
%%%%%%%%%%%%%%%%%%%%%      main.tex


%\batchmode
%%%%%%%%%%%%%%%%%  preamble.tex



% this is the preamble for AMS-Latex  files
\documentstyle[amscd,amstex]{amsart}
\emergencystretch=2cm

\newsymbol\boxtimes 1202

%shorthands

\newcommand{\nc}{\newcommand}
\newcommand{\rc}{\renewcommand}

% symbols we shall use will be defined below using \nc command


\nc{\ad}{{\mbox{\bf{ad}}}}
%\nc{\AJ}{{\mbox{{\cyr ya}}}}
\nc{\AJ}{{\operatorname{aj}}}
\nc{\Aut}{{\operatorname{Aut}}}
\nc{\Bls}{{{\cal B}ls}}
\nc{\Boxtimes}{{\fbox{$\times$}}}
\nc{\blt}{{\bullet}}
\nc{\bSt}{{\mbox{\bf{St}}}}
\nc{\card}{{\operatorname{card}}}
\nc{\Cch}{{\check{C}}}
\nc{\cd}{{\operatorname{cd}}}
\nc{\Ch}{{\operatorname{Ch}}}
\nc{\chara}{{\operatorname{char}}}
\nc{\CHom}{{\cal{H}om}}
\nc{\Coker}{{\operatorname{Coker}}}
\nc{\codim}{{\operatorname{codim}}}
\nc{\Cone}{{\operatorname{Cone}}}
\nc{\cSgn}{{\cal{S}gn}}
\nc{\depth}{{\operatorname{depth}}}
\nc{\dirlim}{{\underset{\rightarrow}{\operatorname{lim}}}}
\nc{\dotbox}{{\overset{\bullet}{\boxtimes}}}
\nc{\dotimes}{{\overset{\bullet}{\otimes}}}
\nc{\Ed}{{\operatorname{Edge}}}
\nc{\emp}{{\emptyset}}
\nc{\Ext}{{\operatorname{Ext}}}
\nc{\Fac}{{\cal{F}ac}}
\nc{\Fun}{{\operatorname{F}}}
\nc{\FS}{{\cal{FS}}}
\nc{\Hom}{{\operatorname{Hom}}}
\nc{\had}{{{\hat{\mbox{\bf{ad}}}}}}
\nc{\hgt}{{\operatorname{ht}}}
\nc{\Id}{{\operatorname{Id}}}
\nc{\id}{{\operatorname{id}}}
\nc{\Ima}{{\operatorname{Im}}}
\nc{\ind}{{\operatorname{ind}}}
\nc{\Ind}{{\operatorname{Ind}}}
\nc{\infi}{{\operatorname{inf}}}
\nc{\infh}{{\frac{\infty}{2}}}
\nc{\invlim}{{\underset{\leftarrow}{\operatorname{lim}}}}
\nc{\Jac}{{{\cal J}ac}}
\nc{\Ker}{{\operatorname{Ker}}}
\nc{\lcm}{{\operatorname{lcm}}}
\nc{\Locsys}{{{\cal L}ocsys}}
\nc{\Map}{{{\cal M}ap}}
\nc{\modul}{{\operatorname{mod}}}
\nc{\Mor}{{\operatorname{Mor}}}
\nc{\MS}{{\cal{MS}}}
\nc{\Ob}{{\operatorname{Ob}}}
\nc{\opp}{{\operatorname{opp}}}
\nc{\Or}{{{\cal O}r}}
\nc{\Ord}{{{\cal O}rd}}
\nc{\Part}{{{\cal P}art}}
\nc{\PGL}{{\operatorname{PGL}}}
\nc{\Pic}{{\operatorname{Pic}}}
\nc{\Rep}{{{\cal{R}}ep}}
\nc{\rk}{{\operatorname{rk}}}
\nc{\Sets}{{{\cal{S}}ets}}
\nc{\Sew}{{{\cal{S}}ew}}
\nc{\sgn}{{\operatorname{sgn}}}
\nc{\Sh}{{{\cal S}h}}
\nc{\Sign}{{{\cal S}ign}}
\nc{\Spe}{{\mbox{\bf{Sp}}}}
\nc{\supr}{{\operatorname{sup}}}
\nc{\Supp}{{\operatorname{Supp}}}
\nc{\supp}{{\operatorname{supp}}}
\nc{\Teich}{{{\cal{T}}eich}}
\nc{\tFS}{{\widetilde{\cal{FS}}}}
\nc{\Tor}{{\operatorname{Tor}}}
\nc{\totimes}{{\tilde{\otimes}}}
\nc{\tr}{{\operatorname{tr}}}
\nc{\tRep}{{\widetilde{{\cal R}ep}}}
\nc{\tTeich}{{\widetilde{{\cal T}eich}}}
\nc{\Vect}{{{\cal V}ect}}
\nc{\Ve}{{\operatorname{Vert}}}
\nc{\wt}{{\widetilde}}

