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Sam Lewallen
Sam Lewallen
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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 33 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.

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, 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.

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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Sam Lewallen
Sam Lewallen
  • 1.1k
  • 12
  • 21

Rank 2 flat bundles on an elliptic curve, via extensions

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, 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.