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%  Twisted Deformation Quantization of Algebraic Varieties
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%  Amnon Yekutieli
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% \title[Deformation Quantization]
% {Twisted Deformation Quantization of Algebraic Varieties}
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% \author[Amnon Yekutieli]{Amnon Yekutieli}
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\begin{document}

%% top matter for notes %%
\mbox{}
\vspace{2cm}

\begin{center}
{\large \bfseries Twisted Deformation Quantization of 
Algebraic Varieties}


\vspace{1ex}
Lecture Notes
\footnote[1]{written 17 June 2009; corrected 11 Sep 2010}

\vspace{3ex}
Amnon Yekutieli\\
Ben Gurion University, ISRAEL

\texttt{\small http://www.math.bgu.ac.il/$\sim$amyekut}
\end{center}

\vspace{2ex}
%% ** end of top matter for notes **


% \begin{frame}[label=title]
% \titlepage
% \end{frame}

%\begin{frame}[label=toc]
Here is the plan of my lecture: %\pause

\begin{enumerate}
\item[1.] Some background on Deformation  Quantization
\item[2.] Poisson Deformations of Algebraic Varieties
\item[3.] Associative Deformations of Algebraic Varieties
\item[4.] Deformation Quantization 
\item[5.] Twisted Deformations of Algebraic Varieties
\item[6.] Twisted Deformation Quantization
\end{enumerate}
%\pause

Notes are available online. The notes also contain four
appendices, and a bibliography.
%\pause

Part of this work is joint with Frederick Leitner. 

%\end{frame}


\section{Some background on Deformation Quantization}
%\begin{frame}[label=sec1]

%\sect{1. Some background on Deformation Quantization}

%\pause
Let $\K$ be a field of characteristic $0$, and let $C$ be a 
commutative $\K$-algebra. 

%\pause
Recall that a \hilite{drkblue}{Poisson bracket} on $C$ is a 
$\K$-bilinear function
\[ \{-,-\} : C \times C \to C \]
which makes $C$ into a Lie algebra, and is a biderivation (i.e.\ 
a derivation in each argument). 

%\pause
The pair $\bigl( C, \{-,-\} \bigr)$ is called a Poisson algebra. 

%\pause
Poisson algebras arise in several ways, e.g.\ classical Hamiltonian
mechanics, or Lie theory.
%\end{frame}

%\begin{frame}
Let $\K[[\hbar]]$ be the ring of formal power series in the
variable $\hbar$. 

%\pause
Let $C[[\hbar]]$ be the set of formal power series with
coefficients in $C$, which we view only as a 
$\K[[\hbar]]$-module. 

%\pause
A \hilite{drkblue}{star product} on $C[[\hbar]]$ is a function
\[ \star : C[[\hbar]] \times C[[\hbar]] \to C[[\hbar]]  \]
which makes $C[[\hbar]]$ into an associative 
$\K[[\hbar]]$-algebra, with unit $1 \in C$, and  such that 
\[ f \star g \equiv f g \ \opn{mod}\ \hbar \]
for any $g, f \in C$. 

%\pause
The pair $\bigl( C[[\hbar]], \star \bigr)$ is called an 
\hilite{rust}{associative deformation} of $C$.
%\end{frame}

%\begin{frame}
\begin{exa} \label{exa3}
Suppose $\bigl( C[[\hbar]], \star \bigr)$ is an associative
deformation of $C$. 

%\pause
Given $f, g \in C$, we know that 
\[ f \star g - g \star f \equiv 0 \ \opn{mod}\ \hbar . \]

%\pause
Hence there is a unique element 
\[ \{ f, g \}_{\star} \in C \]
such that 
\[ \smfrac{1}{\hbar} \bigl( f \star g - g \star f \bigr)
\equiv \{ f, g \}_{\star}  \ \opn{mod}\ \hbar . \]

%\pause
It is quite easy to show that $\{ -,- \}_{\star}$ is a Poisson
bracket on $C$. We call it the \hilite{drkblue}{first order bracket}
of $\star$. 
\end{exa}
%\end{frame}

%\begin{frame}
Deformation quantization seeks to reverse Example \ref{exa3}.

%\pause
\begin{dfn}
Given a Poisson bracket $\{ -,- \}$ on the algebra $C$, a 
\hilite{drkblue}{deformation quantization} of $\{ -,- \}$ is an
associative
deformation $\bigl( C[[\hbar]], \star \bigr)$ of $C$ 
whose first order bracket is $\{ -,- \}$.
\end{dfn}
%\end{frame}

%\begin{frame}
In physics $\hbar$ is the \hilite{rust}{Planck constant}. For a
quantum
phenomenon depending on $\hbar$, the limit as $\hbar \to 0$ is thought
of the as the classical limit of this phenomenon.
%\pause

The original idea by the physicists Flato et.\ al.\ (\cite{BFFLS},
1978) was that deformation quantization should model the transition
from classical Hamiltonian mechanics to quantum mechanics.
%\pause

Special cases (like the Moyal product) were known. The problem 
arose: \hilite{rust}{does any Poisson bracket admit a 
deformation quantization?}
%\pause 

For a symplectic manifold $X$ and $C = \mrm{C}^{\infty}(X)$ 
the problem was solved by De Wilde and Lecomte  (\cite{DL}, 1983). 
A more geometric solution was discovered by Fedosov (\cite{Fe},
1994).
%\pause

The general case, i.e.\ $C = \mrm{C}^{\infty}(X)$ for a 
Poisson manifold $X$, was solved by Kontsevich  (\cite{Ko1}, 1997).
See surveys in the book \cite{CKTB}.

%\hyperlink{toc<8>}{\beamergotobutton{toc}}
%\end{frame}




\section{Poisson Deformations of Algebraic Varieties}
%\begin{frame}[label=sec2]
%\sect{2. Poisson Deformations of Algebraic Varieties}
%\pause

In algebraic geometry we have to consider deformations as sheaves. 
%\pause

Let $X$ be a smooth algebraic variety over $\K$, with structure 
sheaf $\mcal{O}_{X}$. 
%\pause

We view $\mcal{O}_{X}$ as a Poisson $\K$-algebra with zero bracket.
%\pause

\begin{dfn}
A \hilite{rust}{Poisson deformation} of $\mcal{O}_X$ 
is a sheaf $\mcal{A}$ of flat, $\hbar$-adically complete,
commutative Poisson $\K[[\hbar]]$-algebras on $X$, with an 
isomorphism of Poisson algebras 
\[ \psi : \mcal{A} / (\hbar) \iso \mcal{O}_X , \]
called an augmentation. 
\end{dfn}
%\end{frame}

%\begin{frame}
A \hilite{rust}{gauge equivalence} $\mcal{A} \to \mcal{A}'$
between 
Poisson deformations is a $\K[[\hbar]]$-linear 
isomorphism of sheaves of Poisson algebras, that commutes with the 
augmentations to $\mcal{O}_X$. 
%\pause

Given a Poisson deformation $\mcal{A}$ of $\mcal{O}_X$, we 
may define the \hilite{drkblue}{first order bracket}
\[ \{ -,- \}_{\mcal{A}} :  \mcal{O}_X \times \mcal{O}_X \to \mcal{O}_X
. \]
%\pause

