1$ vectors in the $n$-dimensional space over the
field $\Bbb Z/2\Bbb Z$ that is invariant under all linear
transformations, and is zero whenever the rank of $n+1$ vectors is
less than $n$.

(About function $D$ see [6].)

\example{Example (Parshin symbol and Weil symbol)} Consider two
monomials $c_1x^{k_1}$, $c_2x^{k_2}$. Parshin symbol $[c_1x^{k_1},
c_2x^{k_2}]$ equals to $(-1)^{k_1k_2}c_1^{-k_2}c_1^{k_1}$. So it
is equal to the Weil symbol of functions $c_1x^{k_1}, c_2x^{k_2}$
at the origin $x=0$.
\endexample

Parshin symbol is

1) skew symmetric, for example

$[c_1\bold x^{\bold k_1}, c_2\bold x^{\bold k_2}, \dots,
c_{n+1}\bold x^{\bold k_{n+1}}] = [c_2\bold x^{\bold k_2},
c_1\bold x^{\bold k_1}, \dots, c_{n+1}\bold x^{\bold
k_{n+1}}]^{-1}$,

2) multiplicative, i.e.

$[a_1 b_1\bold x^{\bold l_1+\bold m_1},\dots, c_{n+1}\bold
x^{\bold k_{n+1}}] =[a_1\bold x^{\bold l_1},\dots, c_{n+1}\bold
x^{\bold k_{n+1}}][b_1\bold x^{\bold m_1},\dots, c_{n+1}\bold
x^{\bold k_{n+1}}].$

\subhead{2.3. The Value of a Character on the Product of Roots of
the System}
\endsubhead
Let $\chi_{\bold k}:(\Bbb C^*)^n\rightarrow \Bbb C^*$ be a
character corresponding to a point $\bold k \in \Bbb Z^n$.

\definition{Definition} For every vertex $A$ of the polyhedron
$\Delta =\Delta_1+\dots +\Delta_n $, where $\Delta_i$ is a Newton
polyhedron of Laurent polynomial $P_i$,  we associate  a number
$[P_1,\dots,P_n,\chi_{\bold k}]_A\in \Bbb C^*$ in the following
way: Let $A=A_1+\dots+A_n$ be the decomposition of the vertex
$A\in \Delta $, $A_i\in \Delta _i$. Assume that at the vertex
$A_i$ polynomial $P_i$ corresponds to a monomial $c_i\bold
x^{\bold k_i}$. Then by definition the  number
$[P_1,\dots,P_n,\chi_{\bold k}]_A$ is the Parshin symbol
$[c_1\bold x^{\bold k_1},\dots, c_n\bold x^{\bold k_n},
\chi_{\bold k}]$.
\enddefinition

\proclaim {Theorem \rm[4] }{For the system of equations (1) the
value of the character $\chi _ {\bold k}$ on the product
$M(P_1,\dots, P_n)$ of roots of the system is computed by $$
\chi_{\bold k}(M(P_1,\dots,P_n))=\prod _{A\in \Delta
}([P_1,\dots,P_n,\chi _{\bold k}]_A)^{(-1)^nC_A}, $$ where the
product is conducted over all the vertices $A$ of the polyhedron
$\Delta =\Delta _1+\dots+ \Delta _n$, and $C_A$ is the
combinatorial coefficient at the vertex $A$.}
\endproclaim


\head{3.  Parshin--Kato Theory for Parshin symbols}
\endhead

\subhead{3.1  Flags, Parameters, Parshin Symbols and Reciprocity
Laws \rm{[2, 3]}}
\endsubhead

Let $F=( C_0\subset C_1\subset \dots \subset C_{n-1})$ be a flag
of the  subvarieties of $X$ with $\dim C_k =k$ and $C_k$ are
locally irreducible about the point $C_0$. Definition by
induction: a collection of rational functions $u_1,\dots,u_n$ is
{\bf a set of parameters about $F$} if:

1) restrictions of $u_1,\dots, u_{n-1}$ on $C_{n-1}$ are
well-defined

2) a set of restrictions of $u_1,\dots, u_{n-1}$ on $C_{n-1}$ is a
set of parameters for a flag $F_{n-1}=(C_0,\dots, C_{n-2})$ on
$C_{n-1}$.

3) a principal divisor $(u_n)$ (on the normalization $\tilde X$ of
$X$) contains the subvariety $C_{n-1}$ with the coefficient 1.

