# Friday, April 08

## Vanishing theorems

:::{.remark}
Setup: $X\in \Proj\Var\slice k, \mcf \in \Sh(X; \Ab\Grp)$.
What is $H^0(X; \mcf)$?
Note that if 
\[
\chi(X; \mcf) \da \sum_k (-1)^k h^k(X; \mcf)
,\]
if $\tau_{\geq 1}\cocomplex{H}(X;\mcf) = 0$ then this $\chi(X;\mcf) = h^0(X; \mcf)$.
By Serre duality, $h^n(X;\mcf) = h^0(\omega_X \tensor \mcf\inv)$ which holds if $X$ is Gorenstein, e.g. a locally complete intersection.

Recall that $\OO_X(D)(U) = \ts{\phi \st (\phi) + D \geq 0}$. 
Note that if $\mcf = \OO(D)$ then $h^0(X;\mcf)\neq 0 \iff D\sim D'$ where $D' > 0$ is effective.
:::

:::{.remark}
If $D\sim D'$ where $-D' > 0$ is effective, then $h^0(X; \OO(D)) = 0$.
Note that if $D \subseteq X \subseteq \PP^N$ is projective, take $H \subseteq \PP^N$ and $\OO_{\PP^N}(1) = \OO_{\PP^N}(H)$ and intersect to obtain $D \cdot H^{n-1} = \deg D$.
:::

:::{.example title="?"}
If $X$ is a smooth projective curve and $\mcf = \OO_X(D)$ is a line bundle.
Riemann-Roch yields \[
h^0(X;\mcf) - h^1(X;\mcf) = \deg D -g + 1
\] 
and \[
\deg D = h^0(D) - h^0(K_X - D)
\implies \deg(K_X - D) = 2g-2 - \deg D
.\]
:::

:::{.example title="?"}
If $X$ is a smooth projective curve,

- $\OO_X(D)$ is ample $\iff D > 0$ (some large multiple is a hyperplane section).
- $\OO_X(D)$ is very ample $\impliedby \deg D \geq 2g-2+3$ (very ample: some multiple is ample).

There exists an embedding $X\injects \PP^N$, and $\OO_X(D) = \OO_X(1) = \OO_{\PP^N}(1)\mid_X$.
One can show $h^0(D - \pt) < h^0(D)$.
:::

:::{.example title="?"}
An effective but not ample divisor: take two lines in $\PP^1\times \PP^1$ which do not intersect.
:::

:::{.theorem title="Kodaira"}
Suppose $X\in\smooth\Proj\Var\slice k$ where $k=\CC$ or $\characteristic k = 0$ with $k=\kbar$ and let $\mcf = \omega_X(L)$ with $L$ ample.
Then \[
\tau_{\geq 1} \cocomplex{h}(X;\mcf) = 0
.\]
:::

:::{.remark}
A note on the proof: uses Deligne-Illusie and liftability from Witt vectors.
This liftability holds for all curves, all K3s, and some Calabi-Yau threefolds.
:::

:::{.remark}
For curves, $h^1(X; \omega_X(L)) = h^0(-L)$.
:::

:::{.theorem title="Kawamata-Viehweg vanishing (generalized Kodaira vanishing)"}
Let $X \in \smooth\Proj\Var\slice \CC$ with $D = \union_k D_K$ normal crossing union of smooth divisors and write its formal boundary as $\Delta \da \sum a_i D_i$ with $0 < a_i < 1$ and $a_i \in \QQ$.
Suppose $\mcf \equiv K_X + \Delta + A$ for $A$ ample, then 
\[
\tau_{\geq 1}\cocomplex{h}(X; \mcf) = 0
.\]
:::

:::{.remark}
Say $X$ has **klt singularities** (Kawamata log terminal) iff there exists a projective morphism $Y \mapsvia{f} X$ with $Y\contains \union_i D_i$ with each $D_i$ snc, and $f^* K_X = K_Y + \Delta$.
Generally $Y$ is smooth and $f$ is a resolution.
:::

:::{.remark}
A note on the MMP: take $X_0$ a variety, produce a variety $X$ with $K_X$ nice, e.g. $-K_X > 0$ or $K_X \geq 0$ numerically.
At each stage, contract a curve (the result is a $-1$ curve) are perform a **flip**.
So if $C \in X_0$, produce $X_0 \to X_1$ with $CK_X < 0$.
:::