Let M be a 2-dimensional (complex dimension) Kähler manifold and $\phi$ be a real $(1,1)$-form. Is it possible that there exists a function $u$ such that $\phi=u\sqrt{-1}\partial\bar{\partial}u$?



Can a real (1,1) form $\phi$ be represented by $u\sqrt{-1}\partial\bar{\partial}u$ on a Kähler manifold?
If $M$ is compact this is obviously false: if we can write $\phi=u\sqrt{-1}\partial\overline{\partial}u$ then $\int\_M \omega\wedge\phi\leq 0$ integrating by parts. So for every $\phi$ which does not satisfy this (for example $\phi=\omega$) it will not be possible to find such $u$. 