\negskip{2}
This is a Poisson bracket whose formula is
\[ \{ f,g \}_{\mcal{A}} := \psi \bigl( \smfrac{1}{\hbar} \{ \til{f},
\til{g} 
\} \bigr) , \]
where $f, g \in \mcal{O}_X$ are local sections, and 
$\til{f}, \til{g} \in \mcal{A}$ are arbitrary local lifts. %%\pause
%\pause

The first order bracket  is invariant under gauge equivalence. 
%\end{frame}

%\begin{frame} [label=EXA.10]
\begin{exa} \label{exa10}
Let $\{ -,- \}_1$ be some Poisson bracket on $\mcal{O}_{X}$. 
%\pause

Define 
\[ \mcal{A} := \mcal{O}_X [[\hbar]] . \] %%\pause
This is a sheaf of $\K[[\hbar]]$-algebras,
with the usual commutative multiplication, and the obvious 
augmentation $\mcal{A} / (\hbar) \cong \mcal{O}_X$. 
%\pause

Put on $\mcal{A}$ the $\K[[\hbar]]$-bilinear Poisson bracket 
$\{ -,- \}$ such that
\[ \{ f, g \} = \hbar \{ f, g \}_1  \]
for $f, g \in \mcal{O}_X$. 
%\pause

Then $\mcal{A}$ is a Poisson deformation of $\mcal{O}_X$. The first
order 
bracket in this case is just
\[ \{ -,- \}_{\mcal{A}} = \{ -,- \}_{1} . \]
\end{exa}

%\end{frame}

%\begin{frame}
Poisson deformations are controlled by a sheaf of DG (differential
graded) Lie algebras 
$\mcal{T}^{}_{\mrm{poly}, X}$,
called the \hilite{drkblue}{poly derivations}. %%\pause
%\pause

This is explained in Appendix A.
%\end{frame}




\section{Associative Deformations of Algebraic Varieties}
%\begin{frame}[label=sec3]
%\sect{3. Associative Deformations of Algebraic Varieties}
%\pause

\begin{dfn}
An \hilite{drkblue}{associative deformation of $\mcal{O}_X$} 
is a sheaf $\mcal{A}$ 
of flat, $\hbar$-adically complete, associative, unital 
$\K[[\hbar]]$-algebras on $X$, with an isomorphism of algebras
\[ \psi : \mcal{A} / (\hbar) \iso \mcal{O}_X , \]
called an augmentation.
\end{dfn}
%\pause

There is a suitable notion of gauge equivalence between associative 
deformations. 
%\end{frame}

%\begin{frame}
Given an associative deformation $\mcal{A}$ we 
may define the first order bracket
\[ \{ -,- \}_{\mcal{A}} :  \mcal{O}_X \times \mcal{O}_X \to \mcal{O}_X
. \]
%\pause

\negskip{2}
The formula is
\[ \{ f,g \}_{\mcal{A}} :=
\psi \bigl( \smfrac{1}{\hbar} (\til{f} \star 
\til{g} - \til{g} \star \til{f}) \bigr) . \]

%\pause
The first order bracket is invariant under gauge equivalence. %%\pause
%\pause

Note that both kinds of deformations -- Poisson and associative --
include as
special cases the classical commutative deformations of $\mcal{O}_X$.
%\pause

Associative deformations are controlled by a quasi-coherent sheaf of
DG
Lie algebras $\mcal{D}^{}_{\mrm{poly}, X}$,
called the \hilite{drkblue}{poly differential operators}. %%\pause
%\pause

This is explained in Appendix A.
%\end{frame}





\section{Deformation Quantization}
%\begin{frame}[label=sec4]
%\sect{4. Deformation Quantization}
%\pause

Kontsevich \cite{Ko1} proved that any Poisson deformation of a real 
$\mrm{C}^{\infty}$ manifold $X$ can be canonically quantized. 
%\pause

In this section 
we present an algebraic version of this result. But first a
definition. 
%\pause

\begin{dfn}
Let $\mcal{A}$ be a Poisson deformation of $\mcal{O}_X$. %%\pause
A \hilite{drkblue}{quantization} of $\mcal{A}$ is an 
associative deformation $\mcal{B}$, such that the first order
brackets satisfy
\[ \{ -,- \}_{\mcal{B}} = \{ -,- \}_{\mcal{A}} . \]
\end{dfn} 
%\pause

\negskip{2}
Recalling Example 2.2,
%\hyperlink{EXA.10<4>}{\blue \ul{Example \ref{exa10}}}, 
%\hyperlink{EXA.10<4>}{\blue \ul{Example 2.2}}, 
we see that this definition captures the essence
of deformation quantization, namely quantizing a Poisson bracket on 
$\mcal{O}_X$.
%\end{frame}

%\begin{frame}
\begin{thm}\tup{(\cite{Ye1})} \label{thm1} 
%\pause
\hilite{brtgreen}{
Let $\K$ be a field containing $\mbb{R}$, and let $X$ be a smooth
affine algebraic variety over $\K$.} 
%\pause

%\negskip{1}
\hilite{brtgreen}{There is a canonical bijection
\[ \begin{aligned}
& \opn{quant} : \frac{ \{ \tup{Poisson deformations of 
$\mcal{O}_{X}$} \} }
{\tup{ gauge equivalence}} \\
&  \hspace{10ex}  \iso
\frac{ \{ \tup{associative deformations of 
$\mcal{O}_X$} \} }{\tup{ gauge equivalence}} ,
\end{aligned} \] 
%\pause
which is a quantization as defined above. }
\end{thm} 
%\end{frame}

%\begin{frame}
By ``canonical'' I mean that this quantization map 
commutes with \'etale morphisms $X' \to X$.
%\pause

Actually our result in \cite{Ye1} is stronger -- it holds for a wider
class of varieties, not just affine varieties. 
However all these cases are subsumed in Corollary \ref{cor_new}
below. 
%\pause

Theorem \ref{thm1} is a consequence of the following more general
result.
%\end{frame}

%\begin{frame}
\begin{thm} \tup{(\cite{Ye1})} \label{thm2} 
%\pause
\hilite{brtgreen}{Let $\K$ be a field containing $\mbb{R}$, and let
$X$ be a
smooth algebraic variety over $\K$.} 
%\pause

\hilite{brtgreen}{Then there is a diagram 
\[ \UseTips \xymatrix @C=5ex @R=3.5ex {
\mcal{T}^{}_{\mrm{poly}, X} 
\ar[d]
& \mcal{D}^{}_{\mrm{poly}, X}
\ar[d] \\
\opn{Mix}^{}(\mcal{T}^{}_{\mrm{poly}, X} ) 
\ar[r]
&
\opn{Mix}^{}(\mcal{D}^{}_{\mrm{poly}, X} )
} \] } 
%\pause

\negskip{1}
\hilite{brtgreen}{where:} 
%\pause
\begin{itemize}
\item \hilite{brtgreen}{$\opn{Mix}^{}(\mcal{T}^{}_{\mrm{poly}, X})$
and
$\opn{Mix}^{}(\mcal{D}^{}_{\mrm{poly}, X})$
are sheaves of DG Lie algebras on $X$, called {\em mixed
resolutions};} 
%\pause
\item \hilite{brtgreen}{the vertical arrows are DG Lie algebra 
quasi-isomorphisms;} 
%\pause
\item \hilite{brtgreen}{and the horizontal arrow is 
an $\mrm{L}_{\infty}$ quasi-isomorphism.} 
\end{itemize}   
\end{thm}
%\end{frame}

