# Friday, April 29

## Applications of derived categories 

:::{.remark}
Some major work in this area:

- Beilinson-Gelfand-Gelfand (BGG)
- Bendel-Kapranov
- Mukai
- Bendal-Orlov
- Orlov
- Kutznatsov
- Kontsevich, Fukaya (homological mirror symmetry)
- Beilinson-Bernstein-Gabber-Deligne on perverse sheaves
- Bridgeland

:::

:::{.remark}
Results:

- BGG: $\Db(\Coh \PP^n) \cong \Db(\rmod)$ for $R$ a certain ring.
- BK, K: same for quadrics and grassmannians.

Recall that given $\cat T\in\triang\Cat$ with an exceptional collection $\ts{\mce_i}$, they generate a triangulated subcategory $\gens{\mce_i} \leq \cat T$.
It turns out that $\gens{\mce_i}\cong \rmod$ for $R = \bigoplus \Endo \mce_i$.
Beilinson produces a collection $\ts{\OO, \Omega^1,\cdots, \Omega^{n-1}}$, but an easier alternative is $\ts{\OO, \OO(1), \cdots, \OO(n-1)}$.
If the collection is full, then $\cat T \cong \gens{\mce_i}$.
As an alternative to $R$, one can take the corresponding quiver: make a directed graph $\mce_1\to\mce_2\to\cdots$ where each node has $\Endo \mce_i$ attached and each edge $\mce_i\to\mce_j$ is assigned $\oplus_n \Hom(\mce_i, \mce_j[n])$.
So the derived category corresponds to representations of this quiver.

Example: for $\PP^1$, one obtains the following quiver:

\begin{tikzcd}
	\CC && \CC \\
	\OO && {\OO(1)}
	\arrow["{\CC\oplus \CC}"', shift right=3, from=2-1, to=2-3]
	\arrow[shift left=3, from=2-1, to=2-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwxLCJcXE9PIl0sWzIsMSwiXFxPTygxKSJdLFswLDAsIlxcQ0MiXSxbMiwwLCJcXENDIl0sWzAsMSwiXFxDQ1xcb3BsdXMgXFxDQyIsMix7Im9mZnNldCI6M31dLFswLDEsIiIsMix7Im9mZnNldCI6LTN9XV0=)

:::

:::{.proposition title="?"}
If $X\in\Alg\Var\slice k$ admits a full exceptional collection, then the following also admit a full exceptional collection:

- Any $\PP^n\dash$bundle $Y = \PP(V) \to X$, and
- Any blowup $Y = \Bl_Z X$ for $Z$ a smooth subvariety.

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:::{.corollary title="?"}
Any rational smooth projective surface admits a full exceptional collection, by running the MMP.
:::

:::{.conjecture}
Given a smooth surface admitting a full exceptional collection, is it rational?
For a threefold, is it a blowup of something rational?
:::

:::{.definition title="Semiorthogonal decompositions"}
Given $\cat T\in \triang\Cat$ and $A\leq \cat T$ a full triangulated subcategory, one can define two subcategories ${}^\perp A$ and $A^\perp$:
\[
A^\perp = \ts{F \st \Hom(F, A) = 1}
.\]
:::

:::{.remark}
For $\cat C \in \triang\Cat$, one can take $\HoH \cat C$.
For $\cat C = \Dc(X)$, the $\HoH_0 D(X) \cong \ZZ^n \oplus A$ as a group, for $A$ some finite torsion group.
If one has a full exceptional collection, then $A = \HoH_0( \gens{\mce_1, \cdots, \mce_n}^\perp )$.
As a corollary, the length $m$ of an exceptional collection satisfies $m\leq \rank_\ZZ \HoH_0 \Dc(X)$.
:::

:::{.conjecture title="Kaznutsov"}
If $\ts{\mce_1,\cdots, \mce_n}$ is an exceptional collection and $n = \rank_\ZZ \HoH_0 \Dc(X)$, then this is a *full* exceptional collection.
:::

:::{.remark}
For surfaces of general type, a special Godeaux surface produces a counterexample.
There is a much easier counterexample coming from a Burnist (?) surface -- generally fake $\PP^2$, fake Fanos, etc.
See A-Orlov, Orlov-Gorheaise, Katgerov-?, ??
:::

:::{.remark}
Phantoms: categories with zero $\HoH$, so no full exceptional collections.
:::

## Well-known classical results

:::{.theorem title="Bondal-Orlov (very important!)"}
If $X\in \smooth\proj\Var$ where either $K_X$ or $-K_X$ is ample, then $X$ can be recovered from $\Dc(X)$.
:::

:::{.remark}
Having $-K_X$ ample yields **Fano** varieties, and $K_X$ ample yields **general-type** surfaces.
:::

:::{.theorem title="Mukai"}
If $A \in \Ab\Var$, then $\Dc(A) \cong \Dc(A\dual)$.
Such pairs are referred to as **Mukai partners**.
:::

:::{.remark}
How to construct the equivalence $\Dc(A) \to \Dc(A\dual)$: take the Fourier-Mukai transform.
Use the Poincare bundle $P_A \to A\times A\dual$, and construct the functor as a push-pull over the span $(A \from_{p_1} A\times A\dual \to_{p_2} A\dual)$, so 
\[
\mcf \mapsto (p_2)_* \qty{ (p_1)^* \mcf \tensor P_A) }
.\]
:::

:::{.remark}
The next in line: K3 surfaces. 
An easy example: take Kummer surfaces, so $A\to A/\pm 1$ and then blow up the 16 nodes.
:::