% letters

\nc{\bo}{{\mbox{\bf{0}}}}
\nc{\One}{{\mbox{\bf{1}}}}
\nc{\one}{{\mbox{\bf{1}}}}

\nc{\BA}{{\Bbb A}}
\nc{\bA}{{\overline{A}}}
\nc{\ba}{{\mbox{\bf{a}}}}
\nc{\baB}{{\overline{B}}}
\nc{\baeta}{{\bar{\eta}}}
\nc{\baJ}{{\bar{J}}}
\nc{\BB}{{\Bbb B}}
\nc{\bB}{{\mbox{\bf{B}}}}
\nc{\bc}{{\mbox{\bf{c}}}}
\nc{\bC}{{\overline{C}}}
\nc{\BC}{{\Bbb{C}}}
\nc{\bCC}{{\overline{\cal{C}}}}
\nc{\bCM}{{\overline{\cal{M}}}}
\nc{\bD}{{\bar{D}}}
\nc{\BD}{{\overline{D}}}
\nc{\bd}{{\mbox{\bf{d}}}}
\nc{\BE}{{\overline{E}}}
\nc{\BF}{{\overline{F}}}
\nc{\bF}{{\mbox{\bf{F}}}}
\nc{\bg}{{\mbox{\bf{g}}}}
\nc{\bG}{{\mbox{\bf{G}}}}
\nc{\BG}{{\Bbb G}}
\nc{\bGamma}{{\overline{\Gamma}}}
\nc{\bbH}{{\bar{\mbox{\bf{H}}}}}
\nc{\bH}{{\mbox{\bf{H}}}}
\nc{\bI}{{\mbox{\bf{I}}}}
\nc{\bL}{{\mbox{\bf{L}}}}
\nc{\BL}{{\Bbb{L}}}
\nc{\blambda}{{\bar{\lambda}}}
\nc{\bM}{{\mbox{\bf{M}}}}
\nc{\bmu}{{\vec{\mu}}}
\nc{\bN}{{\mbox{\bf{N}}}}
\nc{\BN}{{\Bbb{N}}}
\nc{\bnu}{{\mbox{\boldmath{${\nu}$}}}}
\nc{\bof}{{\mbox{\bf{f}}}}
\nc{\BP}{{\Bbb P}}
\nc{\bP}{{\mbox{\bf{P}}}}
\nc{\BPO}{{\overset{\circ}{\BP}}}
\nc{\BQ}{{\Bbb Q}}
\nc{\bq}{{\mbox{\bf{q}}}}
\nc{\BR}{{\Bbb{R}}}
\nc{\bR}{{\mbox{\bf{R}}}}
\nc{\br}{{\mbox{\bf{r}}}}
\nc{\breta}{{\bar{\eta}}}
\nc{\bs}{{\mbox{\bf{s}}}}
\nc{\bS}{{\mbox{\bf{S}}}}
\nc{\bt}{{\mbox{\bf{t}}}}
\nc{\bU}{{\mbox{\bf{U}}}}
\nc{\bV}{{\mbox{\bf{V}}}}
\nc{\bu}{{\mbox{\bf{u}}}}
\nc{\BUpsilon}{{\bar{\Upsilon}}}
\nc{\bw}{{\mbox{\bf{w}}}}
\nc{\bx}{{\mbox{\bf{x}}}}
\nc{\bX}{{\mbox{\bf{X}}}}
\nc{\BZ}{{\Bbb{Z}}}
\nc{\bz}{{\mbox{\bf{z}}}}
\nc{\bZ}{{\mbox{\bf{Z}}}}
\nc{\bzero}{\mbox{\boldmath{$0$}}}


\nc{\CA}{{\cal A}}
\nc{\CAD}{{\overset{\bullet}{\cal{A}}}}
\nc{\CAO}{{\overset{\circ}{\cal{A}}}}
\nc{\CB}{{\cal B}}
%\nc{\CC}{{\cal C}}
\nc{\CalD}{{\cal D}}
\nc{\CE}{{\cal E}}
\nc{\CF}{{\cal F}}
\nc{\CG}{{\cal G}}
\nc{\CH}{{\cal H}}
\nc{\CI}{{\cal I}}
\nc{\CID}{{\overset{\bullet}{\cal{I}}}}
\nc{\CJ}{{\cal J}}
\nc{\CK}{{\cal K}}
\nc{\CL}{{\cal L}}
\nc{\CM}{{\cal M}}
\nc{\CN}{{\cal N}}
\nc{\CO}{{\cal O}}
\nc{\CP}{{\cal P}}
\nc{\CPO}{{\overset{\circ}{\cal{P}}}}
\nc{\CQ}{{\cal Q}}
\nc{\CR}{{\cal R}}
\nc{\CS}{{\cal S}}
\nc{\CT}{{\cal T}}
\nc{\CTD}{{\overset{\bullet}{\cal{T}}}}
\nc{\CTPO}{{\overset{\circ}{\cal{T}\cal{P}}}}
\nc{\CU}{{\cal{U}}}
\nc{\CV}{{\cal V}}
\nc{\CW}{{\cal W}}
\nc{\CX}{{\cal X}}
\nc{\CY}{{\cal Y}}
\nc{\CZ}{{\cal Z}}


\nc{\dCL}{{\overset{\bullet}{\cal{L}}}}
\nc{\dd}{{\operatorname{d}}}
\nc{\ddelta}{{\overset{\bullet}{\delta}}}
\nc{\dfu}{{\overset{\bullet}{\frak{u}}}}
\nc{\dlambda}{{\overset{\bullet}{\lambda}}}
\nc{\DO}{{\overset{\circ}{D}}}
\nc{\dpar}{{\partial}}
\nc{\dS}{{\overset{\bullet}{S}}}
\nc{\dT}{{\overset{\bullet}{T}}}

%\nc{\fA}{{\frak{A}}}
%\nc{\fb}{{\frak{b}}}
%\nc{\fC}{{\frak{C}}}
%\nc{\fE}{{\frak{E}}}
%\nc{\fF}{{\frak{F}}}
%\nc{\ff}{{\frak{f}}}
%\nc{\fg}{{\frak{g}}}
%\nc{\fH}{{\frak{H}}}
%\nc{\fl}{{\frak{l}}}
%\nc{\fL}{{\frak{L}}}
%\nc{\fM}{{\frak{M}}}
%\nc{\fn}{{\frak{n}}}
%\nc{\fp}{{\frak{p}}}
%\nc{\fu}{{\frak{u}}}
%\nc{\fZ}{{\frak{Z}}}






\nc{\hCH}{{\hat{\cal{H}}}}
\nc{\hCI}{{\hat{\cal{I}}}}
\nc{\hfC}{{\hat{\frak{C}}}}
\nc{\hfg}{{\hat{\frak{g}}}}
\nc{\hL}{{\hat{L}}}
\nc{\HO}{{\overset{\circ}{H}}}
\nc{\hpsi}{{\hat{\psi}}}
\nc{\hx}{{\hat{x}}}

