A question about holomorphic structure in Atiyah Bott's paper.

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I am reading Atiyah & Bott's The Yang-Mills equations over Riemann surfaces. I meet a question about the holomorphic structure.



Let $Eto M$ be a complex vector bundle over a compact Riemann surface. There is a Hermitian metric on $E$.



Then
beginequation
mathscr Acongmathscr B
endequation
for
beginequation
beginsplit
mathscr A=&textall smooth hermitian connections\
mathscr B=&textall holomorphic structures on E
endsplit
endequation



The identification is described as: given a connection in $mathscr A$
beginequation
d_AcolonOmega^0(M;E)toOmega^1(M;E)
endequation
then complexify it
beginequation
d_AcolonOmega^0_mathbb C(M;E)toOmega^1_mathbb C(M;E)
endequation
and we have
beginequation
Omega^1_mathbb C(M;E)=Omega^1,0(M;E)oplusOmega^0,1(M;E)
endequation



Then
beginequation
d_A=d_A'oplus d_A''
endequation
for
beginequation
beginsplit
d_A'colon & Omega^0_mathbb C(M;E)toOmega^1,0(M;E)\
d_A''colon & Omega^0_mathbb C(M;E)toOmega^0,1(M;E)
endsplit
endequation
and the operator $d_A''$ determines a holomorphic structure of $E$.



Questions:



  1. The decomposition $Omega^1_mathbb C(M;E)=Omega^1,0(M;E)oplusOmega^0,1(M;E)$ is due to a Hodge star operator. Is there a canonical Hodge star operator here? I know there is another decomposition using the canonical almost complex structure $J$, and I do not understand why not use this.

  2. How to recover $d_A$ from $d_A''$? I guess here we need the Hermitian metric.

Thanks in advance.




differential-geometry connections




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    up vote
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    1












    I am reading Atiyah & Bott's The Yang-Mills equations over Riemann surfaces. I meet a question about the holomorphic structure.



    Let $Eto M$ be a complex vector bundle over a compact Riemann surface. There is a Hermitian metric on $E$.



    Then
    beginequation
    mathscr Acongmathscr B
    endequation
    for
    beginequation
    beginsplit
    mathscr A=&textall smooth hermitian connections\
    mathscr B=&textall holomorphic structures on E
    endsplit
    endequation



    The identification is described as: given a connection in $mathscr A$
    beginequation
    d_AcolonOmega^0(M;E)toOmega^1(M;E)
    endequation
    then complexify it
    beginequation
    d_AcolonOmega^0_mathbb C(M;E)toOmega^1_mathbb C(M;E)
    endequation
    and we have
    beginequation
    Omega^1_mathbb C(M;E)=Omega^1,0(M;E)oplusOmega^0,1(M;E)
    endequation



    Then
    beginequation
    d_A=d_A'oplus d_A''
    endequation
    for
    beginequation
    beginsplit
    d_A'colon & Omega^0_mathbb C(M;E)toOmega^1,0(M;E)\
    d_A''colon & Omega^0_mathbb C(M;E)toOmega^0,1(M;E)
    endsplit
    endequation
    and the operator $d_A''$ determines a holomorphic structure of $E$.



    Questions:



    1. The decomposition $Omega^1_mathbb C(M;E)=Omega^1,0(M;E)oplusOmega^0,1(M;E)$ is due to a Hodge star operator. Is there a canonical Hodge star operator here? I know there is another decomposition using the canonical almost complex structure $J$, and I do not understand why not use this.

    2. How to recover $d_A$ from $d_A''$? I guess here we need the Hermitian metric.

    Thanks in advance.




    differential-geometry connections




    share|cite|improve this question





















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      I am reading Atiyah & Bott's The Yang-Mills equations over Riemann surfaces. I meet a question about the holomorphic structure.



      Let $Eto M$ be a complex vector bundle over a compact Riemann surface. There is a Hermitian metric on $E$.



      Then
      beginequation
      mathscr Acongmathscr B
      endequation
      for
      beginequation
      beginsplit
      mathscr A=&textall smooth hermitian connections\
      mathscr B=&textall holomorphic structures on E
      endsplit
      endequation



      The identification is described as: given a connection in $mathscr A$
      beginequation
      d_AcolonOmega^0(M;E)toOmega^1(M;E)
      endequation
      then complexify it
      beginequation
      d_AcolonOmega^0_mathbb C(M;E)toOmega^1_mathbb C(M;E)
      endequation
      and we have
      beginequation
      Omega^1_mathbb C(M;E)=Omega^1,0(M;E)oplusOmega^0,1(M;E)
      endequation



      Then
      beginequation
      d_A=d_A'oplus d_A''
      endequation
      for
      beginequation
      beginsplit
      d_A'colon & Omega^0_mathbb C(M;E)toOmega^1,0(M;E)\
      d_A''colon & Omega^0_mathbb C(M;E)toOmega^0,1(M;E)
      endsplit
      endequation
      and the operator $d_A''$ determines a holomorphic structure of $E$.



      Questions:



      1. The decomposition $Omega^1_mathbb C(M;E)=Omega^1,0(M;E)oplusOmega^0,1(M;E)$ is due to a Hodge star operator. Is there a canonical Hodge star operator here? I know there is another decomposition using the canonical almost complex structure $J$, and I do not understand why not use this.

      2. How to recover $d_A$ from $d_A''$? I guess here we need the Hermitian metric.

      Thanks in advance.




      differential-geometry connections




      share|cite|improve this question











      I am reading Atiyah & Bott's The Yang-Mills equations over Riemann surfaces. I meet a question about the holomorphic structure.



      Let $Eto M$ be a complex vector bundle over a compact Riemann surface. There is a Hermitian metric on $E$.



      Then
      beginequation
      mathscr Acongmathscr B
      endequation
      for
      beginequation
      beginsplit
      mathscr A=&textall smooth hermitian connections\
      mathscr B=&textall holomorphic structures on E
      endsplit
      endequation



      The identification is described as: given a connection in $mathscr A$
      beginequation
      d_AcolonOmega^0(M;E)toOmega^1(M;E)
      endequation
      then complexify it
      beginequation
      d_AcolonOmega^0_mathbb C(M;E)toOmega^1_mathbb C(M;E)
      endequation
      and we have
      beginequation
      Omega^1_mathbb C(M;E)=Omega^1,0(M;E)oplusOmega^0,1(M;E)
      endequation



      Then
      beginequation
      d_A=d_A'oplus d_A''
      endequation
      for
      beginequation
      beginsplit
      d_A'colon & Omega^0_mathbb C(M;E)toOmega^1,0(M;E)\
      d_A''colon & Omega^0_mathbb C(M;E)toOmega^0,1(M;E)
      endsplit
      endequation
      and the operator $d_A''$ determines a holomorphic structure of $E$.



      Questions:



      1. The decomposition $Omega^1_mathbb C(M;E)=Omega^1,0(M;E)oplusOmega^0,1(M;E)$ is due to a Hodge star operator. Is there a canonical Hodge star operator here? I know there is another decomposition using the canonical almost complex structure $J$, and I do not understand why not use this.

      2. How to recover $d_A$ from $d_A''$? I guess here we need the Hermitian metric.

      Thanks in advance.





      differential-geometry connections





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