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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}
%
\author[Amnon Yekutieli]{Amnon Yekutieli}
\institute[BGU]
{Department of Mathematics\\
Ben Gurion University\\[2em]
Notes available at 
{\blue http://www.math.bgu.ac.il/$\sim$amyekut/lectures} 
\\[2ex]
written 17 June 2009; corrected 11 Sep 2010}
\date[]{}
%\tiny Written 18 Nov 2008}
%\subject{Talks}
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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: 15 Sept 2007}
%
%\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.] \hyperlink{sec1<1>}{\blue \ul{Some background on Deformation 
Quantization}} \pause
\item[2.] \hyperlink{sec2<1>}{\blue \ul{Poisson Deformations of Algebraic 
Varieties}} \pause
\item[3.] \hyperlink{sec3<1>}{\blue \ul{Associative Deformations of 
Algebraic Varieties}} \pause
\item[4.] \hyperlink{sec4<1>}{\blue \ul{Deformation Quantization}} \pause
\item[5.] \hyperlink{sec5<1>}{\blue \ul{Twisted Deformations of Algebraic 
Varieties}} \pause
\item[6.] \hyperlink{sec6<1>}{\blue \ul{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{1. 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{2. 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}

%\hyperlink{sec4<5>}{\beamergotobutton{back}}
\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.

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




\section{3. 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.

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





\section{4. 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-isomorphism. 
\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.
\pause 

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

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

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

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



\section{5. 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 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 
\[ \{ g \in A \mid g \equiv 1 \opn{mod} \hbar \} . \]
\pause

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{frame}
%\begin{figure}
\mbox{ }

\only<1>{\includegraphics[scale=0.4]{fig1a.eps}}
\pause
\only<2>{\includegraphics[scale=0.4]{fig1b.eps}}
\pause
\only<3>{\includegraphics[scale=0.4]{fig1c.eps}}
\pause
\only<4>{\includegraphics[scale=0.4]{fig1d.eps}}
%\label{fig1}
%\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}

\begin{frame}
%\begin{figure}
\includegraphics[scale=0.4]{fig1e.eps}
\end{frame}



\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}.
\pause 

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

\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}

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



\section{6. 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{Ye6})}  \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{Ye6})}  \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}

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



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\end{thebibliography}

\end{document}