\nc{\jo}{{\overset{\circ}{j}}}

\nc{\phid}{{\overset{\bullet}{\phi}}}

\nc{\tA}{{\tilde{A}}}
\nc{\ta}{{\tilde{a}}}
\nc{\tB}{{\tilde{B}}}
\nc{\tb}{{\tilde{b}}}
\nc{\tBP}{{\tilde{\BP}}}
\nc{\tC}{{\tilde{C}}}
\nc{\tc}{{\tilde{c}}}
\nc{\tCA}{{\tilde{\cal{A}}}}
\nc{\tCC}{{\tilde{\cal{C}}}}
\nc{\tCH}{{\tilde{\cal{H}}}}
\nc{\tCI}{{\tilde{\cal{I}}}}
\nc{\tCO}{{\tilde{\cal{O}}}}
\nc{\tCP}{{\tilde{\cal{P}}}}
\nc{\tCT}{{\tilde{\cal{T}}}}
\nc{\tD}{{\tilde{D}}}
\nc{\tDelta}{{\tilde{\Delta}}}
\nc{\tE}{{\tilde E}}
\nc{\tF}{{\tilde F}}
\nc{\tfD}{{\tilde{\frak{D}}}}
\nc{\tfF}{{\tilde{\frak{F}}}}
\nc{\tff}{{\tilde{\frak{f}}}}
\nc{\tfu}{{\tilde{\frak{u}}}}
\nc{\tJ}{{\tilde{J}}}
\nc{\tj}{{\tilde{j}}}
\nc{\tK}{{\tilde K}}
\nc{\tL}{{\tilde{L}}}
\nc{\tM}{{\tilde{M}}}
\nc{\tP}{{\tilde{P}}}
\nc{\tPhi}{{\tilde{\Phi}}}
\nc{\tpi}{\tilde{\pi}}
\nc{\TPO}{{\overset{\circ}{T\BP}}}
\nc{\tR}{{\tilde{R}}}
\nc{\tS}{{\tilde S}}
\nc{\tT}{{\tilde{T}}}
\nc{\ttau}{{\tilde{\tau}}}
\nc{\ttheta}{{\tilde{\theta}}}
\nc{\tU}{{\tilde{U}}}
\nc{\tUpsilon}{{\tilde{\Upsilon}}}
\nc{\tW}{{\tilde W}}
\nc{\ty}{{\tilde y}}
\nc{\tY}{{\tilde Y}}
\nc{\txi}{{\tilde{\xi}}}

\nc{\UD}{{\overset{\bullet}{U}}}
\nc{\UO}{{\overset{\circ}{U}}}

\nc{\vA}{{\vec{A}}}
\nc{\valpha}{{\vec{\alpha}}}
\nc{\vbeta}{{\vec{\beta}}}
\nc{\vc}{{\vec{c}}}
\nc{\vD}{{\vec{D}}}
\nc{\vd}{{\vec{d}}}
\nc{\vgamma}{{\vec{\gamma}}}
\nc{\vK}{{\vec{K}}}
\nc{\vlambda}{{\vec{\lambda}}}
\nc{\vmu}{{\vec{\mu}}}
\nc{\vnu}{{\vec{\nu}}}
\nc{\vo}{{\vec{0}}}
\nc{\vu}{{\vec{u}}}
\nc{\vx}{{\vec{x}}}
\nc{\vy}{\vec{y}}
\nc{\vzero}{\vec{0}}

\nc{\XO}{{\overset{\circ}{X}}}

%\nc{\ya}{{\mbox{\cyr{ya}}}}
\nc{\ya}{{\operatorname{aj}}}


% symbols we shall use

\nc{\nen}{\newenvironment}
\nc{\ol}{\overline}
\nc{\ul}{\underline}
\nc{\ra}{\rightarrow}
\nc{\lra}{\longrightarrow}
\nc{\Lra}{\Longrightarrow}
\nc{\lla}{\longleftarrow}
\nc{\Llra}{\Longleftrightarrow}
\nc{\hra}{\hookrightarrow}
\nc{\iso}{\overset{\sim}{\lra}}
\nc{\rlh}{\rightleftharpoons}

%commands for semiinfinite flags

\nc{\IC}{{\cal{IC}}}
\nc{\PS}{{\cal{PS}}}
\nc{\oCG}{{\overline{\cal G}}}
\nc{\oCQ}{{\overline{\cal Q}}}
\nc{\oCZ}{{\overline{\cal Z}}}
\nc{\dZ}{{\overset{\bullet}{\cal Z}}{}}
\nc{\ddZ}{{\ddot{\cal Z}}{}}
\nc{\oZ}{{\overset{\circ}{\cal Z}}{}}

\nc{\dP}{{\overset{\bullet}{\cal P}}{}}
\nc{\oP}{{\overset{\circ}{\cal P}}{}}
\nc{\oQ}{{\overset{\circ}{\cal Q}}{}}
\nc{\obp}{{\overset{\circ}{{\bf p}}}{}}
\nc{\tbj}{{\tilde{\bf j}}{}}
\nc{\tbp}{{\tilde{\bf p}}{}}
\nc{\tfC}{{\widetilde{\frak C}}{}}
\nc{\tfE}{{\widetilde{\frak E}}{}}
\nc{\tfj}{{\widetilde{\frak j}}{}}
\nc{\tfQ}{{\widetilde{\frak Q}}{}}
\nc{\tfp}{{\widetilde{\frak p}}{}}
\nc{\ofQ}{{\overset{\circ}{{\frak Q}}}{}}
\nc{\tGQ}{{\widetilde{\cal{GQ}}}{}}
\nc{\oGQ}{{\overset{\circ}{\cal{GQ}}}{}}
\nc{\ooGQ}{{\overset{\circ\circ}{\cal{GQ}}}{}}
\nc{\oGZ}{{\overset{\circ}{\cal{GZ}}}{}}
\nc{\tGZ}{{\widetilde{\cal{GZ}}}{}}

