# Thursday, November 03

:::{.remark}
Today: semistable sheaves on a projective variety $X \subseteq \PP^N$, where $\OO_X(1)$ is the pullback of $\OO_{\PP^N}(1)$.
Let $\mcf\in \Coh(X)$, e.g. a vector bundle (locally free of rank $r$) or a line bundle (vector bundle with $r=1$).
Note $X$ is covered by affine varieties $\spec R_i$ corresponding to rings $R_i$, and on affine varieties,

- quasicoherent sheaves $\ro{\mcf}{\spec R_i}$ correspond to modules $M\in \mods{R_i}$,
- coherent sheaves $\ro{\mcf}{\spec R_i}$ correspond to modules $M\in \mods{R_i}^\fg$ which are finitely generated.

For vector bundles, $M\cong R^r$.
By Serre vanishing,
\[
H^{> 0}(X; \mcf(n)) = 0\,\, n \gg 0, \qquad \dim_k H^0(X; \mcf(n)) < \infty\,\,\forall n
,\]
and Grothendieck vanishing yields $H^{> \dim X}(X; \mcf) = 0$ for any $\mcf\in \QCoh(X)$.
By Hirzebruch-Riemann-Roch, the Hilbert series
\[
h_\mcf(n) \da \chi(\mcf(n)) \da \sum (-1)^i h^i(\mcf(n)) = h^0(\mcf(n)),\,\, n\gg 0
\]
is a polynomial in $n$.
This is proved by writing $Y \da X \intersect H$ to get $\mcf(-1)\injects \mcf\surjects \ro{\mcf}{Y}$ and  $h_\mcf(n) - h_\mcf(n-1) = h_Y(n)$.
Thus it suffices to know $h_Y$ is a polynomial, since the LHS is the discrete derivative of $h_\mcf$, and $\dim Y < \dim X$.
:::

:::{.remark}
We define the **reduced Hilbert polynomial** as $\bar h_\mcf(n) \da h_\mcf(n)/h_n$ where $h_n$ is the leading coefficient of $h_\mcf(n)$.
:::

:::{.example title="?"}
Let $\dim X = 1$ be a smooth curve of genus $g$ and $\mcf\in \Coh(X)$ be locally free of rank $r$
Then Riemann-Roch yields
\[
\chi(\mcf) = \deg(\mcf) +r (1-g) = \deg(\mcf) + \chi(\OO_X\sumpower{r})
,\]
using that $h^0(\OO_X) = 1, h^1(\OO_X) = g \implies \chi(\OO_X) = g-1$.
Twist by $n$ to obtain
\[
\chi(\mcf(n)) = \deg(\mcf) + nr \deg(\mcf) + r(1-g) = h_\mcf(n)
,\]
which yields 
\[
\bar h_\mcf(n) 
&= n + {\deg \mcf \over \rank \mcf}{1\over \deg(\OO_X(1))} + { r(1-g)\over r\deg \OO_X(1) } \\
&\da n + \mu(\mcf) c_1 + c_2
\]
where $\mu(\mcf)$ is the **slope** and the $c_i$ are constants that do not depend on $\mcf$.
:::

:::{.definition title="Hilbert stable"}
A sheaf $\mcf\in \Coh(X)$ is **Hilbert stable** (resp. semistable) iff for any nonzero subsheaf $\mce \leq \mcf$ satisfies

a. $\dim \supp \mce = \dim \supp \mcf$, noting the LHS equals $\bar\deg h_{\mce}$ and the RHS equals $\bar\deg h_\mcf$ where $\supp \mcf\da \ts{x\in X\st \mcf_x\neq 0}$ (which is always closed).
b. $\bar h_{\mce} \leq \bar h_\mcf$, resp. $\bar h_{\mce} \leq \bar h_{\mcf}$, where $f < g$ iff $f(n) < g(n)$ for $n \gg 0$ iff $f<g$ in the lexicographic order.

:::

:::{.remark}
Note that condition (a) is automatic for locally free sheaves, since $\mcf_x = \OO_X\sumpower{r}$ for every $x$ and thus $\supp \mcf = X$.
An example where this won't hold: let $i: p\injects X$ for $X$ a curve and take the skyscraper sheaf $i_* \OO_p$.
More generally, for a closed embedding $i: Z\injects X$ one gets $\supp(i_* \OO_Z) \subseteq Z \subseteq X$.
If $X$ is a smooth curve, then any $\mcf\in \Coh(X)$ decomposes as $\mcf = \mcf_\tors \oplus \mcf_{\free}$ where $\mcf_\tors$ is a torsion sheaf which is a sum of skyscrapers supported at points and $\mcf_\free$ is locally free.
The Hilbert polynomial will be constant on $\mcf_\tors$.
:::

:::{.remark}
Hilbert stability for smooth curves $X$: (a) holds iff $\mcf$ is torsionfree.
If this holds, then since $0\neq \mce \subseteq \mcf$ with $\mcf$ torsionfree, $\mce$ is locally free.
Then condition (b) is equivalently to $\mu(\mce) < \mu(\mcf)$, respectively $\mu(\mce) \leq \mu(\mcf)$.
Thus Hilbert stability for curves is **slope stability**.
:::

:::{.example title="$g=0$"}
For $X=\PP^1$, Grothendieck splitting yields $\mcf = \bigoplus_{i=1}^r \OO(n_i)$ and $\mcf$ is stable iff $r=1$.
Then $\mu(\mcf) = {\deg\mcf\over\rank \mcf} = {\sum_{i=1}^r n_i \over r}$, and this is semistable iff $n_1 = \cdots = n_r$.
E.g. $\mcf = \OO \oplus \OO(1)$ has slope $1/2$ but contains $\OO(1)$ of slope $1$.
:::

