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\section*{Bondal-Orlov reconstruction theorem}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{higher_geometry}{}\paragraph*{{Higher geometry}}\label{higher_geometry}

[[!include higher geometry - contents]]

\hypertarget{bondalorlov_reconstruction_theorem}{}\section*{{Bondal-Orlov reconstruction theorem}}\label{bondalorlov_reconstruction_theorem}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{statement}{Statement}\dotfill \pageref*{statement} \linebreak
\noindent\hyperlink{sketch_of_proof}{Sketch of proof}\dotfill \pageref*{sketch_of_proof} \linebreak
\noindent\hyperlink{see_also}{See also}\dotfill \pageref*{see_also} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Thomason and Balmer showed that the [[triangulated categories of sheaves|derived category of coherent sheaves]] on a [[smooth variety]], when considered as a [[monoidal category]] (i.e. with the tensor product) in addition to its [[triangulated category]] structure (i.e. as a [[tensor triangulated category]]), completely determines the variety uniquely; see [[spectrum of a triangulated category]]. However, the derived category still turns out to be an interesting invariant when considered without the monoidal category structure; moreover triangulated equivalences and autoequivalences are also important in relation to the [[homological mirror symmetry]] and similar phenomena. In fact, Bondal and Orlov show how to reconstruct a smooth [[variety]] from its [[triangulated categories of sheaves|derived category of coherent sheaves]] when its [[canonical sheaf]] is ample or anti-ample, using only the graded structure (i.e. the translation functor).

\hypertarget{statement}{}\subsection*{{Statement}}\label{statement}

\begin{theorem}
\label{}\hypertarget{}{}
\end{theorem}
\hypertarget{sketch_of_proof}{}\subsection*{{Sketch of proof}}\label{sketch_of_proof}

The idea of the proof is to characterize ``cohomologically'' the closed points, invertible sheaves, and Zariski topology.

The main tool we use is the [[Serre functor]]. Due to [[Grothendieck-Serre duality]], both $D^b(X)$ and $D^b(Y)$ have Serre functors $S_X = (\cdot \otimes \omega_X)[n]$ and $S_Y = (\cdot \otimes \omega_Y)[m]$, $n$ and $m$ being the dimensions of $X$ and $Y$ respectively.

For simplicity assume $\omega_X$ is ample; the anti-ample case is analogous.

\textbf{Step 1.} First we characterize the closed points of $X$ and $Y$.

\begin{defn}
\label{}\hypertarget{}{}
Let $\mathcal{C}$ be a [[linear category|k-linear]] [[graded category]]. A \textbf{point object} of $\mathcal{C}$ is an object $P$ satisfying

\begin{itemize}%
\item (PO-1) $S(P) \simeq P[n]$,
\item (PO-2) $\Hom^i(P, P) = 0$ for $i \lt 0$,
\item (PO-3) $k(P) := \Hom(P, P)$ is a field.

\end{itemize}
\end{defn}
It is clear that the [[skyscraper sheaf]] $\mathcal{O}_x = \mathrm{Sky}_x(k(x))$ of the [[residue field]] at any closed point $x \in X$ is a point object of $D^b(X)$, as is any translation $\mathcal{O}_x[r]$. In fact, when we impose the ampleness condition on $\omega_X$, it turns out that all point objects are of this form.

\begin{prop}
\label{}\hypertarget{}{}
If $\omega_X$ is ample, then a complex $\mathcal{F}^\bullet$ in $D^b(X)$ is a point object iff $\mathcal{F}^\bullet \simeq \mathcal{O}_x[r]$ for some closed point $x \in X$, $r \in \mathbb{Z}$.

\end{prop}
In fact, even though we don't know that $\omega_Y$ is also ample, we get the same result for $D^b(Y)$. Let $G$ be the inverse equivalence to $F$ and note that $F$ and $G$ preserve point objects. Suppose there was a point object $P$ in $D^b(Y)$ not isomorphic to any $\mathcal{O}_y[s]$, and note $G(P) \simeq \mathcal{O}_{x_0}[r]$ for some closed $x_0$. Now for any $y \in Y$, $\Hom(P, \mathcal{O}_y) = \Hom(G(P), G(\mathcal{O}_y)) = \Hom(\mathcal{O}_{x_0}[r], \mathcal{O}_x[r']) = 0$ since $x_0 \ne x$. But $\{\mathcal{O}_y : y \in Y\}$ form a [[spanning class]] of $D^b(Y)$ which implies that $P = 0$.

\textbf{Step 2.} Next we characterize the invertible sheaves on both varieties.

\begin{defn}
\label{}\hypertarget{}{}
Let $\mathcal{C}$ be a $k$-linear graded category. An \textbf{invertible object} of $\mathcal{C}$ is an object $L$ satisfying for some $s \in \mathbb{Z}$

\begin{itemize}%
\item (IO-1) $\Hom^s(L, P) = k(P)$ for all point objects $P$,
\item (IO-2) $\Hom^i(L, P) = 0$ for $i \ne s$.

\end{itemize}
\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
If all point objects of $D^b(X)$, for a smooth projective variety $X$, are translations of skyscraper sheaves of closed points, then all invertible objects of $D^b(X)$ are translations of invertible sheaves on $X$.

\end{prop}
This means that the invertible objects of both $D^b(X)$ and $D^b(Y)$ are translations of invertible sheaves, by step 1.

\textbf{Step 3.} Now we establish a bijection between the sets of points of the varieties.