\nc{\Ue}{{U_\varepsilon}}
\nc{\Upe}{{\Upsilon_\varepsilon}}
\nc{\crho}{{\check{\rho}}}
\nc{\ctheta}{{\check{\theta}}}













%commands to refer to ths, props etc (see below)

\nc{\Thm}[1]{Theorem~\ref{#1}}
\nc{\Prop}[1]{Proposition~\ref{#1}}
\nc{\Lem}[1]{Lemma~\ref{#1}}
\nc{\Cor}[1]{Corollary~\ref{#1}}
\nc{\Conj}[1]{Conjecture~\ref{#1}}
\nc{\Claim}[1]{Claim~\ref{#1}}
\nc{\Defn}[1]{Definition~\ref{#1}}
\nc{\Exa}[1]{Example~\ref{#1}}
\nc{\Rem}[1]{Remark~\ref{#1}}
\nc{\Note}[1]{Note~\ref{#1}}

% environments for thms, props etc

\nen{thm}[1]{\label{#1}{\bf Theorem.\ } \em}{}
\nen{prop}[1]{\label{#1}{\bf Proposition.\ } \em}{}
\nen{lem}[1]{\label{#1}{\bf Lemma.\ } \em}{}
\nen{cor}[1]{\label{#1}{\bf Corollary.\ } \em}{}
\nen{conj}[1]{\label{#1}{\bf Conjecture.\ } \em}{}
\nen{claim}[1]{\label{#1}{\bf Claim.\ } \em}{}

% environments for defs, examples

\nen{defn}[1]{\label{#1}{\bf Definition.\ } }{}
\nen{exa}[1]{\label{#1}{\bf Example.\ } }{}

% environments for rem,note, ack

\nen{rem}[1]{\label{#1}{\em Remark.\ } }{}
\nen{note}[1]{\label{#1}{\em Note.\ } }{}
\nen{exer}[1]{\label{#1}{\em Exercise.\ } }{}


% set lengths
%\setlength{\unitlength}{0.8cm}
\unitlength=1pt
\setlength{\baselineskip}{18pt}
\setlength{\parindent}{0cm}
\setlength{\parskip}{6pt}
\setlength{\textwidth}{16cm}
\setlength{\textheight}{21cm}
\setlength{\oddsidemargin}{0.1in}
\setlength{\evensidemargin}{0.1in}
\setlength{\headheight}{30pt}
\setlength{\headsep}{40pt}
\setlength{\topmargin}{-60pt}
\setlength{\marginparwidth}{0pt}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%MINE%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%FOUNDATIONS
%       1. Graphical Features
%       2. ARROWS:
%       3. VARIOUS SPECIAL SIGNS
%       4. LARGE OPERATORS
%       5. ENDINGS
%       6. Special ``Words'' in TEX:          
%       7. FONT
%%%%%SYMBOLS:
%REDEFINE-the same format as define!: \redefine\gg{frak g}
%\predefine\greater{\gg}
%

%1. SHORTHAND for Graphical Features:          
                                %Indentation & 
\nc{\nn}{{\newline}}
\nc{\np}{{\newpage}}
\nc{\nnn}{{\newpage}}
\nc{\noi}{{\noindent}}
\nc{\nop}{{\noindent {\bf Proof.}} }
                                %BREAKS
\nc{\sbr}{{\smallpagebreak}}
\nc{\mbr}{{\medpagebreak}}
\nc{\bbr}{{\bigpagebreak}}
				%BOLDSYMBOL
\nc{\bbb}{ \boldsymbol         }  




%2. ARROWS:

%%%\nc{\ra}{{\rightarrow}}
\nc{\raa}{{\longrightarrow}}
\nc{\laa}{{\leftarrow}}
\nc{\laaa}{{\longleftarrow}}
\nc{\lrax}{{\leftrightarrow}}     %x-added since Misha uses this one already
\nc{\lr}{{\leftrightarrow}}     % Much shorter version

         				%Implication and Equivalence.
\nc{\imp}{{\Rightarrow}}         %Implication.
\nc{\eq}{{\Leftrightarrow}}        %Equivalence.
							%OLD Version:
\nc{\Ra}{{\Rightarrow}}         %Implication.
\nc{\LRa}{{\Leftrightarrow}}        %Equivalence.

					%MAPS:
\nc{\inj}{{	\hookrightarrow	}}    		%Injective map-right.
\nc{\injj}{{	\hookleftarrow	}}    		%Injective map.
\nc{\sur}{{	\bb{surjective}\longrightarrow	}}	%Surjective map-right.
			%\nc{\sur}{{\twoheadrightarrow}} 
\nc{\surr}{{	\bb{surjective}\longleftarrow	}}	%Surjective map-left
			%\nc{\surr}{{\twoheadleftarrow}} 
\nc{\mm}{{\mapsto}}             		%Map on elements.
\nc{\va}{{\uparrow}}              		%Vertical Up-arrow

%3. VARIOUS SPECIAL SIGNS

\nc{\bb}{\underset}           %Write Belllow
\rc{\aa}{\overset}            %and Above


\nc{\bss}{{\backslash}}           %%%xxx

\nc{\barr}{ \overline }         %Long bar accent
\nc{\ud}{\underline}		%Underline



\nc{\sub}{{\subseteq}}          %Inclusions
\nc{\subb}{{\supseteq}}          %%%xxx
	\nc{\suppx}{{\supseteq}}          %%%xxx

\nc{\ti}{\tilde}              %Tilde
\nc{\tii}{\widetilde}         %Tilde: wide
\nc{\hatt}{\widehat}		%Hat: wide

\nc{\ch}{\check}              %Check

\nc{\lb}{\langle}             %Brackets:left and right.
\nc{\rb}{\rangle}

\nc{\yy}{\infty}                        %(Semi)infinity
\nc{\ys}{{  \frac{\infty}{2}  }}