Using a set of parameters with a rational function $f$ one can
associate a {\bf leading monomial $f_{F}$ } on a flag $F$.
Definition by induction: Let the order of the function $f$ at
$C_{n-1}$ be $k_n$. Then the restriction $\varphi $ of
$fu_n^{-k_n}$ on $C_{n-1}$ is well-defined. Let $\varphi_{F_{n-1}}
$ be a leading monomial of $\varphi $ on flag $F_{n-1}$ with
parameters $u_1,\dots, u_{n-1}$. Then by definition  $f_{F}=
\varphi_{F_{n-1}}u_n^{k_n}$.


Let $f_1,\dots,f_{n+1}$ be $(n+1)$ rational functions on an
$n$-dimensional algebraic variety $X$. With a flag $F$  one can
associate the {\bf Parshin symbol $[f_1,\dots,f_{n+1}]_F$ } : by
definition it is equal to the Parshin symbol of a set of leading
monomials of the function with respect to a set of parameters:

$$[f_1,\dots,f_{n+1}]_F= [(f_1)_F,\dots,(f_{n+1})_F].$$

One can proof that {\bf Parshin symbol is independent of a set of
parameters} and depends only $f_1,\dots,f_{n+1}$ and on the flag
$F$.

\example{Example (Parshin symbol and Weil symbol)} Let $X$ be an
algebraic curve, let flag $F$ be a point $a$ and let $u$ be a
local parameter about the point $a$. For a couple of rational
functions $f_1,f_2$ Weil symbol $[f_1,f_2]_a$ is coincide with
Parshin symbol $[f_1,f_2]_F$.
\endexample


Fix a collection $f_1,\dots,f_{n+1}$ of rational functions on $X$
and fix a flag $G=C_0,\dots, C_{n-1}$ on $X$.  For each $0 \leq i
< n$ denote by $\Psi^i(G)$ set of all flags $\tilde F =\tilde
C_0,\dots, \tilde C_{n-1}$ such that $\tilde C_j =C_j$ if $j\neq
i$ and $\tilde C_j$ is any $j$-dimensional locally irreducable
subvariety.

\proclaim {\bf Theorem (Parshin reciprocity laws)}.  For each $0
\leq i < n$ $$\prod_{F\in \Psi^i(G)}[f_1,\dots,f_{n+1}]_F =1$$.
\endproclaim

 \subhead{3.2. Brilinski--McLaughrin topological version of Parshin
symbols theory~\rm{[1]}}
\endsubhead
Let $f_1,\dots,f_{n+1}$ be $(n+1)$ meromorphic functions on an
$n$-dimensional complex analytic space $X$. Denote by $U$ the
complement in $X$ of the divisors of $f_1,\dots,f_{n+1}$ and the
singular locus of $X$. Brilinski and McLaughlin defined a certain
cohomology class $(f_1,\dots,f_{n+1})\in H^k(U, \Bbb C^*)$.

{\bf Class $(f_1,\dots,f_{n+1})$ is skew symmetric in set of
functions $f_1,\dots,f_{n+1}$ and is  multiplicative in each
component}.


Let $F=( C_0\subset C_1\subset \dots \subset C_{n-1})$ be a flag
of the irreducible subspaces of $X$ with $\dim C_k =k$. Brilinski
and McLaughlin associated with the flag $F$ a {\it flag-localized
homology class $\gamma_F\in H_k(U,\Bbb Z)$)}.

\proclaim {\bf Theorem (Topological reciprocity laws)} For each $0
\leq i < n$ $$\sum_{F\in \Psi^i(G)}\gamma_F =0 \in H_n(U,\Bbb
Z)$$.
\endproclaim



\proclaim {\bf Theorem (Topology and Parshin Symbol) } The Parshin
symbol $[f_1,\dots,f_{n+1}]_F$ can be  obtained by pairing of the
cohomology class $(f_1,\dots,f_{n+1})\in H^n(U, \Bbb C^*)$ and the
flag-localized homology class $\gamma_F\in H_n(U,\Bbb Z).$
\endproclaim

\head{4. A Multidimensional Logarithmic function and topology of
Parshin symbols [7]}
\endhead

\subhead{4.1. A Functional  $\Log $ on $n$-Dimensional Cycles}
\endsubhead
Let $(\Bbb C^*)^{n+1}$ be a complex $n+1$ dimensional torus and
Let $\omega$ be an $(n+1)$-form $\left(\dfrac{1}{2\pi
i}\right)^{n+1} \dfrac {dx_1}{x_1}\wedge \dots \wedge \dfrac
{dx_{n+1}}{x_{n+1}}$. Denote by $E$ the following subset in $(\Bbb
C^*)^{n+1}$: $(x_1,\dots, x_{n+1})\in E \Leftrightarrow \{\exists
j, 1\leq j \leq n+1\} : \{x_j=1\}$.