Since $\mathcal{O}_X$ is an invertible object, we can assume without loss of generality that $F(\mathcal{O}_X)$ is isomorphic to some invertible sheaf $\mathcal{L}_0$ on $Y$. For any $x \in X$,

\begin{displaymath}
k(x) = \Hom(\mathcal{O}_X, \mathcal{O}_x) = \Hom(\mathcal{L}_0, \mathcal{O}_y[s])
\end{displaymath}
for some $y \in Y$, which implies $s=0$ and $F(\mathcal{O}_x) \simeq \mathcal{O}_y$, $k(x) \simeq k(y)$. We get a bijection of sets $\phi : X \to Y$ by mapping $x \mapsto y$.

\textbf{Step 4.} Now we can get an isomorphism of the [[canonical ring]]s of the varieties.

Assume WLOG $F(\mathcal{O}_X) \simeq \mathcal{O}_Y$. Note that $S_X^k(\mathcal{O}_X)[-nk] \simeq \omega_X^{nk}$ for any $k$. By commutativity of $F$ with the Serre functor, $F(\omega_X^{nk}) = S_Y^k(\mathcal{O}_Y)[-nk] = \omega_Y^{nk}$. Therefore

\begin{displaymath}
H^0(X, \omega_X^{nk}) = \Hom(\mathcal{O}_X, \omega_X^{nk}) = \Hom(\mathcal{O}_Y, \otimes_Y^{nk}) = H^0(Y, \omega_Y^{nk})
\end{displaymath}
and $\bigoplus_k H^0(X, \omega_X^{nk}) \simeq \bigoplus_k H^0(Y, \omega_Y^{nk})$, that is we have an isomorphism of the canonical rings $A(X)$ and $A(Y)$.

\textbf{Step 5.} Since $X = \mathrm{Proj}(A(X))$ iff $\omega_X$ is ample, it remains therefore only to show that $\omega_Y$ is ample. To do this we characterize the Zariski topology ``cohomologically''.

For any invertible sheaf $\mathcal{L}$ on $X$, $\alpha \in \Hom(\mathcal{O}_X, \mathcal{L})$, and $x \in X$, let $\alpha_X^* : \Hom(\mathcal{L}, \mathcal{O}_x) \to \Hom(\mathcal{O}_X, \mathcal{O}_x)$ be the induced map that takes $f$ to $f \circ \alpha$, and define the subset $U_\alpha = \{ x \in X : \alpha_x^* \ne 0 \} \subset X$. Now $\omega_X$ being ample is equivalent to the subcollection $\{ U_\alpha : \alpha \in \Hom(\mathcal{O}_X, \omega_X^m), m \in \mathbb{Z} \}$ forming a basis of the Zariski topology on $X$. Now the equivalence $F$ maps any $\mathcal{O}_X \stackrel{\alpha}{\to} \mathcal{L} \to \mathcal{O}_x$ to

\begin{displaymath}
\mathcal{O}_Y \stackrel{F(\alpha)}{\to} F(\mathcal{L}) \to \mathcal{O}_{\phi(x)}[s].
\end{displaymath}
$F(\alpha)$ corresponds bijectively to some $\beta \in \Hom(\mathcal{O}_Y, \omega_Y^m)$ and in fact our bijection $f : X \to Y$ becomes a homeomorphism mapping $U_\alpha$ to $V_{F(\alpha)}$. It follows that $\{ V_\beta : \beta \in \Hom(\mathcal{O}_Y, \omega_Y^m), m \in \mathbb{Z} \}$ forms a basis of the Zariski topology on $Y$, which implies $\omega_Y$ is ample.

\hypertarget{see_also}{}\subsection*{{See also}}\label{see_also}

\begin{itemize}%
\item [[triangulated categories of sheaves]]
\item [[2-ring]]
\item [[spectrum of a triangulated category]]
\item [[spectrum of an abelian category]]
\item [[Tannaka duality for geometric stacks]]
\item [[higher geometry]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

The original paper:

\begin{itemize}%
\item [[Aleksei Bondal]], [[Dmitri Orlov]], \emph{Reconstruction of a variety from the derived category and groups of autoequivalences}, Compositio Mathematica \textbf{125} (03), 327-344 \href{http://dx.doi.org/10.1023/A:1002470302976}{doi}, \href{http://www.mi.ras.ru/~orlov/papers/Compositio2001.pdf}{pdf}\emph{\href{http://www.mi.ras.ru/~orlov/papers/Compositio2001.pdf}{Reconstruction of a variety from the derived category and groups of autoequivalences}}, Compositio Mathematica 125 (03), 327-344.

\end{itemize}
A nice exposition of the proof with more details can be found in section 3.1 of the course notes of [[Igor Dolgachev]] on [[derived categories]]:

\begin{itemize}%
\item [[Igor Dolgachev]], \href{http://www.math.lsa.umich.edu/~idolga/derived9.pdf}{\emph{Derived categories}}.

\end{itemize}
The theorem is also discussed in:

\begin{itemize}%
\item [[Raphaël Rouquier]], \emph{\href{http://arxiv.org/abs/math/0503548}{Cat\'e{}gories d\'e{}riv\'e{}es et g\'e{}om\'e{}trie birationnelle}}, S\'e{}minaire Bourbaki, no 947, March 2005.


\item [[Daniel Huybrechts]], \emph{Fourier-Mukai transforms in algebraic geometry}, 2006.



\end{itemize}


\end{document}