# Monday, May 02

## Calabi-Yau Categories

:::{.remark}
Recall that a collection $\mce_i$ is *exceptional* iff $[\mce_j, \mce_i[n]] = 0$ if $n\geq0$ and $j > i$.
If there exists a full exceptional collection, $\derivedcat X \cong \derivedcat(\rmod)$ for some $R$.
Recall that a variety is *Fano* if $-K_X$ is ample.
:::

:::{.question}
Do full exceptional collections exist for Fano $n\dash$folds for $n=3$ or $4$?
:::

:::{.answer}
Typically no.
:::

:::{.remark}
Let 
\[
X_3 \da V(f_3(x_0,\cdots, x_5)) \subseteq \PP^5
,\]
then which $X_3$ are rational?
Note that $K_{X_3} = \OO(-6+3) = \OO(-3)$.
Kuznatsov shows that $H^i(\OO_X) = \CC[0]$ and $\Ext^i(\mcl, \mcl) = H^i(\OO_X)$.
One could look for exceptional collections of line bundles, so $\Ext^i(\mcl_j, \mcl_i) = H^m(\mcl_i \tensor \mcl_j\inv) = 0$ for all $m$.
On $\PP^n$, take $\OO(-k)$ for $1\leq k\leq n$ since $K_{\PP^n} = \OO(-n-1)$.
For $X_3$, there is enough vanishing that $\OO, \OO(1), \OO(2)$ are exceptional (everything below the index 3 from above).
Kuznatsov shows that the "Kuznatsov component" $K = \gens{\OO, \OO(1), \OO(2)}^\perp$ is a **Calabi-Yau category** of dimension 2.
:::

:::{.remark}
If $Y$ is a Calabi-Yau variety of dimension $n$, so $K_Y = 0$, there is a Serre functor
\[
S: \Db(Y) &\to \Db(Y) \\
F &\mapsto F\tensor\omega_Y[n]
.\]
Then (probably) $S = T^n$, a shift by $n$.
A category is a Calabi-Yau category of dimension $n$ iff

- It has a Serre functor
- $S = T^n$

One can also define fractional dimension using $S^q$.
:::

:::{.conjecture}
$X$ is rational iff $K = \Db(Y)$ for $Y\in \mathsf{K3}$.
:::

:::{.remark}
A technique due to Clemens-Griffith for cubic threefolds.
Let $X \subseteq \PP^4$ be a smooth non-nodal curve.
Consider the intermediate Jacobian $J_3(X)$, which is a PPAV for any smooth 3-fold.
Basic operations: blowing up a point $p$ or a curve $C$, since blowing up a surface is the identity.
Blowing up a point: $J_3(\Bl_p X) = J_3(X)$, so it doesn't change.
For a curve, $J_3(\Bl_C X) = J(C) \oplus J_3(X)$.
As a corollary, if $X$ is rational then $J_3(X) = \bigoplus J(C_i)$ for some curves $C_i$.
For non-rationality, show it's not the Jacobian of a curve by considering the theta divisor.
:::

:::{.remark}
For 4-folds $X$, one can now also blow up surfaces.
The intermediate cohomology carries a Hodge structure.
Conjecture: $X$ is rational iff its Hodge structure looks like a K3.
:::

:::{.remark}
Older techniques for checking rationality: see log thresholds, generally birational geometry e.g. due to Manin.
E.g. groups of birational automorphism for quartic 4-folds are small.
See another approach due to Mumford using torsion in cohomology.
:::

## T-Structures and Hearts

:::{.remark}
Note that it's possible for $\cat A,\cat B\in \Ab\Cat$ to satisfy $\derivedcat{\cat A} \iso \derivedcat{\cat B}$.
:::

:::{.example title="?"}
Some examples:

- In the presence of a full exceptional collection $\Db(\Coh X) \iso \Db(\rmod)$.
- Fourier-Mukai: $\Db(\Coh A) \iso \Db(\Coh A\dual)$ for dual AVs.
:::

:::{.example title="Perverse sheaves (BBD)"}
Start with $X\in \Var\slice \CC$ Hausdorff paracompact and constructible sheaves which come with stratifications into closed subsets on which they restrict to locally constant sheaves. 
Note that one can realize these sheaves as pullbacks from a poset associated to the stratification.
There are categories $\Const$ and $\Perv$ with $\Db(\Const) \iso \Db(\Perv)$ -- here perverse sheaves are complexes of constructible sheaves with support conditions $h^j(\cocomplex\mcf) \leq -j$ and $h^j(D\cocomplex\mcf) \leq -j$ for $D$ the Verdier dual; this is a category closed under duality.
:::

:::{.remark}
On $T\dash$structures: write $D = \derivedcat{\cat A}$, then there are subcategories

- $D^{\leq n} = D^{\leq 0}[n]$, complexes such that $H^{> n} = 0$.
- $D^{\geq n} = D^{\geq 0}[n]$, complexes such that $H^{< n} = 0$.

Then $D^{\geq 0} \intersect D^{\leq 0} = A$ is a category equivalent to complexes supported in degree zero, since any such bounded complex is quasi-isomorphic to such a complex.
Some properties:

- $A_0\in D^{<0}$ and $A_1 \in D^{> 1}$ satisfy $[A_0, A_1] = 0$.
- For all $C\in \Db(\cat A)$, there exists $A_0\to C\to A_1 \to A_0[1]$.

Note that there is a canonical truncation
\[
\tau_{\leq 0}(\cdots \to C\inv \mapsvia{d^0} C^0\to C^1\to \cdots) = (\cdots\to C\inv \to \ker d^0 \to 0)
.\]
:::

## Bridgeland stability

:::{.remark}
Take $X$ a smooth projective curve, let $D = \Db(\Coh X) = \Db\VectBundle$.
There is a notion of a semistable sheaf (all subsheaves have smaller slopes $\mu(\mcf) \da \deg \mcf/\rank \mcf$) and an HN filtration where the quotients are semistable and the slopes decrease.
Bridgeland observed there is a central charge 
\[
Z: \Coh X &\to \CC \\
\mcf &\mapsto -\deg \mcf + i \rank \mcf
,\]
which can be used to recovered the heart $\cat A = \Coh X$.
Idea: vary $Z$ to get different hearts, and $\ts{Z_i}$ form a complex analytic variety, and one can form a new category of tilted complexes (complexes sitting in two degrees).
:::