local coordinate expression if an equation.

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I have some basic questions. The settings are following.



Let $M$ be a complex manifold (Actually, it is a kahler maniold.), $alpha=$Im$barpartial phi$, and $omega=-dalpha$. Define a vector field $V$ by $i(V)omega=-alpha$. (here, $i$ : interior product).



Let $phi$ be a real analytic function satisfying $V phi=2phi$. (*)



My question is



  1. how the equation (*) can be written in local holomorphic coordinates as $$phi_a phi^a=2phi$$
    or $$ phi^abarbphi_a phi_barb=2phi$$

where $phi_a = fracpartial phipartial z^a$ and $phi^a$ is defined by $phi_barb=sqrt-1phi_abarbphi^a$ .



and 2. Why this equation is called "Monge-Ampere type" equation. I know just the definition about 'Monge-Ampere equation' : some PDE with the term of determinant of hessian matrix.



Thanks.
(If there is some recommended book, please let me know.)






differential-equations







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    up vote
    2
    down vote

    favorite












    I have some basic questions. The settings are following.



    Let $M$ be a complex manifold (Actually, it is a kahler maniold.), $alpha=$Im$barpartial phi$, and $omega=-dalpha$. Define a vector field $V$ by $i(V)omega=-alpha$. (here, $i$ : interior product).



    Let $phi$ be a real analytic function satisfying $V phi=2phi$. (*)



    My question is



    1. how the equation (*) can be written in local holomorphic coordinates as $$phi_a phi^a=2phi$$
      or $$ phi^abarbphi_a phi_barb=2phi$$

    where $phi_a = fracpartial phipartial z^a$ and $phi^a$ is defined by $phi_barb=sqrt-1phi_abarbphi^a$ .



    and 2. Why this equation is called "Monge-Ampere type" equation. I know just the definition about 'Monge-Ampere equation' : some PDE with the term of determinant of hessian matrix.



    Thanks.
    (If there is some recommended book, please let me know.)






    differential-equations







    share|cite|improve this question

























      up vote
      2
      down vote

      favorite









      up vote
      2
      down vote

      favorite











      I have some basic questions. The settings are following.



      Let $M$ be a complex manifold (Actually, it is a kahler maniold.), $alpha=$Im$barpartial phi$, and $omega=-dalpha$. Define a vector field $V$ by $i(V)omega=-alpha$. (here, $i$ : interior product).



      Let $phi$ be a real analytic function satisfying $V phi=2phi$. (*)



      My question is



      1. how the equation (*) can be written in local holomorphic coordinates as $$phi_a phi^a=2phi$$
        or $$ phi^abarbphi_a phi_barb=2phi$$

      where $phi_a = fracpartial phipartial z^a$ and $phi^a$ is defined by $phi_barb=sqrt-1phi_abarbphi^a$ .



      and 2. Why this equation is called "Monge-Ampere type" equation. I know just the definition about 'Monge-Ampere equation' : some PDE with the term of determinant of hessian matrix.



      Thanks.
      (If there is some recommended book, please let me know.)






      differential-equations







      share|cite|improve this question















      I have some basic questions. The settings are following.



      Let $M$ be a complex manifold (Actually, it is a kahler maniold.), $alpha=$Im$barpartial phi$, and $omega=-dalpha$. Define a vector field $V$ by $i(V)omega=-alpha$. (here, $i$ : interior product).



      Let $phi$ be a real analytic function satisfying $V phi=2phi$. (*)



      My question is



      1. how the equation (*) can be written in local holomorphic coordinates as $$phi_a phi^a=2phi$$
        or $$ phi^abarbphi_a phi_barb=2phi$$

      where $phi_a = fracpartial phipartial z^a$ and $phi^a$ is defined by $phi_barb=sqrt-1phi_abarbphi^a$ .



      and 2. Why this equation is called "Monge-Ampere type" equation. I know just the definition about 'Monge-Ampere equation' : some PDE with the term of determinant of hessian matrix.



      Thanks.
      (If there is some recommended book, please let me know.)






      differential-equations




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      edited Sep 5 at 7:31

























      asked Sep 5 at 7:26









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