Question on Kahler geometry: Kahler form and $mathcalO_X(1)$

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Given a projective surface (2 complex dimensions) $X$ we can equip it with the line bundle $mathcalO_X(1)$. Let us fix this line bundle. Recall that all projective surfaces are Kahler surfaces. Then, I heard in a seminar that for the class of the associated Kahler form $J$ it is true that
$$
[J] = c_1(mathcalO_X(1))
$$
Why is this the case (if I formulated correctly)? Could you explain such a relation?










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    Given a projective surface (2 complex dimensions) $X$ we can equip it with the line bundle $mathcalO_X(1)$. Let us fix this line bundle. Recall that all projective surfaces are Kahler surfaces. Then, I heard in a seminar that for the class of the associated Kahler form $J$ it is true that
    $$
    [J] = c_1(mathcalO_X(1))
    $$
    Why is this the case (if I formulated correctly)? Could you explain such a relation?










    share|cite|improve this question























      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      Given a projective surface (2 complex dimensions) $X$ we can equip it with the line bundle $mathcalO_X(1)$. Let us fix this line bundle. Recall that all projective surfaces are Kahler surfaces. Then, I heard in a seminar that for the class of the associated Kahler form $J$ it is true that
      $$
      [J] = c_1(mathcalO_X(1))
      $$
      Why is this the case (if I formulated correctly)? Could you explain such a relation?










      share|cite|improve this question













      Given a projective surface (2 complex dimensions) $X$ we can equip it with the line bundle $mathcalO_X(1)$. Let us fix this line bundle. Recall that all projective surfaces are Kahler surfaces. Then, I heard in a seminar that for the class of the associated Kahler form $J$ it is true that
      $$
      [J] = c_1(mathcalO_X(1))
      $$
      Why is this the case (if I formulated correctly)? Could you explain such a relation?







      algebraic-geometry complex-geometry vector-bundles






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      asked Sep 10 at 13:31









      Gorbz

      24419




      24419




















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          Because you're restricting the hyperplane class (which corresponds to the sheaf $mathscr O_Bbb P^n(1)$) in $Bbb P^n$ to $X$ to get the the sheaf $mathscr O_X(1)$, and $c_1$ of the hyperplane class bundle is precisely the Kähler form on $Bbb P^n$. So it's just naturality, pulling back by the inclusion map in both cases.






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            Because you're restricting the hyperplane class (which corresponds to the sheaf $mathscr O_Bbb P^n(1)$) in $Bbb P^n$ to $X$ to get the the sheaf $mathscr O_X(1)$, and $c_1$ of the hyperplane class bundle is precisely the Kähler form on $Bbb P^n$. So it's just naturality, pulling back by the inclusion map in both cases.






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              Because you're restricting the hyperplane class (which corresponds to the sheaf $mathscr O_Bbb P^n(1)$) in $Bbb P^n$ to $X$ to get the the sheaf $mathscr O_X(1)$, and $c_1$ of the hyperplane class bundle is precisely the Kähler form on $Bbb P^n$. So it's just naturality, pulling back by the inclusion map in both cases.






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                up vote
                0
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                Because you're restricting the hyperplane class (which corresponds to the sheaf $mathscr O_Bbb P^n(1)$) in $Bbb P^n$ to $X$ to get the the sheaf $mathscr O_X(1)$, and $c_1$ of the hyperplane class bundle is precisely the Kähler form on $Bbb P^n$. So it's just naturality, pulling back by the inclusion map in both cases.






                share|cite|improve this answer












                Because you're restricting the hyperplane class (which corresponds to the sheaf $mathscr O_Bbb P^n(1)$) in $Bbb P^n$ to $X$ to get the the sheaf $mathscr O_X(1)$, and $c_1$ of the hyperplane class bundle is precisely the Kähler form on $Bbb P^n$. So it's just naturality, pulling back by the inclusion map in both cases.







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                answered Sep 20 at 18:57









                Ted Shifrin

                61k44388




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