# Wednesday, January 27

## Bundles and Connections

:::{.definition title="Connections"}
Let \( \mathcal{E}\to X  \) be a vector bundle, then a **connection** on \( \mathcal{E}  \) is a map of sheaves of abelian groups
\[
\nabla: \mathcal{E}\to \mathcal{E}\tensor \Omega^1_X  
\]
satisfying the *Leibniz rule*: 
\[ 
\nabla (fs) = f \nabla s + s\tensor ds 
\] 
for all opens $U$ with $f\in \OO(U)$ and $s\in \mathcal{E}(U)$. 
Note that this works in the category of complex manifolds, in which case $\nabla$ is referred to as a **holomorphic connection**.
:::

:::{.remark}
A connection $\nabla$ induces a map 
\[
\tilde{\nabla}: \mathcal{E}\tensor \Omega^p &\to \mathcal{E}\tensor \Omega^{p+1} \\
s \otimes \omega &\mapsto \nabla s \wedge w + s\tensor d \omega
.\]
where $\wedge: \Omega^p \tensor \Omega^1 \to \Omega^{p+1}$.
The standard example is
\[
d: \OO &\to \Omega^1 \\
f &\mapsto df
.\]
where the induced map is the usual de Rham differential.
:::

:::{.exercise title="?"}
Prove that the *curvature* of $\nabla$, i.e. the map
\[
F_{\nabla} \da \nabla \circ \nabla: \mathcal{E}\to \mathcal{E}\tensor \Omega^2  
\]
is $\OO\dash$linear, so $F_{\nabla}(fs) = f\nabla \circ \nabla(s)$.
Use the fact that $\nabla s \in \mathcal{E}\tensor \Omega^1$ and \( \omega \in \Omega^p \) and so \( \nabla s \otimes \omega \in \mathcal{E} \Omega^1 \otimes \Omega^p  \) and thus reassociating the tensor product yields $\nabla s \wedge \omega \in \mathcal{E}\tensor \Omega^{p+1}$. 
:::

:::{.remark}
Why is this called a connection?

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This gives us a way to transport $v\in \mathcal{E}_p$ over a path \( \gamma \) in the base, and $\nabla$ provides a differential equation (a flow equation) to solve that lifts this path.
Solving this is referred to as **parallel transport**.
This works by pairing \( \gamma'(t) \in T_{ \gamma(t) } X \) with \( \Omega^1 \), yielding \( \nabla s = ( \gamma'(t)) = s( \gamma(t)) \) which are sections of \( \gamma \).

Note that taking a different path yields an endpoint in the same fiber but potentially at a different point, and $F_\nabla = 0$ if and only if the parallel transport from $p$ to $q$ depends only on the homotopy class of \( \gamma \).

> Note: this works for any bundle, so can become confusing in Riemannian geometry when all of the bundles taken are tangent bundles!

:::

:::{.example title="A classic example"}
The Levi-Cevita connection $\nabla^{LC}$ on $TX$, which depends on a metric $g$.
Taking $X=S^2$ and $g$ is the round metric, there is nonzero curvature:

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\end{tikzpicture}

In general, every such transport will be rotation by some vector, and the angle is given by the area of the enclosed region.

:::

:::{.definition title="Flat Connection and Flat Sections"}
A connection is **flat** if $F_\nabla = 0$.
A section \( s \in \mathcal{E}(U)  \) is **flat** if it is given by 
\[
L(U) \da \ts{ s\in \mathcal{E}(U) \st \nabla s = 0}
.\]
:::

:::{.exercise title="?"}
Show that if $\nabla$ is flat then $L$ is a *local system*: a sheaf that assigns to any sufficiently small open set a vector space of fixed dimension.
An example is the constant sheaf $\ul{\CC^d}$.
Furthermore $\rk(L) = \rk(\mathcal{E})$. 
:::

:::{.remark}
Given a local system, we can construct a vector bundle whose transition functions are the same as those of the local system, e.g. for vector bundles this is a fixed matrix, and in general these will be constant transition functions.
Equivalently, we can take $L\tensor_\RR \OO$, and $L\tensor 1$ form flat sections of a connection.
:::

## Sheaf Cohomology

:::{.definition title="Čech complex"}
Let \( \mathcal{F}  \) be a sheaf of abelian groups on a topological space $X$, and let \( \mathfrak{U} \da \ts{U_i} \covers X  \) be an open cover of $X$.
Let $U_{i_1, \cdots, i_p} \da U_{i_1} \intersect U_{i_2} \intersect\cdots \intersect U_{i_p}$.
Then the **Čech Complex** is defined as 
\[
C_{\mathfrak{U}}^p(X, \mathcal{F}) \da \prod_{i_1 < \cdots < i_p} \mathcal{F}(U_{i_1, \cdots, i_p})   
\]
with a differential
\[
\bd^p: C_{\mathfrak{U}}^p(X, \mathcal{F}) &\to C_{\mathfrak{U}}^{p+1}(X \mathcal{F}) \\
\sigma &\mapsto (\bd \sigma)_{i_0, \cdots, i_p} \da \prod_j (-1)^j \ro{\sigma_{i_0, \cdots, \hat{i_j}, \cdots, i_p}}{U_{i_0, \cdots, i_p}}
\]
where we've defined this just on one given term in the product, i.e. a $p\dash$fold intersection.

:::

:::{.exercise title="?"}
Check that $\bd^2 = 0$.
:::

:::{.remark}
The Čech cohomology $H^p_{\mathfrak{U}}(X, \mathcal{F})$ with respect to the cover \( \mathfrak{U}  \) is defined as $\ker \bd^p/\im \bd^{p-1}$.
It is a difficult theorem, but we write $H^p(X, \mathcal{F})$ for the Čech cohomology for any sufficiently refined open cover when $X$ is assumed paracompact.
:::

:::{.example title="?"}
Consider $S^1$ and the constant sheaf $\ul{\ZZ}$:

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\end{tikzpicture}

ere we have 
\[
C^0(S^1, \ul{\ZZ}) = \ul{\ZZ}(U_1) \oplus \ul{\ZZ}(U_2) = \ul{\ZZ} \oplus \ul{\ZZ}
,\]
and 
\[
C^1(S^1, \ZZ) = \bigoplus_{\substack{ \text{double} \\ \text{intersections}} } \ul{\ZZ}(U_{ij})  \ul{\ZZ}(U_{12}) = \ul{\ZZ}(U_1 \intersect U_{2}) = \ul{\ZZ} \oplus \ul{\ZZ}
.\]
We then get
\[
C^0(S^1, \ul{\ZZ}) &\mapsvia{\bd} C^1(S^1, \ul{\ZZ}) \\
\ZZ \oplus \ZZ &\to \ZZ\oplus \ZZ \\
(a, b) &\mapsto (a-b, a-b)
,\]

Which yields $H^*(S^1, \ul{\ZZ}) = [\ZZ, \ZZ, 0, \cdots]$.
:::