\nc{\ca}{		\left\{		\smallmatrix		}	%%Cases
\nc{\eca}{		\endsmallmatrix	\right\}	}
			
				% The following do not work
				%I need to open and close the bracket 
\nc{\sus}{ 	\subsection		}
\nc{\sss}{ 	\subsubsection		}

\nc{\lema}{ 	\subsection{Lemma}		}
\nc{\lemm}{ 	\subsubsection{Lemma}		}
\nc{\coro}{ 	\subsection{Corollary}		}
\nc{\corr}{ 	\subsubsection{Corollary}	}


\nc{\tim}{{\times}}             %Times
	\nc{\btim}{{\boxtimes}}
				%Semidirect products
%??	\nc\ltim{\ltimes}                  	%Triangle pointing left
	%\nc\rtim{\rtimes}			%Triangle pointing right
%??	\nc\rtim{\righttimes}			%Triangle pointing right


\nc{\ten}{{\otimes}}            %TENSORING:
\nc{\tenA}{	\bb{A}\ten	}
\nc{\tenB}{	\bb{B}\ten	}
\nc{\bten}{{\boxtimes}}         %Tensoring: outer   
\nc{\pl}{{\oplus}}              %Direct sum
\nc{\con}{{ @>\cong>> }}  %Isomorphism with a right arrow
\nc{\conn}{{     @<\cong<<  }}    %Isomormphism with a left arrow

\nc{\half}{{ \frac{1}{2} }}     %%%xxxxxxxx  %Half
\nc{\onetwo}{{ \frac{1}{2} }}       %Half
%%%\nc{\12}{{ \frac{1}{2} }}       %Half

                                %DOTS:
\nc{\ci}{{\circ}}               %Circle dot
\nc{\cdd}{{\cdot}}            	%Multiplication dot
		\nc{\cdx}{{\cdot}}	          %old
\nc{\cddd}{{\cdot\cdot\cdot}}	%Three dots
%\dot : one dot above
%\ddot : two dots above
%\ddots : thre dots slanted


%4. LARGE OPERATORS                                %such as \sum

\nc{\cupp}{\bigcup}             %cup,cap
\nc{\capp}{\bigcap}
\nc{\tenn}{\bigotimes}          %times,plus
\nc{\pll}{\bigoplus}

                                %PRODUCTS  & coproducts:
\nc{\pii}{\prod}                %Product 
\nc{\ppii}{\bigprod}            %Big product
\nc{\cci}{\sqcup}              %Coprodduct
\nc{\ccii}{\bigsqcup}
                                %Semi-direct product
\nc{\sd}{\triangleleft}         %Semi-direct product
\nc{\sdd}{\triangleright}       %Semi-direct product

\nc{\wwe}{\bigwedge}            %wedge         
\nc{\cce}{\bigcoprod}           %cowedge





%5. ENDINGS
\nc{\edd}{{ \end{document}	}} 
%\nc{\pp}{\endproclaim}        %Endproclaim
%\nc{\hh}{\endheading}         %Endheading


                                %SHORTHAND for 



%6. SPECIAL ``Words'' in TEX:          
                                        %Special Symbols
\nc{\df}{{ \overset{ \text{def}}= }}		%definition
\nc{\inv}{{ {}^{-1}      }}			%inverse
\nc{\thh}{	^{th}	}                     	%nth

%%%\nc{\emp}{   \emptyset}}      		%Emptyset (Exists above)
\nc{\we}{{\wedge}}				%wedge
\nc{\wee}{{     \overset{2}\wedge       }}
                                        	%Limits: proj and ind
\nc{\limp}{{\underset {\leftarrow}\lim}}
\nc{\limi}{{\underset {\rightarrow}\lim}}

                                       	%Particular Varieties
\nc{\aaa}{{  {\Bbb A}^1  }}          		%A1
\nc{\ppp}{{ {\Bbb P}^1 }}            		%P1
\nc{\ppn}{{ {\Bbb P}^n }}            		%Pn
\nc{\pt}{	{ \text{pt} }	}		%point
				
		                       %Particular Fields:
\nc{\qlb}{{ \barr{{\Bbb Q}_l} }}      		%Q-el-bar
\nc{\ffq}{{  {\Bbb F}_q  }}           		%Ef-q (finite field)
\nc{\ffp}{{  {\Bbb F}_p  }}           		%Ef-q (finite field)
\nc{\tw}{   ^{(1)}	}

                                        %Actual ``Words''
\nc{\Spec}{{ \text{Spec} \ {}     }} 
\nc{\aand}{{ \ \text{and}\ }}
\nc{\hk}{{       \mathrm{hyperk\ddot{a}hler}    }}




					%Linear algebra
%\nc{\Ker}{\text{Ker}}
%\nc{\Coker}{\text{Coker}}
%\rc{\Im}{{ \text{Im} }}
%\nc{\codim}{\text{codim}}
\nc{\rank}{{\ \text{rank}\ }}
						%Hom-versions
%\nc{\Hom}{\text{Hom}}
\nc{\End}{\text{End}}
\nc{\RHom}{\text{RHom}}
	\nc{\HHom}{\text{$\HH$om}}
	\nc{\EEnd}{\text{$\EE nd$}}
	\nc{\RHHom}{\text{R$\HH$om}}
%\nc{\Ext}{\text{Ext}}

\nc{\ord}{{\ \text{ord}\ }}			%order of zero
\nc{\divv}{{\ \text{div}\ }}			%divisor


                                        %Various
                                        %Relative Products, abelianization
                                        %and affinization
\nc{\timA} {{   \underset{A}\tim             }}
\nc{\timB} {{   \underset{B}\tim             }}
\nc{\timC} {{   \underset{C}\tim             }}
\nc{\timG} {{   \underset{G}\tim             }}
\nc{\timH} {{   \underset{H}\tim             }}
\nc{\timN} {{   \underset{N}\tim             }}
\nc{\timP}{{    \underset{P}\tim             }}
\nc{\timQ}{{    \underset{Q}\tim             }}
\nc{\timS} {{   \underset{S}\tim             }}
\nc{\ab}{{       ^{ab}   }}
\nc{\af}{{       ^{aff}  }}