Consider an $n$-dimensional simplicial complex $X$. Let
$\gamma=\sum k_j\Delta_j, k_j\in \Bbb Z, \Delta_j \subset X$ be an
$n$ dimensional cycle. Consider a smooth map $\bold f:X\rightarrow
(\Bbb C^*)^{n+1}$, where $\bold f =f_1,\dots,f_{n+1}$.

\definition{Definition of a $\Log $  functional} For a smooth map $\bold
f:X\rightarrow (\Bbb C^*)^{n+1}$ and a cycle $\gamma=\sum
k_j\Delta_j$ let $\Log (\bold f,\gamma)$ be an element of the
group $\Bbb C/\Bbb Z$ defined by a formula $$\Log (\bold
f,\gamma)=\int_{\sigma}\omega,$$ where $\sigma$ is an
$n+1$-dimensional chain in $(\Bbb C^*)^{n+1}$ such that $\partial
\sigma -\bold f(\gamma) \subset E$.
\enddefinition

The $\Log $  functional is well-defined:

1) Form $\omega$ equals to zero on $E$

2) Let $\varphi_i$ be the argument of the coordinate function
$x_i$. Then $\omega=d\varphi_1\wedge\dots\wedge d\varphi_{n+1} + d
\alpha$, where $\alpha$ is a $n$ form on $(\Bbb C^*)^{n+1}$

3) if $\partial \sigma_1 -\bold f(\gamma) \subset E$ and $\partial
\sigma_2 -\bold f(\gamma) \subset E$ then $\partial
\sigma_1-\partial \sigma_2=\partial \sigma_3$ where $\sigma_3$ is
a $n+1$-chain in $E$. Let $\sigma$ be an $n+1$ cycle
$\sigma_1-\sigma_2 -\sigma_3$. Then $\int_{\sigma_1}\omega-
\int_{\sigma_2}\omega=\int_{\sigma}\omega=\int_{\sigma}d\varphi_1\wedge\dots\wedge
d\varphi_{n+1}\in \Bbb Z$.

\example{Example ($\ln$ and $\Log $  functional)} Let $X$ be a
point $\{ a \}$ and let $\gamma$ be the point $\{ a \}$ with a
coefficient 1. Then for a map $f:X\rightarrow \Bbb C^*$ we have
$$\Log ( f,\gamma)=\dfrac{1}{2\pi i} \ln f(a). $$
\endexample

\example{Example (Beilinsin integral and $\Log $  functional)}
Consider a complex algebraic curve $X$ and  two nonzero
meromorphic functions $f$ and $g$ on $X$. Let $A$ be the union of
the divisors  $(f)$ and $(g)$ and let  $U$ be $X\setminus A$. Take
a loop $\gamma$ in $U$. Then for a map $(f,g):\gamma\rightarrow
(\Bbb C^*)^2$ we have $$\Log ( f,g,\gamma)= I_{\gamma}(f,g) $$
\endexample



{\bf $\Log $ functional} is

1)  {\bf skew symmetric}, for example

$\Log  (f_1, f_2, \dots, f_{n+1},\gamma) = - \Log (f_2, f_1,
\dots, f_{n+1},\gamma)$;

2)  {\bf additive}, i.e.,

$\Log (\varphi_1 \psi_1, f_2,\dots, f_{n+1}, \gamma) =\Log
(\varphi_1, f_2, \dots, f_{n+1}, \gamma)+ \Log (\psi_1, f_2,
\dots, f_{n+1}, \gamma)$.

Let $M$ be a smooth real manifold and let $\bold f$ be a smooth
map $\bold f:M\rightarrow (\Bbb C^*)^{n+1}$. With each
$n$-dimensional cycle on can associate an element $\Log  (\bold
f,\gamma)$ in the group $\Bbb C/\Bbb Z$.

\proclaim{Theorem} If the form $\bold f^* \omega$ is identically
equal to zero on $M$, then a functional $\gamma \rightarrow \Log
(\bold f, \gamma)$ is a cohomology class in $H^n (M,\Bbb C/\Bbb
Z)$.
\endproclaim

\example {Main Example} Let $X$ be an $n$-dimensional complex
algebraic variety, $f_1,\dots,f_{n+1}$ --- rational functions on
$X$ and $M = X\setminus \cup(f_j)$. Consider a map $\bold
f:M\rightarrow (\Bbb C^*)^{n+1}$, $\bold f=f_1,\dots,f_{n+1}$.
Then $\bold f^*\omega\equiv 0$. So a functional $\gamma
\rightarrow \Log (\bold f, \gamma)$ is a cohomology class in $H^n
(M,\Bbb C/\Bbb Z)$.
\endexample

\subhead{4.2. Multiplicative version}
 \endsubhead
Using exponential function one can define a multiplicative version
of $\Log $  functional.

