Kahler Form Under a Rational Map

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I am sure this is well known but I am having trouble finding the appropriate literature.



I am in interested in the following question:



Given a rational map $f:M --rightarrow mathbb C P^n$ what can we say about the pullback of the Fubini-Study form $f^* omega_textFS$ which is a smooth positive $(1,1)$ form on $M setminus V$ for some subvariety $V subset M$ of codimension at least 2.



For example I am interested in the case of $(M^n,omega)$ a Kahler manifold and when the above form has finite mass, i.e. when $$int_M f^*omega_textFS wedge omega^n-1 < infty$$




differential-geometry algebraic-geometry complex-geometry




share|cite|improve this question























    up vote
    0
    down vote

    favorite
    1












    I am sure this is well known but I am having trouble finding the appropriate literature.



    I am in interested in the following question:



    Given a rational map $f:M --rightarrow mathbb C P^n$ what can we say about the pullback of the Fubini-Study form $f^* omega_textFS$ which is a smooth positive $(1,1)$ form on $M setminus V$ for some subvariety $V subset M$ of codimension at least 2.



    For example I am interested in the case of $(M^n,omega)$ a Kahler manifold and when the above form has finite mass, i.e. when $$int_M f^*omega_textFS wedge omega^n-1 < infty$$




    differential-geometry algebraic-geometry complex-geometry




    share|cite|improve this question





















      up vote
      0
      down vote

      favorite
      1









      up vote
      0
      down vote

      favorite
      1






      1





      I am sure this is well known but I am having trouble finding the appropriate literature.



      I am in interested in the following question:



      Given a rational map $f:M --rightarrow mathbb C P^n$ what can we say about the pullback of the Fubini-Study form $f^* omega_textFS$ which is a smooth positive $(1,1)$ form on $M setminus V$ for some subvariety $V subset M$ of codimension at least 2.



      For example I am interested in the case of $(M^n,omega)$ a Kahler manifold and when the above form has finite mass, i.e. when $$int_M f^*omega_textFS wedge omega^n-1 < infty$$




      differential-geometry algebraic-geometry complex-geometry




      share|cite|improve this question











      I am sure this is well known but I am having trouble finding the appropriate literature.



      I am in interested in the following question:



      Given a rational map $f:M --rightarrow mathbb C P^n$ what can we say about the pullback of the Fubini-Study form $f^* omega_textFS$ which is a smooth positive $(1,1)$ form on $M setminus V$ for some subvariety $V subset M$ of codimension at least 2.



      For example I am interested in the case of $(M^n,omega)$ a Kahler manifold and when the above form has finite mass, i.e. when $$int_M f^*omega_textFS wedge omega^n-1 < infty$$





      differential-geometry algebraic-geometry complex-geometry





      share|cite|improve this question










      share|cite|improve this question




      share|cite|improve this question









      asked Aug 2 at 10:19









      ben

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