%Some already DEFINED shorthand ``words'':
%\nc{\mod}{\text{mod{\}}}
%\nc{\dim}{\text{dim}}
%\nc{\max}{\text{max}}
%\nc{\and}
%\nc{\or}{\text{or}}







%7. FONTS





%FONT: SCRIPT,      Cal by a double letter \??
\rc{\AA}{{\cal A}}
\rc{\BB}{{\cal B}} 
\nc{\CC}{{\cal C}}
\nc{\DD}{{\cal D}}
\nc{\EE}{{\cal E}}
\nc{\FF}{{\cal F}}
\nc{\GG}{{\cal G}}
\nc{\HH}{{\cal H}}
\nc{\II}{{\cal I}}
\nc{\JJ}{{\cal J}}
\nc{\KK}{{\cal K}}
\nc{\LL}{{\cal L}}
\nc{\MM}{{	\cal M	  }}
\nc{\NN}{{\cal N}}
\nc{\OO}{{\cal O}}
\nc{\PP}{{\cal P}}
\nc{\QQ}{{\cal Q}}
\nc{\RR}{{\cal R}}
\rc{\SS}{{\cal S}}
\nc{\TT}{{\cal T}}
\nc{\UU}{{\cal U}}
\nc{\VV}{{\cal V}}
\nc{\WW}{{\cal W}}
\nc{\ZZ}{{\cal Z}}
\nc{\XX}{{\cal X}}
\nc{\YY}{{\cal Y}}


% FONT: BLACKBOARD    version that uses a single letter such as "\A"
			%Except in some cases bellow where one uses:
			%	\hH,\lL,\oO,\pP,\sS
			
		%Also: some old versions are written indented
\nc{\A}{{\Bbb A }}
\nc{\B}{{\Bbb B}}
\nc{\C}{{\Bbb C}}
		\nc{\cc}{{\Bbb C}}
\nc{\Cs}{{\Bbb C^*}}
		\nc{\cs}{{\Bbb C^*}}
		\nc{\ccs}{{\Bbb C^*}}
\nc{\D}{{\Bbb D}}
\nc{\E}{{\Bbb E}}
\nc{\F}{{\Bbb F}}
\nc{\G}{{\Bbb G}}
	\nc{\hH}{{\Bbb H}}
\nc{\I}{{\Bbb I}}
\nc{\J}{{\Bbb J}}
\nc{\K}{{\Bbb K}}
	\nc{\lL}{{\Bbb L}}
				\nc{\lll}{{\Bbb L}}
\nc{\M}{{\Bbb M}}
\nc{\N}{{\Bbb N}}
	\nc{\oO}{{\Bbb O}}
	\nc{\pP}{{\Bbb P}}      
				%\nc{\pp}{{\Bbb P}}
\nc{\Q}{{\Bbb Q}}
\nc{\R}{{\Bbb R}}
	\nc{\sS}{{\Bbb S}}
\nc{\T}{{\Bbb T}}
\nc{\U}{{\Bbb U}}
\nc{\V}{{\Bbb V}}
\nc{\W}{{\Bbb W}}
\nc{\Z}{{\Bbb Z}}
\nc{\X}{{\Bbb X}}
\nc{\Y}{{\Bbb Y}}

%FONT: FRAKTUR        by a single letter with "f", such as "\fA"
			%except the case 	"\fiI" instead of "\fi"
\nc{\fA}{{\frak A}}
\nc{\fB}{{\frak B}}
\nc{\fC}{{\frak C}}
\nc{\fD}{{\frak D}}
\nc{\fE}{{\frak E}}
\nc{\fF}{{\frak F}}
\nc{\fG}{{\frak G}}
\nc{\fH}{{\frak H}}
\nc{\fI}{{\frak I}}
\nc{\fJ}{{\frak J}}
\nc{\fK}{{\frak K}}
\nc{\fL}{{\frak L}}
\nc{\fM}{{\frak M}}
\nc{\fN}{{\frak N}}
\nc{\fO}{{\frak O}}
\nc{\fP}{{\frak P}}
\nc{\fQ}{{\frak Q}}
\nc{\fR}{{\frak R}}
\nc{\fS}{{\frak S}}
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\nc{\fiI}{{\frak i}}  %!!!
	\nc{\ffi}{{\frak i}}  %!!!
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% Change of meaning: \let\A\B means that A now means B ?

% 1. INTRODUCTION:
%\input{intro}
%\section{Introduction}

% 2. (sub)(sub)SECTIONS:
%\section{Introduction}

%\subsection{} 

%\subsection{} 
%\label{group}{\bf [[Group $\bG$ and its Weyl group $\CW_f$.]]}

%\subsubsection{Lemma}
%     a) The evident inclusion $\qc^\alpha\subset\CQ^\alpha$ is an open
%          embedding;
%     b) $\CQ^\alpha$ is a projective variety.
%{\em Proof.}  $\Box$

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%\end{thebibliography}

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%LOCAL DEFINITIONS
%HOME
%\end{document}



 



%		~\ref{Satake}   reference to the text
%		~\cite{gk} 	reference to the literature
%		~\S4		(section 4)

%Semiinfinite DEFINITIONS
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                                        %Superindices
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%TITLE and AUTHORS

\title[]{
Maxim's ICM talks
}

%\author{Michael Finkelberg}
%\address{Independent Moscow University, 65-3-86, }
%\email{	fnklberg@@ium.ips.ras.ru	}

%\author{	Ivan Mirkovi\'c			}
%\address{Dept. of Mathematics and Statistics, University of Massachusetts               at Amherst, Amherst MA 01003-4515, USA}
%\email{		mirkovic@@math.umass.edu	}


%\thanks{}

\date{Nov 1997}

\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%PAPER%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%Local definitions

%HOME

%\end{document}




Maxim Kontsevich gave two talks yesterday at the ICM, a "secret"one
with 15 people and a public one with many hundreds... this is a VERY
informal summary, with many disclaimers etc etc - 
just my impressions/notes

I'll start from the public one (and probably not get to the small
one, where he proposed a better alternative to triangulated
categories, relating to quivers, mirror symmetry
for Fano varieties and singularity theory..)