%\begin{frame}
The mixed resolutions combine the \hilite{drkblue}{commutative
\v{C}ech 
resolution} associated to an affine open covering of $X$, and the
\hilite{rust}{Grothendieck sheaf of jets}. 
%\pause

An \hilite{rust}{$\mrm{L}_{\infty}$ quasi-isomorphism} is a
generalization of a
DG Lie algebra quasi-iso\-mor\-phism. 
%\pause

Theorem \ref{thm2} is proved using the \hilite{drkblue}{Formality
Theorem}
of Kontsevich \cite{Ko1} and \hilite{rust}{formal geometry}.
%\pause

More on the proof of Theorem \ref{thm2} in Appendices B and C.
%\end{frame}

\begin{rem}
Calaque and Van den Bergh \cite{CV} proved (using ideas of Tamarkin and 
Halbout) that a global quantization map exists over $\mbb{Q}$. 

According to Kontsevich, changing the local formality isomorphism 
(i.e.\ changing the Drinfeld associator) can have a 
nontrivial effect on the global quantization process. 
 

This phenomenon is very intriguing and should be studied.
\end{rem}



\section{Twisted Deformations of Algebraic Varieties}
%\begin{frame}[label=sec5]
%\sect{5. Twisted Deformations of Algebraic Varieties}
%\pause

What can be done in general, when the variety $X$ is not affine? Can
we still make use of Theorem \ref{thm2}? 
%\pause

In the paper \cite{Ko3} Kontsevich suggests that in general the 
deformation quantization of a Poisson bracket might have to be a 
\hilite{rust}{stack of algebroids}. This is a generalization of
the notion of sheaf of algebras. 
%\pause

Actually stacks of algebroids appeared earlier, 
under the name \hilite{drkblue}{sheaves of twisted modules},
in the work of Kashiwara \cite{Ka}. See also \cite{DP}, \cite{PS},
\cite{KS}. 
%\end{frame}

%\begin{frame}
I will use the term \hilite{rust}{twisted associative deformation},
and present an approach that treats the Poisson case as well. 
%\pause

This approach was suggested to us by Kontsevich. A similar point of
view is taken in \cite{BGNT}. 
%\pause

Here I will explain only a naive definition of twisted 
deformations. A more sophisticated definition, involving gerbes, 
may be found in Appendix D.
%\pause

The fact that the two definitions agree follows from our work on
central extensions of gerbes and obstructions classes \cite{Ye5}.
%\end{frame}

%\begin{frame}
Let $U \subset X$ be an affine open set, and let
$C := \Gamma(U, \mcal{O}_X)$. 
%\pause

Suppose $A$ is an associative or Poisson deformation of the
$\K$-algebra $C$.
%\pause

One may assume that 
$A = C[[\hbar]]$, and it is either endowed with a Poisson bracket
$\{ -,- \}$, or with a star product $\star$.  
%\pause

In either case $A$ becomes a pronilpotent Lie algebra, and 
$\hbar A$ is a Lie subalgebra.
%\pause

In the Poisson
case the Lie bracket is $\{ -,- \}$, and in the associative case the
Lie
bracket is the commutator
\[ [a,b] := a \star b - b \star a . \] 
%\pause

%\negskip{2}
Let us denote the corresponding pronilpotent group by
\[ \opn{IG}(A) := \opn{exp}(\hbar A) , \]
%\negskip{1}
and call it the \hilite{drkblue}{group of inner gauge transformations}
of $A$.
%\end{frame}

%\begin{frame}
The group $\opn{IG}(A)$ acts on the deformation $A$ by gauge
equivalences.
We denote this action by $\opn{Ad}$. 
%\pause

In the Poisson case the gauge
transformation $\opn{Ad}(g)$, for $g \in \opn{IG}(A)$,
can be viewed as a formal hamiltonian flow. 
%\pause

In the associative case the intrinsic exponential function
\[ \exp(a) =  \sum_{i \geq 0} \smfrac{1}{i!} a^i , \] 
for $a \in \hbar A$, allows us to identify the group $\opn{IG}(A)$
with the multiplicative subgroup 
\[ \{ g \in A \mid g \equiv 1 \opn{mod} \hbar \} . \]
Under this identification the operation $\opn{Ad}(g)$ is just
conjugation by the invertible element $g$. 
%\end{frame}

%\begin{frame}
The above can be sheafified: to a deformation $\mcal{A}$ of
$\mcal{O}_X$ we
assign the sheaf of groups $\opn{IG}(\mcal{A})$. 
%\pause

Let us fix an affine open covering
$\{ U_0, \ldots, U_m \}$ of $X$. 
We write
\[ U_{i,j, \ldots} := U_i \cap U_j \cap \cdots . \]
%\end{frame}

%\begin{frame}
\begin{dfn} \label{dfn1} 
%\pause
A \hilite{drkblue}{twisted associative \tup{(}resp.\ Poisson\tup{)}
deformation} $\bsym{\mcal{A}}$ of $\mcal{O}_X$
consists of the following data: 
%\pause
\begin{enumerate}
\item For any $i$, a deformation $\mcal{A}_i$ of $\mcal{O}_{U_i}$. 
%\pause
\item For any $i < j$, a gauge equivalence
\[ g_{i,j} : \mcal{A}_i|_{U_{i,j}} \to \mcal{A}_j|_{U_{i,j}} .\] 
%\pause
\negskip{1}
\item For any $i < j <k$, an element
\[ a_{i,j,k} \in \Gamma \bigl( U_{i,j,k}, \opn{IG}(\mcal{A}_i) \bigr)
. \]
\end{enumerate} 
\end{dfn}
%\end{frame}

\begin{figure}
\includegraphics[scale=0.4]{fig1d.eps}
%\label{fig1}
\caption{}
\end{figure}
%\end{frame}


%\begin{frame}
The conditions are: 
%\pause
\begin{enumerate}
\rmitem{i} For any $i <j <k$ one has
\[ g_{i,k}^{-1} \circ g_{j,k} \circ g_{i,j} = \opn{Ad}(a_{i,j,k}^{-1})
. \]
%\pause 
\negskip{1}
%
\rmitem{ii} For any $i <j < k < l$ one has
\[ a_{i,j,l}^{-1} \cdot a_{i,k,l} \cdot a_{i,j,k} = 
g_{i,j}^{-1}(a_{j,k,l}^{}) . \]
\end{enumerate}
%\end{frame}


See Figure 1 for an illustration. 


