I have some hopefully elementary questions about rank 2 flat bundles on an elliptic curve $E$.
Take $p\in E$, and consider the exact sequence
$$0\to \mathcal{O}(-p) \to V \to \mathcal{O}(p)\to 0$$
so $V$ is a rank 2 holomorphic vector bundle on $E$ with deg$(E)=0$. RR says that $H^1(E;\mathcal{O}(-2p))$ is 2-dimensional, so there should be 3 distinct extensions a $P^1 = P(H^1(E;\mathcal{O}(-2p))$'s worth of distinct extensions up to iso, but I am only interested in those which admit a FLAT holomorphic connection.
Question: is there any way to determine which $V$ fit in the above sequence, and to determine which are flat? Can I also determine the holonomy of the flat connections which occur?
Another question: How many gauge-inequivalent holomorphic flat connections can any $V$ admit? I think a bundle can admit at most one unitary connection? Is the same true for a general connection?
I don't know much about this stuff, so I was just going to start futzing around with theta functions. But I thought I'd ask to see if there's a better way.
EDIT: For any one interested in the answer, all the non-trivial extensions admit flat connections, and they are parameterized in a very nice way, see below.
Best Answer
The extensions of this form that admit a flat holomorphic connection are precisely the non-split ones.
The work of Hitchin, Donaldson, Corlette, and Simpson from the late 1980's shows that a bundle admits a flat holomorphic connection if and only if it has a Higgs field, i.e. a section $\phi \in H^0(K \otimes \mbox{End } V)$, making the Higgs pair $(V,\phi)$ semistable. See Hitchin's "The self-duality equations on a Riemann surface," for example. On an elliptic curve, $K \cong O$.
The classification of vector bundles on a smooth elliptic curve is rather easy and is accomplished in an early paper of Atiyah. Using this, it is not hard to show that every non-split extension of the form you state is isomorphic to $L \oplus L^{-1}$ for some $L \in \mbox{Pic}_0$, except on four lines in the two-dimensional extension space $H^1(O(-2p))$, where it is a non-split extension
$$0 \longrightarrow L \longrightarrow V \longrightarrow L \longrightarrow 0$$
for some $L$ with $L^2 \cong O$.
In either case, it is easy to find a Higgs field making the pair semistable: a diagonal Higgs field (i.e. an endomorphism preserving the splitting) in the case of $L \oplus L^{-1}$, and a nilpotent Higgs field (i.e. the composite map $V \to L \to V$ ) in the case of the non-split extension.
On the other hand, $O(p) \oplus O(-p)$ will not admit a Higgs field making it semistable, for $O(p)$ will always be an invariant, hence destabilizing, subbundle.
Regarding your question about what holonomies arise, let's first recall the answer for line bundles. The flat $C^\times$-connections, or representations $\pi_1 (E) \to C^\times$, are parametrized by $C^\times \times C^\times$. This is analytically, but not algebraically, isomorphic to the moduli space of flat $C^\times$ connections on $E$, which is a $C$-bundle over $\mbox{Pic}_0 E \cong E$. In particular, the projection to $E$ gives a holomorphic map $C^\times \times C^\times \to E$. If I remember correctly, this turns out to be nothing but $(w,z) \mapsto (\log w + \tau \log z)/(2 \pi i)$, where $E = C/\langle 1, \tau \rangle$. That tells you very explicitly which holonomies map to which line bundles. All this is a small fragment of the work of Simpson. I think it is spelled out explicitly in an expository paper by Goldman and Xia.
Now, the moduli space of flat $SL(2,C)$-connections on $E$, modulo gauge equivalence, is the space of two commuting elements of $SL(2,C)$, modulo conjugation. On a dense open set these elements are semisimple, and then (by a result of Borel and Steinberg) they lie in a common maximal torus $\cong C^\times$. They are conjugate in $G$ if and only if they are exchanged by an element of the Weyl group $W \cong Z/2$. So (ignoring the non-semisimple elements) the moduli space will be $(C^\times \times C^\times)/W$. The map from moduli of flat connections to moduli of bundles is then pretty clearly the $W$-quotient $(C^\times \times C^\times)/W \to E/W \cong P^1$ of the map of the previous paragraph. Here $P^1$ is parametrizing bundles of the form $L \oplus L^{-1}$. It is the projectivization of your vector space $H^1(O(-2p))$, since the bundle $V$ only depends on the extension class up to a scalar.
In other words, the holonomies that appear are conjugate to those in the fibers of this map, at least away from those four special points in $P^1$ where the non-split extensions with $L^2 \cong O$ appear. There there will be non-semisimple holonomies. For example, over the identity, there will be pairs of upper-triangular matrices with 1's on the diagonal. But perhaps I will leave this case as an exercise...