Functional  $\exp 2\pi i \Log (\bold f,\gamma)$ takes value in
$\Bbb C^*$. It is:

1)  {\bf skew symmetric}, for example

$\exp 2\pi i \Log  (f_1, f_2, \dots, f_{n+1},\gamma) = (\exp 2\pi
i \Log (f_2, f_1, \dots, f_{n+1},\gamma))^{(-1)};$

2)  {\bf multiplicative}, i.e. $$\exp 2\pi i \Log (\varphi_1
\psi_1, f_2,\dots, f_{n+1}, \gamma) $$ $$
 =\exp 2\pi i \Log (\varphi_1, f_2,
\dots, f_{n+1}, \gamma) \exp2\pi i \Log (\psi_1, f_2, \dots,
f_{n+1}, \gamma).$$

\proclaim{Theorem} If the form $\bold f^* \omega$ is identically
equal to zero on $M$, then $$\gamma \rightarrow \exp 2\pi i \Log
(\bold f, \gamma)$$ is a cohomology class in $H^n (M,\Bbb C^*)$.
\endproclaim


\proclaim{Theorem} Let $X\subset (\Bbb C^*)^n$ be a real
$n$-dimensional torus $|y_1|=\dots=|y_n|=1$ and let $\gamma$ be
the fundamental cycle of the manifold $X$. Consider $n+1$
monomials $c_1\bold y ^{\bold k_1},\dots, c_{n+1} \bold y ^{\bold
k_{n+1}}$. Let $\bold f: X\rightarrow (\Bbb C^*)^{n+1} $ be a
restriction of the map $(c_1\bold y ^{\bold k_1},\dots, c_{n+1}
\bold y ^{\bold k_{n+1}}):(\Bbb C^*)^{n} \rightarrow (\Bbb
C^*)^{n+1}$ on $X$. Then $$\exp (2\pi i \Log  (\bold f,
\gamma))=[c_1\bold y ^{\bold k_1},\dots, c_{n+1} \bold y ^{\bold
k_{n+1}}].$$
\endproclaim

\proclaim{Theorem} Let $f_1,\dots,f_{n+1}$ be $(n+1)$ meromorphic
functions on an $n$-dimensional complex analytic space $X$. Let
$F$ be a flag on $X$ and let $\gamma_F\in H_k(U,\Bbb Z)$) be the
flag-localized homology class. Then

$$\exp (2\pi i \Log  (f_1,\dots,f_{n+1}, \gamma_F
))=[f_1,\dots,f_{n+1} ]_F.$$.
\endproclaim

 \centerline{LITERATURE}

\ref\no1
 \by  Brylinski J.-L., McLaughlin D.A.
 \paper Multidimensional reciprocity laws
 \jour J. reine angew. Math.
 \yr 1996
 \vol481
 \issue
 \pages 125--147
 \endref

 \ref\no2
 \by Parshin~A.~N.
 \paper Local class field theory
 \inbook Trudy Mat. Inst. Steklov
 \eds
 \publ
 \publaddr
 \yr 1984
  \vol 165
 \pages
 \endref

 \ref\no3
 \by Parshin~A.~N.
 \paper Galois cohomology and Brauer group of local fields
 \inbook Trudy Mat. Inst. Steklov
 \eds
 \publ
 \publaddr
 \yr1990; English Transl. Proc. Steklov Inst. Math., vol 183, 191--201, 1991
 \vol183
 \pages
 \endref



\ref\no4
 \by  Khovanskii A.
 \paper  Newton polyhedrons, a new formula for mixed volume, product
  of roots of a system of equations
 \inbook   Proceed. of a Conf. in Honour of V.I. Arnold
 \publ   Fields Inst. Comm.
 \vol  24, Amer. Math. Soc.
 \publaddr USA
 \yr 1999
 \pages 325--364
\endref



  \ref\no5
 \by Khovanskii A.
 \paper  Newton polytopes, curves on
toric surfaces, and inversion of Weil's theorem
 \jour Russian Math. Surveys
 \yr 1997
 \vol 52 \issue 6
 \pages 1251--1279
 \endref

 \ref\no6
 \by Khovanskii A.
 \paper An Analog of Determinant Related to Parshin--Kato Theory
and Integral Polytopes
 \jour Functional analysis and Its
Application
 \yr 2006
 \vol 40
 \issue 2
 \pages 55--64
 \endref



 \ref\no7
 \by Khovanskii~A.
 \paper A Log functional  and multidimensional symbols {\rm (in
 preparation)}
 \jour
 \yr
 \vol
 \issue
 \pages
 \endref


\end