%\begin{frame}
Condition (i) says that the $2$-cochain $\{ \opn{Ad}(a_{i,j,k}) \}$ 
measures the failure of the $1$-cochain $\{ g_{i,j} \}$ to be a
cocycle. 
%\pause

This 
tells us whether the collection $\{ \mcal{A}_i \}$ of local
deformations can be glued into a global deformation of $\mcal{O}_X$. 
%\pause

Condition (ii) -- usually called the tetrahedron equation -- says that
the
$2$-cochain $\{ a_{i,j,k} \}$ satisfies a twisted cocycle condition. 
%\pause

\begin{exa}
If $\mcal{A}$ is a usual deformation of $\mcal{O}_X$, then we obtain
a twisted deformation $\bsym{\mcal{A}}$ by taking
$\mcal{A}_i : = \mcal{A}|_{U_i}$, $g_{i,j} := \bsym{1}$ and
$a_{i,j,k} := 1$.
\end{exa}
%\end{frame}

%\begin{frame}
\begin{rem} \label{rem1}
For a twisted associative deformation $\bsym{\mcal{A}}$
there is a well defined abelian category 
$\cat{Coh} \bsym{\mcal{A}}$ of ``coherent left
$\bsym{\mcal{A}}$-modules'', which is a deformation of the 
abelian category $\cat{Coh} \mcal{O}_X$. See the work of Lowen and
Van den Bergh \cite{LV}.
%\pause

Indeed, there is a
geometric Morita theory, which says that twisted associative
deformations of $\mcal{O}_X$ are the same as deformations of 
$\cat{Coh} \mcal{O}_X$ (as stack of abelian categories). This is explained in
the new book by Kashiwara and Schapira \cite{KS}.

We do not know of a similar interpretation of twisted Poisson
deformations. 
\end{rem}

\begin{rem}
It appears that twisted deformations behave very similarly when $X$ is a
complex analytic manifold.
%\pause 

For a differentiable manifold $X$ there is nothing interesting, since all
obstruction clases vanish.
\end{rem}

%\end{frame}



\section{Twisted Deformation Quantization}
%\begin{frame}[label=sec6]
%\sect{6. Twisted Deformation Quantization}
%\pause

There is a notion of \hilite{drkblue}{twisted gauge equivalence} 
$\bsym{\mcal{A}} \to \bsym{\mcal{B}}$ between twisted
associative (resp.\ Poisson) deformations of $\mcal{O}_X$. 
%\pause

Just as in the case of usual deformations, given a 
twisted (associative or Poisson) deformation $\bsym{\mcal{A}}$ 
of $\mcal{O}_X$, we can define the first order bracket 
$\{ -,- \}_{\bsym{\mcal{A}}}$ on $\mcal{O}_X$. 
%\pause

Let $\bsym{\mcal{A}}$ be a twisted Poisson deformation, and 
let $\bsym{\mcal{B}}$ be a twisted associative deformation. We say 
that $\bsym{\mcal{B}}$ is a \hilite{rust}{twisted quantization} of
$\bsym{\mcal{A}}$ if 
\[ \{ -,- \}_{\bsym{\mcal{B}}} = \{ -,- \}_{\bsym{\mcal{A}}} . \]
%\pause

\negskip{2} 
The next theorem is influenced by ideas of Kontsevich (from 
\cite{Ko3} and private communications).
%\end{frame}

%\begin{frame}
\begin{thm}\tup{(\cite{Ye5})}  \label{thm_new}
%\pause
\hilite{brtgreen}{Let $\K$ be a field containing $\mbb{R}$, and let
$X$ be a
smooth algebraic variety over $\K$.} 
%\pause

\hilite{brtgreen}{Then there is a canonical bijection 
\[ \begin{aligned}
& \opn{quant} : \frac{ \{ \tup{twisted Poisson deformations of 
$\mcal{O}_{X}$} \} }
{\tup{ twisted gauge equivalence}} \\
&  \quad \iso
\frac{ \{ \tup{twisted associative deformations of 
$\mcal{O}_X$} \} }{\tup{ twisted gauge equivalence}} ,
\end{aligned} \] 
%\pause
which is a twisted quantization in the sense above.}
\end{thm} 
%\pause

As before, by ``canonical'' we mean that this quantization map
commutes with \'etale morphisms $X' \to X$.
%\end{frame}

%\begin{frame}
The proof of Theorem \ref{thm_new} relies on a rather complicated
calculation of Maurer-Cartan equations in cosimplicial DG Lie
algebras, and on a new theory of \hilite{rust}{nonabelian integration
on surfaces}.
%\pause

The theorem, together with the results on obstruction classes for
gerbes, implies:
%\pause

\begin{cor}\tup{(\cite{Ye5})}  \label{cor_new}
%\pause

\hilite{brtgreen}{Assume 
\[ \mrm{H}^1(X, \mcal{O}_{X}) = \mrm{H}^2(X, \mcal{O}_{X}) = 0 . \]}
%\pause

\negskip{1}
\hilite{brtgreen}{Then the quantization map of the theorem gives a
bijection
\[ \begin{aligned}
& \opn{quant} : \frac{ \{ \tup{Poisson deformations of 
$\mcal{O}_{X}$} \} }
{\tup{ gauge equivalence}} \\
&  \hspace{10ex}  \iso
\frac{ \{ \tup{associative deformations of 
$\mcal{O}_X$} \} }{\tup{ gauge equivalence}} .
\end{aligned} \] }
\end{cor} 
%\end{frame}

%\begin{frame}
Let me finish with a question. 
%\pause

Given a variety $X$, with Poisson bracket $\{-,-\}_1$ on 
$\mcal{O}_X$, we can form the Poisson deformation 
$\mcal{A} := \mcal{O}_X[[\hbar]]$, with bracket 
$\hbar \{-,-\}_1$. 
%\pause

By viewing $\mcal{A}$ as a twisted Poisson deformation, and applying
Theorem \ref{thm_new}, we get a twisted associative deformation 
$\bsym{\mcal{B}} := \opn{quant}(\mcal{A})$.
%\pause

We say $\bsym{\mcal{B}}$ is \hilite{drkblue}{really twisted} if
it is not
equivalent to any usual deformation $\mcal{B}$.
%\pause

\begin{que}
\hilite{brtgreen}{Does there exist a variety $X$, with a symplectic
Poisson bracket $\{-,-\}_1$, such that the corresponding twisted
associative deformation $\bsym{\mcal{B}}$ is really twisted?}
\end{que}
%\pause

My feeling is that the answer is positive. 
%\pause

And moreover, an example should be when $X$ is any Calabi-Yau surface,
and $\{-,-\}_1$ is any nonzero Poisson bracket on $X$.
%\pause

\medskip
\begin{center}
{\em \bfseries \red - END -}
\end{center}


\appendix
%%\begin{frame}[label=app.toc]
Here are the appendices:

\begin{enumerate}
\item[A.] DG Lie Algebras and Deformations
\item[B.] The Universal Quantization Map
\item[C.] The $\mrm{L}_{\infty}$ quasi-isomorphism of the Level of 
Sheaves 
\item[D.] Twisted Deformations via Stacks of Gluing Groupoids
\end{enumerate}

%%\hyperlink{toc<8>}{\beamergotobutton{toc}}
%%\end{frame}

\section{DG Lie Algebras and Deformations}
%%\begin{frame}[label=appA]
%\sect{A. DG Lie Algebras and Deformations}
%%\pause

The idea that DG (differential graded) Lie algebras control 
deformation problems is attributed to Deligne. See \cite{GM}.
%%\pause

Recall that a DG Lie algebra is a graded $\K$-module 
$\mfrak{g} = \boplus_{p \in \mbb{Z}} \mfrak{g}^{p}$,
with a bracket $[-, -]$ 
satisfying the graded version of the Lie algebra identities, 
together with a graded derivation $\mrm{d}$ 
of degree $1$ and square $0$.
%%\pause

