# Line Bundles on $\mcx^Y$ (Monday, September 27)

:::{.remark}
Notation: $\mcx$ will denote a Kac-Moody flag variety, and $X$ a usual flag variety.
For any $\lambda \in D_Y^0$, define the algebraic line bundle $\mcl(-\lambda) \to \mcx^Y$ to be the pullback of the tautological bundle on $\PP(L^{\max}(\lambda))$ via the morphism $\iota_\lambda: \mcx^Y \to \PP(L^{\max} (\lambda))$.
Recall that we defined $Y\dash$regular weights to get an embedding into a flag variety.

Let $X$ be a finite dimensional variety, then a vector bundle on $X$ is a map $\mce \mapsvia{\pi} X$ with each fiber a $\CC\dash$module and for all $x\in X$ there exists an open $U \subseteq X$ and a homeomorphism $\phi: U \times \CC^n \to \pi\inv(U)$ over $U$, so the following diagram commutes:


\begin{tikzcd}
	{\pi\inv(U)} && {U\times \CC^n} \\
	\\
	& U
	\arrow["\pi"', from=1-1, to=3-2]
	\arrow["{\pr_1}", from=1-3, to=3-2]
	\arrow["\phi", from=1-1, to=1-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsMyxbMCwwLCJcXHBpXFxpbnYoVSkiXSxbMiwwLCJVXFx0aW1lcyBcXENDXm4iXSxbMSwyLCJVIl0sWzAsMiwiXFxwaSIsMl0sWzEsMiwiXFxwcl8xIl0sWzAsMSwiXFxwaGkiXV0=)

We refer to $\phi$ as a trivialization.
Writing $U_{12} \da U_1 \intersect U_2$, given trivializations over $U_i$ we require that the trivializations on $U_{12}$ are related by an element $T_{12} \in \GL_n$, and the induced map $U_{12} \times \CC^n\selfmap$ are essentially given by matrices with entries given by functions on $U_{12}$
The key is that these satisfy the cocycle condition:
\[
T_{kj}\mid_{U_{ijjk}} T_{ji} \mid_{U_{ijk}} = T_{ki}\mid_{U_{ijk}}
.\]

Given a vector bundle, set $\mcf$ to be the sheaf of sections of $\pi: \mce\to X$.
If for example $U \subseteq X$ is trivializable, then $\globsec{U, \mcf}$ are $n\dash$tuples of functions $U\to \CC$, so $\ro{\mcf}{U}\cong \OO_U\sumpower{n}$, making it locally free.

:::

:::{.proposition title="about locally free sheaves"}
Given a vector bundle, set $\mcf$ to be the sheaf of sections of $\pi: \mce\to X$.
Then

1. If $\mcf$ is locally free, then $\Hom_{\Sh\slice X}(\mcf, \OO_X) \in \Sh\slice X$ is locally free.
2. If $n=1$, then $\mcf \tensor \mcf\dual \cong \OO_X$, making it an invertible sheaf under the monoidal tensor product.
3. Pullbacks of locally free sheaves are again locally free:

\begin{tikzcd}
Z\fiberprod{X} \mce 
  \ar[r] 
  \ar[d]
& 
\mce
  \ar[d] 
\\
Z
  \ar[r]
& 
X 
\end{tikzcd}

where we equivalently write $f^* \mcf$.
:::

:::{.remark}
How to think about a flag variety:
given $w\in W_Y'$ and $U_W \subseteq X^Y$, so $U^- \subseteq G/P$.
Then $\ts{U_w}_{w\in W_Y'} \covers X^Y$ is an open cover with $U_w \cong \CC^{\ell(w_0')}$ with $w_0$ the longest element and $w_0'$ is a minimal coset representative.
If $v\in U_w \iff v=w$ for any $T\dash$fixed point $v$, so there's only one such fixed point in every open.
We have elements $wP/p \in X^Y$, so


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:::

:::{.example title="?"}
For $G \da \SL_{n+1}$, we have $Y = \ts{2,\cdots, n}, W = S_{n+1} = \ts{(w_0 \cdots w_n)}$ and the minimal length representatives have increasing coordinates, so we get
\[
W_Y' = \ts{(0 \mid 1,2\cdots, n), (1 \mid 0, 2, \cdots, n), \cdots , (n\mid 0,1\cdots, n-1)}
.\]
For every $i\in W_Y' = \ts{0, \cdots, n}$, we have $U_i \subseteq X^Y \subseteq G/P_Y$. 
We can obtain $\PP^n \cong \leftquotient{\CC\units}{\CC^{n+1}}$, which is $G/P^Y = X^Y$ here.
So we can take $U_i \da \ts{ \tv{x_0,\cdots, x_n} \st x_i\neq 0 } \subseteq \CC^n$, which is dimension $n$ since the longest element is $(n \mid 0,1,\cdots, n-1)$.
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:::{.example title="?"}
Let $k\in \ZZ$, we'll define $\OO_{\PP^n}(k)$, a line bundle on $\PP^n$.
Taking $n=1$ to get $\SL_2$ and $\PP^1$ above, we have $W_Y' = \ts{0, 1}$ and $\CC = U_1 = \spec \CC [x_{0/1}]$ and $U_0 = \spec \CC[x_{0/1}]$, then on their intersection we have $x_{0/1} = x_{1/0}\inv$.
So transitioning $U_0\to U_1$ is given by $x_{0/1}^k = x_{1/0}^{-k}$, and $U_1\to U_0$ by $x_{1/0}^k = x_{0/1}^{-k}$, which defines a line bundle denoted $\mce \da \OO(k)$.
What are the global sections $\globsec{\PP^1; \OO(k)}$?
This requires $f(x_{0/1}\inv) x_{0/1}^k = g(x_{0/1})$, so the global sections are $\CC[x,y]_{k}$ the homogeneous polynomials of degree $k$.
One can check that $\dim \globsec{\PP^n; \OO(k)} = {n+k \choose k}$.
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:::{.remark}
Next time: we'll try to match these up with line bundles of the form $G \mix{P} \CC_\lambda$.
:::