:::{.example title="$g=1$"}
Let $X$ be a smooth elliptic curve.
By a theorem of Atiyah, there is a unique indecomposable semistable sheaf $\mcf$ of degree zero.
When $r=1$, one can take $\mcf = \OO_X$.
For $r=2$, $\OO_X \oplus \OO_X$ has slope zero but has subsheaf $\OO_X$ of slope zero, so this is only semistable.
Instead, it is a sheaf fitting into an extension $\OO_X \injects \mcf \surjects \OO_X$, which is semistable since it contains $\OO_X$.
Such extensions $A\to\mcf\to B$ are classified by $H^1(X; B\tensor A\inv)$, which here is $H^1(X; \OO_X)$ which is dimension 1.
Note that $\deg \mcf = \deg \OO_X + \deg \OO_X = 0+0=0$.
:::

:::{.remark}
For $g\geq 2$, there is a moduli space of stable and semistable vector bundles of fixed rank $r$ and degree $d$
This recovers $\Jac$ for $r=1$, and thus is a noncommutative generalization of $\Pic$.
If $d=r(g-1)$ then one can define a theta divisor, and around 20 years ago there was an analog of the Riemann-Roch formula which computed its sections.
:::

:::{.theorem title="The main theorem"}
These moduli spaces exist, and the proof if by GIT by reducing it to an action of $\SL_n$ on a projective variety and applying the Hilbert-Mumford numerical criterion.
:::

:::{.remark}
Two basic notions to discuss: $S\dash$equivalence on semistable sheaves, and the Harder-Narasimhan filtration.
:::

:::{.definition title="The Harder-Narasimhan filtration"}
The Harder-Narasimhan filtration: if $\mcf$ is stable, do nothing, otherwise pick a maximal destabilizing subsheaf $\mcf_1 \subseteq \mcf_0\da \mcf$ (i.e. a subsheaf of largest slope or reduced Hilbert polynomial).
Continue this to obtain a decreasing filtration $0 = \mcf_k \subseteq \cdots \subseteq \mcf_1 \subseteq \mcf_0 = \mcf$.
Define the associated graded sheaf $\gr \mcf \da \bigoplus \mcf_i/\mcf_{i+1}$.
:::

:::{.remark}
The result: the associated graded pieces $\mcf_i/\mcf_{i+1}$ are semistable with increasing slopes. 
E.g. take $0 \subseteq \OO(1) \subseteq \OO \oplus \OO(1)$ where $\mu(\OO(1)) = 1$ and $\mu(\OO)=0$.
If $\mcf$ is semistable, the subsheaves will all have the same slope and the pieces $\mcf_i / \mcf_{i+1}$ are indecomposable.
:::

:::{.definition title="$S\dash$equivalence"}
Say $\mcf \sim_S \mcf'$ iff $\gr \mcf \cong \gr\mcf'$.
:::

:::{.example title="?"}
If $X$ is an elliptic curve with $\mcf' \da \OO_X \oplus \OO_X$, take $\OO_X \injects \mcf_\alpha \surjects \OO_X$ 
corresponding to $\alpha\in H^1(\OO_X) = \CC$.
There is a filtration $0 \subseteq \OO_X \subseteq \mcf$ with graded pieces $\OO_X, \OO_X$.
:::

:::{.example title="?"}
This theory becomes complicated for singular or reducible varieties.
Let $X$ be two intersecting copies of $\PP^1$, so $X = X_1 \union X_2$ with $X_i= \PP^i$:

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Then $d = \deg \OO_X(1) = \deg \OO_{X_1}(1) + \deg \OO_{X_2}(1) = d_1 + d_2$.
Define a multidegree $\vector{d} = (d_1, d_2)$ and multirank $\vector{r} = (r_1, r_2)$.
Condition (a) requires $\not\exists \mce \subseteq \mcf$ with $\supp \mce = \pt$ (0-dimensional support).
We have
\[
0 \to \OO_{X_2}(-p) = \ker f \injects \OO_X \surjectsvia{f} \OO_{X_1}\to 0
,\]
where the kernel consists of functions on $X$ which restrict to zero on $X_1$, and hence vanish at $p$.
These subsheaves are not torsionfree, despite $\OO_X$ being torsionfree.

One can compute

- $p_\mcf = n(r_1 d_1 + r_2 d_2) + \deg \mcf + \text{const}$
- $\mu(\mcf) = {\deg\mcf\over \sum r_i d_i}$, 

which is no longer just degree over rank, and is called the **Seshadri slope** and generalizes slopes to curves with multiple irreducible components.
This is interesting even in the case $\vector{r}(\mcf) = (1,1)$, since semistability is now nontrivial (whereas previously we used that line bundles have no sub-bundles).
For simple singularities like nodes, there are numerical conditions on multidegrees to guarantee (semi)stability.
Note that there are infinitely many degree zero sheaves, since $\deg \mcf = \deg \ro\mcf{X_1} + \deg\ro\mcf{X_2}$ which can be taken to be $n$ and $-n$ -- however, there are only finitely many (semi)stable multidegrees.
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:::{.remark}
Coming up: setting up the GIT problem matches up the notions of (semi)stability, and $S\dash$equivalence becomes orbit-closure equivalence.
:::