\section{1st
talk:
Motivic Galois Groups and Deformation Quantization 
}


The title was Motivic Galois Groups and DeformationQuantization - 
the one-line summary is that he has understood (at least conjecturally)
the arithmetic nature of his deformation quantization, leading
to a BIG picture and equivalence between quantization problems and
multiple zeta values... or in his words, strange actions of Galois groups
in algebra..

To motivate people he started out with two consequences of his 
conjectures that he can already prove.

\sus{
Hochschild cohomology of a manifold
$X$}
The first involves what he calls the Hochschild cohomology of a manifold
$X$: the direct sum of the cohomology groups of $X$ (smooth variety 
in characteristic zero) with coefficients in polyvector fields. 

These groups carry an algebra structure and many additional operations
(coming for example from contractions with differential forms.)

Now there is a construction of Chern classes
for the tangent bundle $TX$ in this Hochschild cohomology, 
with the kth Chern class lying in the kth cohomology with coefficients 
in $k$-polyvectors (the "$k,k$" part). This construction is based
on the use of Atiyah classes. 

\sss{Theorem
} The odd chern classes $ch1, ch3, ch5$ etc act on $HH(X)$
as derivations.

[My completely uninformed guess is that this is a generalization
of the Lie property of Atiyah classes in his paper onRozansky-Witten
invariants, see also Kapranov.]

\sss{
 Lie (co)algebra
of multiple zeta values
}
In his big picture this action comes from the Lie (co)algebra 
of multiple zeta values: these elements correspond to the (nonvanishing)
odd values of the Riemann zeta function......

\sus{
Duflo Isomorphism
}
Second Application: The Duflo Isomorphism.
Let 
$\fg$
 be afinite dimensional Lie algebra. Duflo shows
there is an algebra isomorphism from 
$\fg$-invariants in the symmetric algebra 
$(S\fg)^\fg$ to
to the center $Z(U\fg)$
of the enveloping algebra.

This isomorphism is obtained by first 
applying to any $f$ in 
$S\fg$,
thought of as a function on $g*$,
a specific differential operator, and then mapping to 
$U\fg$ using PBW.
This differential operator 
is the Fourier transform of a formal power series
on $\fg*,$
arising
from the $A-hat$ genus (coefficients Bernoulli numbers).

I have the formula if someone needs it). 

\sss{}
Kontsevich claims that 

(a) he has an algebraic proof
of this, not using structure theory of Lie algebras.

(b) This proof works in any rigid tensor category
(eg for Lie superalgebras)

(c) Instead of this $A-hat$ genus series you can replace
it by multiplying it with the exponential of ANY 
formal power series with only odd terms

(I refuse to write the formulas on this keyboard! maybe later).
Inparticular he proposes another candidate for the Duflo map,
which has the property that it is given by an entire function,
unlike the usual one..

Well - that's just the warmup!

\sus{
Conjectural isomorphism of torsors
}
Now for the main part of the talk - he will construct two torsors
over infinite dimensional proalgebraic groups over the rationals,
from deformation theory and from number theory, and claim they
should be isomorphic....

\sus{
Deformation theory}
\sss{
Isomorphism of
differential graded Lie algebras
$Assoc_N$ and $Poisson_N$,
controlling deformations of
$\A^n$ in schemes and Poisson schemes
} 
Consider the homotopy category
of dgla (differential graded Lie algebras) over a field $k$
of $char=0$. 
This category contains objects controling
all sorts of deformation theories.
Let $A=k[x1,x2...xN] $
polynomial algebra in $N$ variables
(remember $N$ - it will go to infinity later!)

We may consider $A$ in particular as an associative algebra, 
or as a Poisson algebra
(with bracket zero). Hence we have two dgla, 
$Assoc A$ and $Poisson A$,
controling deformation theory of 
$A$ in these two categories.

Explicit models for these are the Hochschild complex
with Gerstenhaber bracket, for associative deformations;
and polyvector fields on $N$-dim affine space, differential$=0$ and
Schouten-Nijenhuis bracket, for Poisson deformations.

His big theorem from a year ago claims these two are isomorphic
(in this category), and it also constructs an explicit isomorphism over 
the complex numbers.

\sss{
Why complex numbers? 
}
The coefficients were integrals
over configuration spaces of points in hyperbolic space, 
of wedges of "angle forms". He suggests slightly 
modifying the forms he used in the paper:
$\alpha(z,w)=1/2 \pi i times dlog [(z-w)/(\bar z-w)]$
- in his paper there's one more factor with $\bar w$'s -----
the old one produced real numbers, this one complex.

\sss{
Isomorphism space
$Iso(Assoc,Poisson)$
for
differential graded Lie algebras
$Assoc=Assoc_\yy$ and $Poisson=Poisson_\yy$,
controlling deformations of
$\A^\yy$,
as a torsor
} 
Now the set of isomorphisms between $Assoc$ and $Poisson$
is actually the $k$-points of an affine scheme $Iso(Assoc,Poisson).$
This scheme is a projective limit of torsors over affine algebraic 
groups. NOW let $N$ go to infinity - these groups become
a semidirect product of a multiplicative group 
and a connected (pro)unipotent group.

\sss{
Proof of the deformation quantization
}
Once you understand this (which I certainly don't)
the 
proof of the deformation quantization is easy:
Hilbert 90 tells you there is no first cohomology
of k with coefficients in this group, hence our torsor  is
trivial - in particular it has a point....

