A question about Narasimhan-Seshadri Donaldson theorem.

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I see a statement of the theorem in Jonathan Evans' lecture note 13




An indecomposable Hermitian holomorphic vector bundle $mathcal E$ on
a Riemann surface $M$ is stable if and only if there is a compatible
unitary connection on $mathcal E$ with constant central cervature
$$star F_nabla=-2pi imu(mathcal E)$$




Some notations: $F_nabla$ is the curvature of the connection $nabla$, $mu(mathcal E)=fracc_1(mathcal E)textrank(mathcal E)$ is the slope of $mathcal E$.



Now I do not understand the statement, because from Atiyah & Bott's paper, The Yang-Mills Equations over Riemann Surfaces sec 5, we can identify the space of all unitary connection on $E$ and that of all holomorphic structures over $E$, where $E$ is the underlying complex vector bundle of $mathcal E$. So now we are given a holomorphic structure $mathcal E$, I think the unitary connection is uniquely determined. So is this theorem telling us that this unique-determined connection has this property?




connections holomorphic-bundles




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    up vote
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    I see a statement of the theorem in Jonathan Evans' lecture note 13




    An indecomposable Hermitian holomorphic vector bundle $mathcal E$ on
    a Riemann surface $M$ is stable if and only if there is a compatible
    unitary connection on $mathcal E$ with constant central cervature
    $$star F_nabla=-2pi imu(mathcal E)$$




    Some notations: $F_nabla$ is the curvature of the connection $nabla$, $mu(mathcal E)=fracc_1(mathcal E)textrank(mathcal E)$ is the slope of $mathcal E$.



    Now I do not understand the statement, because from Atiyah & Bott's paper, The Yang-Mills Equations over Riemann Surfaces sec 5, we can identify the space of all unitary connection on $E$ and that of all holomorphic structures over $E$, where $E$ is the underlying complex vector bundle of $mathcal E$. So now we are given a holomorphic structure $mathcal E$, I think the unitary connection is uniquely determined. So is this theorem telling us that this unique-determined connection has this property?




    connections holomorphic-bundles




    share|cite|improve this question





















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      I see a statement of the theorem in Jonathan Evans' lecture note 13




      An indecomposable Hermitian holomorphic vector bundle $mathcal E$ on
      a Riemann surface $M$ is stable if and only if there is a compatible
      unitary connection on $mathcal E$ with constant central cervature
      $$star F_nabla=-2pi imu(mathcal E)$$




      Some notations: $F_nabla$ is the curvature of the connection $nabla$, $mu(mathcal E)=fracc_1(mathcal E)textrank(mathcal E)$ is the slope of $mathcal E$.



      Now I do not understand the statement, because from Atiyah & Bott's paper, The Yang-Mills Equations over Riemann Surfaces sec 5, we can identify the space of all unitary connection on $E$ and that of all holomorphic structures over $E$, where $E$ is the underlying complex vector bundle of $mathcal E$. So now we are given a holomorphic structure $mathcal E$, I think the unitary connection is uniquely determined. So is this theorem telling us that this unique-determined connection has this property?




      connections holomorphic-bundles




      share|cite|improve this question











      I see a statement of the theorem in Jonathan Evans' lecture note 13




      An indecomposable Hermitian holomorphic vector bundle $mathcal E$ on
      a Riemann surface $M$ is stable if and only if there is a compatible
      unitary connection on $mathcal E$ with constant central cervature
      $$star F_nabla=-2pi imu(mathcal E)$$




      Some notations: $F_nabla$ is the curvature of the connection $nabla$, $mu(mathcal E)=fracc_1(mathcal E)textrank(mathcal E)$ is the slope of $mathcal E$.



      Now I do not understand the statement, because from Atiyah & Bott's paper, The Yang-Mills Equations over Riemann Surfaces sec 5, we can identify the space of all unitary connection on $E$ and that of all holomorphic structures over $E$, where $E$ is the underlying complex vector bundle of $mathcal E$. So now we are given a holomorphic structure $mathcal E$, I think the unitary connection is uniquely determined. So is this theorem telling us that this unique-determined connection has this property?





      connections holomorphic-bundles





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      share|cite|improve this question




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      asked Jul 26 at 17:18









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