# Cohomology of Certain Line Bundles $Z_w$ (Monday, October 04)

:::{.remark}
Some references:

- Fulton, *Intersection Theory*.
  Similar difficulty to Hartshorne if you're going through it yourself!

  - See Young Tableaux books.

- Eisenbud-Harris, *3264 and All That*.
  A more Vakil-style approach.

:::

:::{.definition title="Chow Group"}
The **Chow group** of $X\in\Var\slice k$ is the quotient $A_*(X) \da Z(X)/ \Rat(X)$, where $Z(X) = \ZZ[\Sub(X)]$, the free \(\ZZ\dash\)module on subvarieties of $X$.
The group $Z(X)$ are **algebraic cycle**, and we mod out by rational equivalence.
:::

:::{.example}
If $G\actson X$, then $Y\sim gY\in A_*(X)$, and something similar happens for many algebraic group actions.
Another example is that in $\PP^1$, $x \sim x'$ for all points $x,x'$ since $\PSL_2\actson \PP^1$.
:::

:::{.remark}
Note that there is also an equivariant Chow group/ring.
In general, $A_*(X)$ is difficult/impossible to compute (according to Harris) unless there is an affine stratification.
In these cases, it coincides with Borel-Moore homology.
:::

:::{.theorem title="?"}
If $X$ is smooth, then $A^*(X)$ forms a ring, where the grading is given by codimension of subvarieties.
Thus there is a multiplication $[A] \cdot [B] = [A \intersect B]$ when $A\transverse B$ generically.
Here transversality refers to an open condition on tangent spaces.
:::

:::{.remark}
We have three ways of thinking about line bundles:

- Local trivializations
- Algebraic morphisms with 1-dimensional fibers
- Invertible sheaves

Now we'll add a fourth in terms of divisors.
Define:

- $A_{n-1}(X)\in \Grp$, **Weil divisors**
- $\Pic(X) \in \Grp$, the group of isomorphism classes of algebraic line bundles on $X$ where $[L_1] \cdot [L_2] \da [L_1 \oplus L_2]$.

:::

:::{.proposition title="?"}
Taking the Chern class yields a group morphism $c_1: \Pic(X) \to A_{n-1}(X)$.
If the line bundle is generated by global sections, take the zero section of the global section.
If $X$ is smooth, $c_1$ is an isomorphism, and we write $c_1(\OO_X(Y)) \da [Y]\in A_{n-1}(X)$.
Note that this is slightly different to the ideal sheaf definition in Vakil.
:::

:::{.remark}
See relation to Schubert varieties and Grassmannians in the referenced books.
Bott-Samelson-Demazure and flag varieties will be smooth, although we'll have to be careful for Schubert varieties.
:::

:::{.proposition title="8.1.2"}
Define the length of a word $w\in W$ to be the number of simple reflections, regardless of whether or not $w$ is reduced.
Let $n\da \ell(w)$, then there is a formula for the canonical bundle $K_{Z_w}$ of any Bott-Samelson-Demazure variety $Z_w$ (even Kac-Moody types):
\[
\mcl_w(-\rho) \tensor \OO_{Z_w}( - \sum_{q=1}^n Z_{w(q)} )
.\]

:::

:::{.remark}
Here $\rho\in \lieh\dual_\ZZ$ (e.g. characters of the torus in the semisimple simply connected case) is any element satisfying $\rho( \alpha_i\dual ) = 1$ for all $1\leq i \leq \ell$.
Recall that
\[
Z_w = P_{i_1}\mix{B} \cdots \mix{B} P_{i_n} / B = \ts{ \tv{p_1, \cdots, p_n B/B }}
,\]
and $Z_w(q)$ means deleting the $q$th factor, so $Z_w(q) = \ts{\tv{p_1,\cdots, 1,\cdots, p_nB/B}}$ has the $q$th coordinate set to 1.
Note that there is a quotient map $Z_w \to Z_{w(n)}$, which has a section, and we can use this to induct.
:::

:::{.proof title="?"}
Consider $G$ connected and reductive and let $X=G/B$ be the flag variety, which is smooth.
Then for $\lambda \in X(T)$ corresponds to the algebraic line bundle $\mcl^{\emptyset}( \lambda) = G\mix{B} \CC_{ - \lambda}$.
This yields a function $X(T) \to \Pic(X) \mapsvia{c_1} A_{n-1}(X)$ given by forgetting the $G\dash$action.
This is a group morphism, where adding characters maps to tensoring bundles.