\sss{
Maxim's explicit formula}
Maxim's explicit formula should be thought of
as a map from $Spec C $
to the scheme 
$Iso(Assoc,Poisson).$

\subsection{
Number Theory}
\sss{
The algebra of effective periods
}
The algebra of effective periods is defined to be
a vector space over the rationals, generated by symbols
"integral of (really $\omega$ over $\gamma$, but
allow me to abuse his notation..).
 
Here $\omega$ ranges over all top forms on somooth varieties $X$,
of dimension $n$, and $\gamma$ is in the nth homology of $X$ relative to
a divisor of normal corssings - both over the complex numbers
(but $\gamma$ is in rational homology.)


We mod out by three relations coming from basic calculus:
\nn
1. additivity in $\omega,\gamma$,
\nn
2. Change of variables for maps $(X,Z$ to $(X',Z')$ ($Z$'s 
are the divisors):
 integrate pullback of $\omega$ on $\gamma
\
=\ $
integrate
$\omega$ on pushforward of $c$.
\nn
3. Integration by parts - the usual Stokes formula.


This space has a ring structure: the product of two periods
is the period of the exterior product form on the exterior product 
cycle on $X \times X'$.

\sss{The algebra $P$ of periods
}
The algebra of periods (rather than effective periods)
is obtained by modding out by 
$2\ pi i$ - in other 
words by the period of $dt/t$ on the multiplicative group.

\subsection{
The spectrum $\Spec P$
of the algebra of periods as a torsor
$Iso(H_B,H_{dR})$ of isomorphisms between two
cohomology theories
}
In fact let us consider the category of smooth schemes over
the rationals, and the two monoidal functors "deRham cohomology"
and"Betti cohomology" 
$H_{dR}$ and $H_B$, from schemes to  $\Z$-graded
vector spaces.

The usual comparison theorem says that these two functors become
isomorphic over the complex numbers: in other words there
is a map from 
$\Spec \C$ to the scheme 
$Iso(H_B,H_{dR}) $
of isomorphisms
between them (as monoidal functors.)

\subsubsection{
CLAIM} 
$\Spec P$ is isomorphic to $Iso(H_B,H_{dR}).$

This $Iso(H_B,H_{dR})$ 
is a bitorsor for two motivic Galois groups, automorphisms
$Aut H_B$ and $Aut H_{dR}$.
For today he suggests looking at motives as
bi-equivariant vector bundles over 
$Spec P$, i.e. finite dimensional
representations of the product of these Galois groups.

\subsection{
}
{\bf
CONNECTION between
deformation thory and number theory:
$
ISO(H_B,H_{dR})
\con
ISO(Assoc,Poisson)
$
}
Now Kontsevitch takes us back to deformation theory.
The point is that the integrals he used to define
the deformation quantization are certain periods.
His proof of
the theorem uses only elementary operations of calculus - namely our 1-3
above. He claims, it then follows that
there is a morphism 
$$
ISO(H_B,H_{dR})-> ISO(Assoc,Poisson)
.$$
Moreover there is a commutative triangle when we consider the distinguished
maps from $\Spec \C$ to both spaces - one from comparing Betti to deRham 
and one from his explicit quantization.

\sss{
CONJECTURE 1}
This morphism is a morphism of TORSORS.

But in fact we can refine this conjecture... the periods
appearing in the quantization are all $dlogs....$, 
so the corresponding motives are 
\nn
A. extensions of Tate motives
(only periods powers of 
$2\pi i$)   
\nn
B. Unramified over $Spec Z$.


\sss{
Functions $\ze(n)$
}
An example of such a period is 
$\zeta(n)$, $\zeta$
being the Riemann zeta function - 
$\zeta(n)$ is an integral
on $t_1$ to 
$t_n$, ordered between $0$ and $1$, of $\frac{dt_1}{1-t_1}$, followed
by $dlogs $
of the other $t$'s.

\sss{Structure of Motivic Galois groups
}
Now the Beilinson conjectures imply that the corresponding quotient
of the motivic Galois groups is indeed a semidirect product
of a multiplicative group (corresponding to the Tate twist) and a 
prounipotent group. Moreover the corresponding Lie algebra is freely
generated by the (duals of) the odd values of the zeta function.

Conjecturally this is isomorphic to the famous Grothendieck-Teichmuller
group, appearing in the work of Drinfeld on quantum groups, Ihara, Deligne..

\sss{
$Iso-Tate$
}
So now we consider 
isomorphisms (over mixed Tate modules, unramified over
$\Z$) between 
$H_B$ and $H_{dR}$ - let's call this 
$Iso-Tate$
(my bad terminology).

Functions on $Iso-Tate$ are (conjecturally) isomorphic to polynoimials
over 
$Q$ in $2\pi i, \frac{1}{(2 \pi i}$,
odd zeta values and multiple zeta values..

\sss{
CONJECTURE 2}
$ Iso-Tate$ is isomorphic to $Iso(Assoc,Poisson).$


\sss{
Flows on
the moduli of Poisson manifolds
}
Now conjecture 1 has among its implications that on 
spaces of equivalence classes of Poisson manifolds there
are infinitely many flows - these are given by the action 
of the (duals of) 
$\zeta(3),\zeta((5) $
etc etc...

He has explicit candidates for the first couple of there - 
complicated differential equations on the Poisson tensor,
which remind him of some kind of KdV hierarchy...

\subsection{
Final note}
As $N$ goes to infinity the Lie algebra of 
$Aut(Poison_N) $
is
the $0$th cohomology of a certain graph complex....

He writes down nice simple graphs corresponding
to 
$\zeta(3), zeta(5)...$

\sss{BIG Question at the end}
What about nonzero cohomology
of this graph complex??? 
Is the motivic Galois
group just the zero component of a 
$Z$-graded motivic Galois 
supergroup??


At which point his time was up, and at which point I 
must stop.


\end{document}



