Consider the following theorem of Atiyah.

> Let $X$ be a connected smooth projective curve over an algebraically closed field $k$. Then a vector bundle $L$ on $X$ admits a connection if and only if the degree of every direct summand of $L$ is divisible by the characteristic of $k$.

My thoughts on this are as follows.

**Round 1.** First of all, a connection over $k$, if $k$ is of positive characteristic, is equivalent to the descent data for the Frobenius, assuming our connection has trivial $p$-curvature. Why does $F^*(\mathcal{F})$ have a connection?

**Round 2.** It is kind of a tricky question to have sort of immediate geometric intuition, as the statement is about characteristic $p$ and relies on a purely cohomological definition of a connection. What does the degree of a vector bundle really mean? It is the order of the poles and zeros of a rational section, since we are just dealing with varieties. What is a connection? It is a way of describing the transportation of tangent vectors and in the case of vector bundles associated vector spaces from one point to another. However, in characteristic $p$, this can be a tricky, as we can have maps that will be an everywhere local isomorphism but add a little bit of extra vanishing to the tangent vectors, i.e. like the Frobenius. Similarly, we can have sections of our vector bundle, which despite being nonzero, can have zero derivative everywhere, which must be respected by the connection (there can not be loss of information at the differential level). Now, if we want a valid connection of our vector bundle, we need some type of control on the differential vanishing of the section. it turns out that this information is enough to control the characteristic $p$ "fluff". This is because by constraining the degree of the vector bundle to always divide the degree, we ensure that we always get some characteristic $p$ fluff everywhere on our vector bundle without jumps, and as such, we can validly transport tangent vectors (also using the smooth and connected hypotheses, which would clearly cause problems if not there). I think projective can be replaced by proper, but it maybe makes the proof more geometric. This is also the idea behind $p$-curvature.

**Round 3.** In the above Round 2, we did not really say anything global. Degree is purely global, but $p$-curvature is local. Maybe we can imagine this as some sort of Gauss-Bonnet theorem. There is an easy direction of Atiyah's theorem, which is that$$\text{connection} \implies \text{degree divisible by }p,$$which is just a Chern class thing. The converse is the hard part. On some level, an ideal answer to the converse would include an answer to "what is the obstruction for a connection to exist in higher dimensions?" This may involve some group in crystalline cohomology that vanishes for curves but not in higher dimensions.

Can anyone help me refine my thoughts on this theorem? Are there are any insights I am missing? Is there anything I have said which is just flat wrong? How do others think about this theorem?