Note that for $T = \CC\units$, we have 
\[
X(T) = \Hom_{\Alg\Grp}(\CC\units, \CC\units)= \ts{z \mapsto z^k \st k\in \ZZ} \iso_{\Ab\Grp} \ZZ
,\]
where negatives are permitted since $0\not\in \CC\units$.
More generally, $X(T) \iso_{\Ab\Grp} \ZZ^n$ for $n=\rank T$, where $\tv{\elts{t}{n}} \mapsvia{\lambda} \lambda_1^{k_1}\cdots \lambda_n^{k_n}$.
Since we have an affine stratification by Schubert cells, we can write $A_*(X) = \bigoplus_{w\in W} [X_w]$, and in fact $A_k(X) = \bigoplus _{\ell(w) = k} [X_w]$.
Considering the lattice for $W$, there are $\ell$ dimension 1 Schubert cells, and identifying them as CW cells and applying Poincare duality, there are $\ell$ codimension 1 cells:

\begin{tikzcd}
	&& w \\
	\\
	{w_0 s_1} && {w_0 s_j} && {w_0 s_n} \\
	\vdots && \vdots && \vdots \\
	\vdots && \vdots && \vdots \\
	{s_1} & \cdots & {s_j} & \cdots & {s_n} \\
	\\
	&& e
	\arrow[from=3-1, to=1-3]
	\arrow[from=3-3, to=1-3]
	\arrow[from=3-5, to=1-3]
	\arrow[from=8-3, to=6-1]
	\arrow[from=8-3, to=6-3]
	\arrow[from=8-3, to=6-5]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsMTYsWzIsNywiZSJdLFswLDUsInNfMSJdLFsyLDUsInNfaiJdLFs0LDUsInNfbiJdLFsxLDUsIlxcY2RvdHMiXSxbMyw1LCJcXGNkb3RzIl0sWzAsMiwid18wIHNfMSJdLFsyLDIsIndfMCBzX2oiXSxbNCwyLCJ3XzAgc19uIl0sWzIsMCwidyJdLFsyLDMsIlxcdmRvdHMiXSxbMiw0LCJcXHZkb3RzIl0sWzAsNCwiXFx2ZG90cyJdLFswLDMsIlxcdmRvdHMiXSxbNCwzLCJcXHZkb3RzIl0sWzQsNCwiXFx2ZG90cyJdLFs2LDldLFs3LDldLFs4LDldLFswLDFdLFswLDJdLFswLDNdXQ==)

It turns out that the map is given as follows:
\[
\ZZ^n \cong X(T) &\too A_{n-1}(X) \cong \ZZ^\ell \\
\lambda&\mapsto \sum_{i=1}^\ell \inner{\lambda}{ \alpha_i\dual}[X_{w_0 s_i}] && n\geq \ell
.\]

:::

:::{.example title="?"}
For $G = \SL_2, \mcl( \lambda_k) = \OO_{\PP^1}(k)$ and $X(T) \cong \ZZ$.
Recall that $\globsec{\PP^1, \OO_{\PP^1}(k)} = \CC[x, y]_k$ are homogeneous polynomials of degree $k$ when $k\geq 0$, otherwise there are no global sections.
For example, $\CC[x, y]_2 = \gens{x^2, xy, y^2}$ is dimension $3 = 2 + 1$.
All points are rationally equivalent, so we can take the basepoint $B/B$, and so the map will need to track the multiplicity of points.
The composition is given by the following:


\begin{tikzcd}
	{X(T)} && {\Pic(X)} && {A_{n-1}(X)} \\
	\\
	{\mcl(\lambda_k)} && {\OO_{\PP^1}(k)} && {k[B/B]}
	\arrow[from=1-1, to=1-3]
	\arrow[from=1-3, to=1-5]
	\arrow[maps to, from=3-1, to=3-3]
	\arrow[maps to, from=3-3, to=3-5]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNixbMCwwLCJYKFQpIl0sWzIsMCwiXFxQaWMoWCkiXSxbNCwwLCJBX3tuLTF9KFgpIl0sWzIsMiwiXFxPT197XFxQUF4xfShrKSJdLFs0LDIsImtbQi9CXSJdLFswLDIsIlxcbWNsKFxcbGFtYmRhX2spIl0sWzAsMV0sWzEsMl0sWzUsMywiIiwwLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoibWFwcyB0byJ9fX1dLFszLDQsIiIsMCx7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Im1hcHMgdG8ifX19XV0=)

The cotangent bundle of $X$ is given by $G\mix{P} u = T\dual G/P$ where $P = LU$.
The canonical bundle is the top wedge power, and here we get $G\mix{B} \lien = G\mix{B} \CC_2 = \mcl(-2)$, noting that the canonical is equal to the cotangent bundle here, and we've identified which equivariant bundle this is.
:::