Given a DG Lie algebra $\mfrak{g}$, let us define a new DG Lie 
algebra 
\[ \mfrak{g}[[\hbar]]^+ := 
\boplus_p \, \hbar \mfrak{g}^{p}[[\hbar]] \subset
\boplus_p \, \mfrak{g}^{p}[[\hbar]] , \] 
in which $\hbar$ is central. 
%%\end{frame}

\begin{rem}
Everywhere in these notes we can replace $\K[[\hbar]]$ with any
complete noetherian local $\K$-algebra $R$, with maximal ideal
$\mfrak{m}$, such that $R / \mfrak{m} = \K$. There would have to be
slight modifications of course; e.g.\ instead of 
$\mfrak{g}[[\hbar]]^+$ we would have to take
$\mfrak{g} \hatotimes{} \mfrak{m}$. 
\end{rem}


%%\begin{frame}
The {\em Maurer-Cartan equation} in 
$\mfrak{g}[[\hbar]]^+$ is
\[ \mrm{d}(\alpha) + \smfrac{1}{2} [\alpha, \alpha] = 0 \]
for
\[ \alpha = \sum_{j = 1}^{\infty} \alpha_j \hbar^j
\in \mfrak{g}^{1}[[\hbar]]^+ . \]
%%\pause

Let 
$\opn{exp}(\mfrak{g}^{0}[[\hbar]]^+)$
be the pro-unipotent group associated to the pro-nilpotent Lie
algebra 
$\mfrak{g}^{0}[[\hbar]]^+$. 
%%\pause

There is an action of the group 
$\opn{exp}(\mfrak{g}^{0}[[\hbar]]^+)$ on  
$\mfrak{g}^{1}[[\hbar]]^+$, and this action preserve the set of
solutions of 
the Maurer-Cartan equation. 
%%\pause

One defines
\[ \opn{MC}(\mfrak{g}[[\hbar]]^+) := 
\frac{ \{ \text{solutions of MC equation in } 
\mfrak{g}[[\hbar]]^+ \} } 
{ \opn{exp}(\mfrak{g}^{0}[[\hbar]]^+) } . \]
%%\end{frame}

%%\begin{frame}
Let us return to our deformation problem, where $X$ is a smooth 
algebraic variety over $\K$. Take an affine open set 
$U \subset X$, and let $C := \Gamma(U, \mcal{O}_X)$. 
%%\pause

One can show that any Poisson (resp.\ associative) deformation of $C$
is 
isomorphic to $C[[\hbar]]$ as $\K[[\hbar]]$-algebra (resp.\
$\K[[\hbar]]$-module). Thus it suffices to understand Poisson brackets
and 
star products on $C[[\hbar]]$.
%%\pause

Let $\mcal{T}_C$ denote the module of derivations.  
For $p \geq -1$ define
\[ \mcal{T}^{p}_{\mrm{poly}}(C) := 
\bwedge_C^{p + 1} \mcal{T}_C . \]
So $\mcal{T}^{-1}_{\mrm{poly}}(C) = C$, 
$\mcal{T}^{0}_{\mrm{poly}}(C) = \mcal{T}_C$ and
$\mcal{T}^{1}_{\mrm{poly}}(C) = \bwedge_C^{2} \mcal{T}_C$.
%%\pause

The direct sum
\[ \mcal{T}_{\mrm{poly}}(C) := \boplus_p
\mcal{T}^{p}_{\mrm{poly}}(C) \]
is a DG Lie algebra, called the algebra of {\em poly derivations} 
of $C$. The Lie bracket is the Schouten-Nijenhuis bracket, and the 
differential is $0$.
%%\end{frame}

%%\begin{frame}
A calculation shows that the solutions of the Maurer-Cartan 
equation in \linebreak $\mcal{T}_{\mrm{poly}}(C)[[\hbar]]^+$ 
are the $\K[[\hbar]]$-bilinear Poisson brackets on $C[[\hbar]]$
that vanish modulo $\hbar$, and that the group 
$\opn{exp} \bigl( \mcal{T}^0_{\mrm{poly}}(C)[[\hbar]]^+ \bigr)$
is the group of gauge equivalences. 
%%\pause

In this sense $\mcal{T}_{\mrm{poly}}(C)$ controls Poisson 
deformations of $C$. 
%%\end{frame}

%%\begin{frame}
The second DG Lie algebra in this picture is that of the 
{\em poly differential operators}. 
%%\pause

For $p \geq -1$ one defines
\[ \mcal{D}^{p}_{\mrm{poly}}(C) :=
\{ \phi : C^{p+1} \to C \mid \phi 
\text{ is a differential operator in each argument} \} . \]
%%\pause

\negskip{2}
So $\mcal{D}^{-1}_{\mrm{poly}}(C) = C$ and
$\mcal{D}^{0}_{\mrm{poly}}(C) = \mcal{D}(C)$,
the ring of differential operators. 
%%\pause

$\mcal{D}_{\mrm{poly}}(C)$ is a sub DG Lie algebra 
of the Hochschild cochain complex of $C$, with the Gerstenhaber 
bracket. 
%%\pause

Solutions 
$\beta = \sum_{j = 1}^{\infty} \beta_j \hbar^j$
of the Maurer-Cartan equation in 
$\mcal{D}_{\mrm{poly}}(C)[[\hbar]]^+$ 
correspond to star products on $C[[\hbar]]$, by the formula
\[ f \star g := f g + 
\sum_{j = 1}^{\infty} \beta_j(f, g) \hbar^j . \]
%%\pause

\negskip{2}
And the group 
$\opn{exp} \bigl( \mcal{D}^0_{\mrm{poly}}(C)[[\hbar]]^+ \bigr)$
is the group of gauge equivalences. 
%%\end{frame}

%%\begin{frame}
\begin{rem}
There is a delicate issue hidden here. One can show that 
any star product is gauge equivalent to a differential star product. 
This follows from the fact that $\mcal{D}_{\mrm{poly}}(C)$ is 
quasi-isomorphic to the full Hochschild cochain complex of $C$.
\end{rem}
%%\pause

Geometrically, there are sheaves of DG Lie algebras
$\mcal{T}_{\mrm{poly}, X}$ and $\mcal{D}_{\mrm{poly}, X}$
on $X$, that are quasi-coherent as $\mcal{O}_X$-modules. 
For any affine open set $U$ as above we have
\[ \Gamma(U, \mcal{T}_{\mrm{poly}, X}) = \mcal{T}_{\mrm{poly}}(C) , 
\]
and likewise for $\mcal{D}_{\mrm{poly}}$.
%%\pause

In order to control global deformations one has to resort to some 
kind of resolution of these sheaves of DG Lie algebras, such as the 
mixed resolutions mentioned in Theorem \ref{thm2}.

