I have some basic questions. The settings are following.
Let $M$ be a complex manifold (Actually, it is a kahler maniold.), $\alpha=$Im$\bar{\partial} \phi$, and $\omega=-d\alpha$. Define a vector field $V$ by $i(V)\omega=-\alpha$. (here, $i$ : interior product).
Let $\phi$ be a real analytic function satisfying $V \phi=2\phi$. (*)
My question is
- how the equation (*) can be written in local holomorphic coordinates as $$\phi_a \phi^a=2\phi$$ or $$ \phi^{a\bar{b}}\phi_a \phi_\bar{b}=2\phi$$
where $\phi_a = \frac{\partial \phi}{\partial z^a}$ and $\phi^a$ is defined by $\phi_\bar{b}=\sqrt{-1}\phi_{a\bar{b}}\phi^a$ .
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.)