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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.

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As for any complex manifold, gauge equivalence of holomorphic flat connections is determined by the holonomy. In this case, this means gauge classes are the same as isomorphism classes of finite dimensional vector spaces equipped with two commuting operators (and these isomorphism classes are easy to compute via Jordan form). –  Moosbrugger Aug 21 '11 at 5:31
    
Yes, and I am wondering which pairs of operators can occur in the above extension. –  Sam Lewallen Aug 21 '11 at 13:05
    
Ah, I'm sorry. I misunderstood what "any $V$" meant. Well, the following general theorem answers your first question (positively): an indecomposable vector bundle on a smooth projective curve over $\mathbb{C}$ admits a flat connection if and only if the vector bundle is semi-stable. A partial answer to the second: adding a non-zero 1-form to a connection on $E$ gives different holonomy since global functions on $E$ are constants. I don't see what operators can arise from these particular bundles. –  Moosbrugger Aug 21 '11 at 15:12
    
Moosbrugger means "admits a flat unitary connection." This is the Narasimhan-Seshadri theorem. However, the appropriate result here is the one pertaining to flat $SL(n,C)$ or $GL(n,C)$-connections mentioned in my answer below. –  Michael Thaddeus Aug 21 '11 at 20:01
    
@Moosbrugger: Actually, any indecomposable degree zero bundle on a curve admits a (flat) holomorphic connection - see Atiyah's theorem below. In particular, it can be strictly unstable. For example, fix a line bundle $L$ with $L^2=K$ on a curve of genus 2 or more, and take the extension of $L^\vee$ by $L$ determined by the Kaehler class. That bundle admits (many) holomorphic connections. –  Peter Dalakov Aug 21 '11 at 20:38
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2 Answers 2

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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...

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Actually, there is a small caveat here. Say $g>1$. The bundle $V=K^{1/2}\oplus K^{-1/2}$ does not admit a holomorphic flat connection, by Atiyah's theorem. However, it does admit a Higgs field making it into a stable pair: take a regular nilpotent Higgs field (map $K^{1/2}$ to $K\otimes K^{-1/2}$ by the identity and send $K^{-1/2}$ to $0$). The Hitchin-Simpson correspondence gives you a holomorphic connection on a different holomorphic bundle with the same underlying smooth structure. In this case this is $J^1(K^{-1/2})$. I am not sure if Sam cares about this. –  Peter Dalakov Aug 21 '11 at 22:37
    
Wow, thanks! This is so beautiful! So this $P^1$ is exactly what $SU(2)$ gauge theorists (among others) would call the pillowcase. I'm still trying to digest everything... I suppose the class of $V$ in the extension $0 \to \mathcal{O}\to V \to\mathcal{O}\to 0$ which shows up, if it's to be non-split, must be that special indecomposable bundle $F_2$ which Atiyah talks about? Then this determines the other non-semisimple bundles, just by tensoring with the appropriate 2-torsion line bundle. –  Sam Lewallen Aug 22 '11 at 17:33
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Here is an answer which doesn't invoke non-Abelian Hodge theory.

In his paper "Complex Analytic Connections in Fibre Bundles" (Trans.AMS, v.85, 1957), Atiyah showed the following: Let $V$ be a holomorphic vector bundle on a (smooth, compact) curve, and let $V=\oplus_i V_i$ be the decomposition of $V$ into indecomposable bundles. Then $V$ admits a holomorphic connection (necessarily flat, for dimension reasons) if and only if $\deg V_i=0, \forall i$. In particular, any indecomposable bundle of degree zero admits a holomorphic connection. See Propositions 17,19 and Theorem 10 from the paper. In particular, for any $L\in Pic^0$, the bundles $L\oplus L^{\vee}$ and $L\otimes I_2$ admit a holomorphic connection, whereas a rank 2 bundle of the form $F\oplus F^{\vee}$, $\deg F\neq 0$ never admits a holomorphic connection. (Here $I_2$ is the unique non-trivial extension of $\mathcal{O}$ by $\mathcal{O}$.) Now you can use the calculation that Michael Thaddeus mentions to see that your $V$ admits a holomorphic connection as long as the extension is non-split.

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Thanks! Is it possible to put all this together into a coherent moduli space of $SL(2,C)$ connections? Naively this seems fraught, because of the semisimple guys where these $L\otimes I_2$'s pop up. Is this what the Higgs moduli space does, somehow? I really should read some of these papers... –  Sam Lewallen Aug 22 '11 at 17:40
    
Yes. Actually every object is strictly semistable. And indeed, this is what the Higgs moduli space does. –  Michael Thaddeus Aug 22 '11 at 18:06
    
Yes! This is the moduli space of $SL(2,\mathbb{C})$ local systems, which is homeomorphic to the moduli space of (top.trivial) Higgs bundles. For an elliptic curve and connected reductive $G$ these are described in a beautiful paper by Michael Thaddeus. The normalisations of the two spaces are respectively $T^\vee E\otimes \Lambda /W$ and $J\otimes \Lambda /W$, where $\Lambda$ is the coweight lattice, $W$ is the Weyl group, and $J$ is the unique group extension of $E$ by $\mathbb{C}$. The normalisation of the rep.variety is $H\times H/W$, $H=\mathbb{C}^\times \otimes \Lambda$. –  Peter Dalakov Aug 22 '11 at 19:01
    
Of course, in my 3-rd sentence "respectively" should be replaced by "in reverse order". –  Peter Dalakov Aug 22 '11 at 19:27
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