%%\hyperlink{toc<8>}{\beamergotobutton{toc}}
%%\end{frame}



\section{The Universal Quantization Map}
%%\begin{frame}[label=appB]
%\sect{B. The Universal Quantization Map}
%%\pause

Let $C$ be a smooth $\K$-algebra. 
%%\pause

There is a canonical map of complexes
\[ \mcal{U}_1 : \mcal{T}_{\mrm{poly}}(C) \to
\mcal{D}_{\mrm{poly}}(C) \]
%%\pause
given by
\[ \mcal{U}_1(\partial_1 \wedge \cdots \wedge \partial_k)
(f_1, \ldots, f_k) := 
\smfrac{1}{k!} \sum_{\sigma \in S_k} \opn{sgn}(\sigma)
\partial_{\sigma(1)}(f_1) \cdots \partial_{\sigma(k)}(f_k) \]
for $f_i \in C$ and $\partial_i \in \mcal{T}_C$. 
%%\pause

It is known that $\mcal{U}_1$ is a quasi-isomorphism --
see \cite{Ko1} for the case $C = \mrm{C}^{\infty}(U)$,
and \cite{Ye1} for the case $C = \mcal{O}(U)$ --
and it induces an isomorphism of graded Lie algebras in cohomology. 
%%\pause

But $\mcal{U}_1$ is not a DG Lie algebra homomorphism!
%%\end{frame}

%%\begin{frame}
\begin{thm}\tup{(Kontsevich Formality Theorem)}
%%\pause

Let $C := \K[[t_1, \ldots, t_n]]$, the formal power series ring. 
Assume $\mbb{R} \subset \K$. 
%%\pause

Then $\mcal{U}_1$ extends to an 
$\mrm{L}_{\infty}$ quasi-isomorphism
\[ \mcal{U} = \{ \mcal{U}_j \}_{j = 1}^{\infty} : 
\mcal{T}_{\mrm{poly}}(C) \to
\mcal{D}_{\mrm{poly}}(C) . \]
%%\pause

In other words, $\mcal{U}_1$ is a DG Lie algebra 
quasi-isomorphism, up to specified higher homotopies 
$\mcal{U}_2, \mcal{U}_3, \ldots$. 
%%\pause

Each of the maps $\mcal{U}_j$ is invariant under linear change of 
coordinates.
\end{thm}
%%\end{frame}

%%\begin{frame}
There is an induced $\mrm{L}_{\infty}$ quasi-isomorphism
\[ \mcal{U} : \mcal{T}_{\mrm{poly}}(C)[[\hbar]]^+ \to
\mcal{D}_{\mrm{poly}}(C)[[\hbar]]^+ , \]
%%\pause
and a calculation shows that we get a bijection
\[ \opn{MC}(\mcal{U}) : 
\opn{MC} \bigl( \mcal{T}_{\mrm{poly}}(C)[[\hbar]]^+ \bigr) \iso
\opn{MC} \bigl( \mcal{D}_{\mrm{poly}}(C)[[\hbar]]^+ \bigr)  \]
with an explicit formula. Therefore:
%%\pause

\begin{cor}
Assume  $\mbb{R} \subset \K$ and
$C = \K[[t_1, \ldots, t_n]]$. Then there is a canonical 
bijection of sets
\[ \opn{quant} : 
\frac{ \bigl\{ \tup{Poisson brackets on $C[[\hbar]]$}
\bigr\} }
{\tup{gauge equivalence}} \iso
\frac{ \bigl\{ \tup{star products on $C[[\hbar]]$} \bigr\} }
{\tup{gauge equivalence}} \]
preserving first order brackets. 
\end{cor}

%%\hyperlink{toc<8>}{\beamergotobutton{toc}}
%%\end{frame}



\section{The $\mrm{L}_{\infty}$ quasi-isomorphism of the Level of 
Sheaves}
%%\begin{frame}[label=appC]
%\sect{C. The $\mrm{L}_{\infty}$ quasi-isomorphism of the Level of 
%Sheaves}
%%\pause

Here is an outline of the proof of Theorem \ref{thm2}.
We assume $\mbb{R} \subset \K$, and $X$ is a smooth $n$-dimensional
algebraic variety over $\K$.
%%\pause

A formal coordinate system at a closed point 
$x \in X$ is an isomorphism of $\K$-algebras
\[ \bsym{k}(x)[[\bsym{t}]] = 
\bsym{k}(x)[[t_1, \ldots, t_n]] \iso \what{\mcal{O}}_{X, x}
, \]
where $\bsym{k}(x)$ is the residue field.
%%\pause
 
There is an infinite dimensional scheme 
$\opn{Coor} X$, with a projection 
$\pi : \opn{Coor} X \to X$, which is a moduli space for
formal coordinate systems. (In \cite{Ko1} the notation for 
$\opn{Coor} X$ is $X^{\mrm{coor}}$.)
%%\pause

In particular, for every closed 
point $x \in X$, the $\bsym{k}(x)$-rational points 
in the fiber $\pi^{-1}(x)$ stand in bijection to 
the set of formal coordinate systems at $x$.
%%\end{frame}

%%\begin{frame}
To get an idea of how the scheme $\opn{Coor} X$ looks, let us note 
that 
$\opn{Coor} X = \opn{lim}_{\leftarrow} \opn{Coor}^i X$,
where each $\opn{Coor}^i X$ is the variety 
parametrizing formal coordinate systems up to order $i$. 
%%\pause

Any function $f$ on $X$ has a universal Taylor 
expansion, when we pull it up to $\opn{Coor} X$ via $\pi$. 
Thus the pullback of the structure sheaf $\mcal{O}_X$ embeds 
inside the power series algebra 
$\mcal{O}_{\opn{Coor} X}[[\bsym{t}]]$.
%%\pause

Likewise the pullbacks to $\opn{Coor} X$ of the sheaves
$\mcal{T}_{\mrm{poly}, X}$ and
$\mcal{D}_{\mrm{poly}, X}$ are embedded inside
$\mcal{O}_{\opn{Coor} X} \hatotimes{} 
\mcal{T}_{\mrm{poly}}(\K[[\bsym{t}]])$
and
$\mcal{O}_{\opn{Coor} X} \hatotimes{} 
\mcal{D}_{\mrm{poly}}(\K[[\bsym{t}]])$
respectively.
%%\end{frame}

%%\begin{frame}
Due to the Formality Theorem we obtain an $\mrm{L}_{\infty}$ 
quasi-isomorphism
\[ \mcal{U} : \mcal{O}_{\opn{Coor} X} \hatotimes{} 
\mcal{T}_{\mrm{poly}}(\K[[\bsym{t}]]) \to
\mcal{O}_{\opn{Coor} X} \hatotimes{} 
\mcal{D}_{\mrm{poly}}(\K[[\bsym{t}]]) . \]
%%\pause

If we had a section $\sigma : X \to \opn{Coor} X$
then we could pull $\mcal{U}$ down to an $\mrm{L}_{\infty}$ 
quasi-isomorphism on $X$. 
However usually there are no global sections of $\opn{Coor} X$.
%%\end{frame}

%%\begin{frame}
The group $\opn{GL}_{n}$ acts on $\opn{Coor} X$ by linear change 
of coordinates. Let us define $\opn{LCC} X$ to be the quotient scheme
$\opn{Coor} X / \opn{GL}_{n}$. (``LCC'' stands for ``linear 
coordinate classes''.) 
%%\pause

Recall that the universal 
deformation of Kontsevich is invariant under linear change 
of coordinates, namely under the action of the group 
$\opn{GL}_{n}$. This implies that the $\mrm{L}_{\infty}$ 
morphism $\mcal{U}$ descends to $\opn{LCC} X$; 
and hence it suffices to work with sections 
$\sigma : X \to \opn{LCC} X$.
%%\pause

In the $\mrm{C}^{\infty}$ context such global sections 
$\sigma : X \to \opn{LCC} X$ do exists (because the fibers of the 
bundle $\opn{LCC} X$ are contractible). But this is not the case 
in algebraic geometry. So we must use a trick.
%%\end{frame}

%%\begin{frame}
Let $G$ be the group of $\K$-algebra automorphisms of 
$\K[[\bsym{t}]]$. So
$G \cong \opn{GL}_n \ltimes N$,
where $N$ is the subgroup of elements that act trivially modulo 
$(\bsym{t})^2$. The group $N$ is pro-unipotent. 
It turns out that $\opn{Coor} X$ is a $G$-torsor over $X$. 
%%\pause

Suppose we are given a finite number of sections
\[ \sigma_0, \ldots, \sigma_q : U \to \opn{LCC} X \]
over some open set $U$. 
%%\pause

Using an averaging process for unipotent 
group actions \cite{Ye4}, we show that there exists a canonical
morphism 
\[ \bsym{\sigma} : \bsym{\Delta}^q_{\K} \times U  \to \opn{LCC} X \] 
which restricts to $\sigma_j$ on the $j$-th vertex of
$\bsym{\Delta}^q_{\K}$. Here $\bsym{\Delta}^q_{\K}$ is the 
$q$-dimensional geometric simplex.
%%\end{frame}

%%\begin{frame}
Since sections exist locally, we can choose an open covering 
$X = \bigcup U_i$ with sections 
$\sigma_i : U_i \to \opn{LCC} X$. For any
$i_0, \ldots, i_q$ we then obtain a morphism
\[ \bsym{\sigma} : \bsym{\Delta}^q_{\K} \times 
(U_{i_0} \cap \cdots \cap U_{i_q}) \to \opn{LCC} X . \]
(See Figure 1 for an illustration of the case $q = 1$.)

%%\pause
As $q$ varies we have a {\em simplicial section}
of $\opn{LCC} X \to X$. See \cite{Ye2}.
%%\end{frame}

%%\begin{frame}
\begin{figure}
\psset{unit=0.3cm}
\[ %\small
\begin{array}{rcc}
\begin{pspicture}(4,-5)(10,12)
\psset{unit=0.3cm}
%\psgrid(0,0)(0,-7)(12,15)
%
\psellipse[fillstyle=none,linecolor=brtgreen,
linewidth=0.15](7.5,-3)(3.0,1.5)
\rput(9,-3.2){$U_0$} 
\psellipse[fillstyle=none,linecolor=drkblue,
linewidth=0.15](4,-3)(3,1.0) 
\rput(2.3,-3){$U_1$} 
%
\psellipse[fillstyle=none,linecolor=rust,
linewidth=0.15](5.5,-3)(6.0,2.8)
\rput(3,-6.2){$X$} 
%
%\psclip{\psframe[linestyle=none](-1,0)(12,4)}
\psellipse[fillstyle=none,linecolor=rust,
linewidth=0.15](5.5,4)(6.0,2.8)
%\endpsclip
\psellipse[fillstyle=none,linecolor=rust,
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%
\rput(3,12){$\opn{LCC} X$} 
%>>>>
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(-0.93,8.0)(0.07,10.0)(-0.43,12)
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linewidth=0.15]{c-c}(11.43,4.0)(11.96,6.0)
(10.96,8.0)(11.96,10.0)(11.43,12)
%<<<<<
%
\psellipse[fillstyle=none,linecolor=brtgreen,
linewidth=0.15](7.5,9)(3.0,1.5)
\psellipse[fillstyle=none,linecolor=drkblue,
linewidth=0.15](4,5)(3,1.0) 
%
\psline[linewidth=0.08]{->}(1,3)(1,-2)
\rput(0,1){$\pi$} 
\psline[linewidth=0.08,linearc=3]{->}(4,-2.5)(5,1)(4,4)
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\psline[linewidth=0.08,linearc=6]{->}(8,-2.5)(9,3)(8,7.5)
\rput(8,3){$\sigma_0$} 
%
\end{pspicture}
%%%
& \qquad &
%%%
\begin{pspicture}(-1,-5)(12,12) %(-2,-5)(14,12)
\psset{unit=0.3cm}
%\psgrid(0,0)(0,-7)(14,15)
%
\psclip{
\psellipse[linestyle=none](7.5,-3)(3.0,1.5)
\psellipse[linestyle=none](4,-3)(3,1.0) }
\psellipse[fillstyle=none,linecolor=drkblue,
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\psellipse[linecolor=drkblue,linestyle=dashed](4,-3)(3,1.0)
\rput(4.2,-4.7){$U_0 \cap U_1$} 
%
\psellipse[fillstyle=none,linecolor=rust,
linewidth=0.15](5.5,-3)(6.0,2.8)
%
\psellipse[fillstyle=none,linecolor=rust,
linewidth=0.15](5.5,4)(6.0,2.8)
%\endpsclip
\psellipse[fillstyle=none,linecolor=rust,
linewidth=0.15](5.5,12)(6.0,2.8)
%>>>>
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linewidth=0.15]{c-c}(11.43,4.0)(11.96,6.0)
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%<<<<<
%\psline[fillstyle=none,linecolor=red,
%linewidth=0.15](11.43,4.0)(11.43,12)
%
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%>>>
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\psline[linewidth=0.08]{->}(13,5)(7.2,8)
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\end{array} \]
\caption{Simplicial sections, $q = 1$. 
We start with sections over two open sets $U_0$ and $U_1$ 
in the left diagram; and we pass to a simplicial 
section $\bsym{\sigma}$ on the right.} 
\label{fig1}
\end{figure}
%%\end{frame}

%%\begin{frame}
Another device we use is {\em mixed resolutions}. The mixed 
resolution $\opn{Mix}(\mcal{T}_{\mrm{poly}, X})$
is a sheaf of DG Lie algebras on $X$ which is quasi-isomorphic to 
$\mcal{T}_{\mrm{poly}, X}$. Likewise for 
$\mcal{D}_{\mrm{poly}, X}$. 
%%\pause

The simplicial section $\bsym{\sigma}$
allows us to pull down $\mcal{U}$, and after twisting (because of 
the Grothendieck differential occurring in the mixed resolution) 
we obtain an $\mrm{L}_{\infty}$ quasi-isomorphism
\[ \Psi_{\bsym{\sigma}} :
\opn{Mix}(\mcal{T}_{\mrm{poly}, X}) \to
\opn{Mix}(\mcal{D}_{\mrm{poly}, X})  \]
between sheaves of DG Lie algebras on $X$.

%%\hyperlink{toc<8>}{\beamergotobutton{toc}}
%%\end{frame}




\section{Twisted Deformations via Stacks of Gluing Groupoids}
%%\begin{frame}[label=appD]
%\sect{D. Twisted Deformations via Stacks of Gluing Groupoids}
%%\pause

Recall that a \hilite{drkblue}{groupoid} $G$ is a category in which
all 
morphisms are invertible. %%\pause

We denote by $G(i,j)$ the set of arrows from the
object $i$ to the object $j$. %%\pause

Note that $G(i,i)$ is a group. %%\pause

Any element $g \in G(i,j)$ defines a group isomorphism
\[ \opn{Ad}(g) : G(i,i) \iso G(j,j) . \]
%%\end{frame}

%%\begin{frame}
A \hilite{drkblue}{stack of groupoids} $\bsym{\mcal{G}}$ on $X$
is the geometrization of the notion of groupoid, %%%\pause
in the same way that a sheaf of groups is the geometrization of the
notion of a group. %%\pause

Thus for any open set $U \subset X$ there is a groupoid
$\bsym{\mcal{G}}(U)$. %%\pause

And there are restriction functors
\[ \bsym{\mcal{G}}(U) \to \bsym{\mcal{G}}(V) \] 
for any inclusion $V \subset U$. %%\pause

These satisfy a rather complicated list of conditions. 
For details see \cite{Gi, BM, KS}. %%\pause

In particular, 
given any open set $U \subset X$ and any object $i \in \opn{ob}
\bsym{\mcal{G}}(U)$,
there is a sheaf of groups $\bsym{\mcal{G}}(i,i)$ on $U$. 

A stack of groupoids $\bsym{\mcal{G}}$ is called a {\em gerbe} if
it is locally nonempty and locally connected.


%%\begin{frame}
\begin{dfn} \label{dfn2}
%%\pause
Let $X$ be a smooth algebraic variety over $\K$. 
A \hilite{drkblue}{twisted associative \tup{(}resp.\ Poisson\tup{)}
deformation} $\bsym{\mcal{A}}$ of $\mcal{O}_X$ is 
the following data: %%\pause
\begin{enumerate}
\item A gerbe $\bsym{\mcal{G}}$ on $X$, called the
\hilite{rust}{gluing gerbe} of $\bsym{\mcal{A}}$. %%\pause
\item For any open set $U \subset X$ and 
$i \in \opn{ob} \bsym{\mcal{G}}(U)$, an associative 
\tup{(}resp.\ Poisson\tup{)} deformation 
$\mcal{A}_i$ of $\mcal{O}_{U_i}$.
\end{enumerate} %%\pause
The conditions are: %%\pause
\begin{enumerate}
\rmitem{a} For any $i \in \opn{ob} \bsym{\mcal{G}}(U)$, the sheaf of
groups
$\bsym{\mcal{G}}(i,i)$ coincides with 
$\opn{IG}(\mcal{A}_i)$, the sheaf of inner gauge transformations of
the
deformation $\mcal{A}_i$. %%\pause
%
\rmitem{b} For any $i \in \opn{ob} \bsym{\mcal{G}}(U)$, any
$j \in \opn{ob} \bsym{\mcal{G}}(V)$, any $W \subset U \cap V$ and any
$g \in \mcal{G}(W)(i,j)$, the isomorphism of sheaves of groups
\[ \opn{Ad}(g) : \mcal{G}(i,i)|_W \iso \mcal{G}(j,j)|_W \]
%%\pause
is induced from a (necessarily unique) gauge equivalence
\[ \mcal{A}_i|_W \iso \mcal{A}_j|_W . \]
\end{enumerate}
\end{dfn}
%%\end{frame}

%%\begin{frame}
\begin{thm}\tup{(\cite{Ye5})}  \label{thm.last}
\hilite{brtgreen}{
Definitions \tup{\ref{dfn1}} and \tup{\ref{dfn2}} are equivalent.}
\end{thm} %%\pause

\begin{rem}
Let $\bsym{\mcal{A}}$ be a twisted deformation, with gluing groupoid 
$\bsym{\mcal{G}}$. %%\pause

It is important to note that the set 
$\opn{ob} \bsym{\mcal{G}}(X)$ could be empty, meaning that
$\bsym{\mcal{A}}$ is really twisted; i.e.\ it is not equivalent to a
deformation in the usual sense.

This can be detected 
by the non-vanishing of suitable obstruction classes in \linebreak
$\mrm{H}^2(X, \mcal{O}_X)$. 
\end{rem} 

Indeed, Theorem \ref{thm.last} is a consequence of the fact that all
relevant obstructions classes vanish on affine open sets. 

Finally let me say a few words on the proof of Theorem \ref{thm_new}
in \cite{Ye5}.

Fix an affine open covering 
$\bsym{U} = \{ U_0, \ldots, U_m \}$ of $X$, such that for each $i$
there is an \'etale morphism 
$U_i \to \mbf{A}^n_{\K}$. 

Consider the cosimplicial DG Lie algebra
\[ \mfrak{t} := \Gamma \bigl( X, 
\opn{C}(\bsym{U}, \mcal{T}_{\mrm{poly}, X}) \bigr) . \]
Here $\opn{C}(\bsym{U}, -)$ denotes the cosimplicial \v{C}ech
resolution based on $\bsym{U}$.

Likewise there is a cosimplicial DG Lie algebra
\[ \mfrak{d} := \Gamma \bigl( X, 
\opn{C}(\bsym{U}, \mcal{D}^{\mrm{nor}}_{\mrm{poly}, X}) \bigr) . \]

Suppose $\mfrak{g}$ is any cosimplicial DG Lie algebra. 
Let $\ol{\opn{DT}}(\mfrak{g}[[\hbar]]^+)$ be the set of equivalence
classes of {\em descent triples} in the cosimplicial DG Lie algebra
$\mfrak{g}[[\hbar]]^+$. 

Now twisted Poisson deformations of
$\mcal{O}_X$ correspond canonically to elements of
$\ol{\opn{DT}}(\mfrak{t}[[\hbar]]^+)$. And similarly, twisted
associative deformations are parametrized by 
$\ol{\opn{DT}}(\mfrak{d}[[\hbar]]^+)$.

On the other hand, let 
$\tilde{\mrm{N}} \mfrak{g}$ denote the Thom-Sullivan normalization of
a cosimplicial DG Lie algebra $\mfrak{g}$. We can look at the set
$\ol{\opn{MC}}(\tilde{\mrm{N}} \mfrak{g}[[\hbar]]^+)$
of equivalence classes of Maurer-Cartan elements in the DG Lie
algebra $\tilde{\mrm{N}} \mfrak{g}[[\hbar]]^+$. 

It is easy to deduce from Theorem
\ref{thm2} that there is a canonical bijection
\[ \opn{quant} : 
 \ol{\opn{MC}}(\tilde{\mrm{N}} \mfrak{t}[[\hbar]]^+) \iso
\ol{\opn{MC}}(\tilde{\mrm{N}} \mfrak{g}[[\hbar]]^+) . \]

To wrap it all up we prove, by a rather complicated calculation, that
for any cosimplicial DG Lie algebra of {\em quantum type}
$\mfrak{g}$, namely for which $\mfrak{g}^i = 0$ for $i < -1$, there is
a functorial bijection
\[ \ol{\opn{dt}} : 
\ol{\opn{MC}}(\tilde{\mrm{N}} \mfrak{g}[[\hbar]]^+) \iso 
\ol{\opn{DT}}(\mfrak{g}[[\hbar]]^+) . \]



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\end{document}