Let $X$ be a compact Riemann surface, is it possible to have a description for the space of all Khaler forms on $X$? More concretely, given two Kahler forms $\Omega$ and $\Omega'$ is it known what is the relationship bethween them?
I know for sure that any compact Riemann surface is a hermitian manifold, which in turns is always Kalher (in dimension $1$); but I am asking about the relationship about two different Kahler structures.
Best Answer
$\newcommand{\dd}{\partial}$So this has an answer: The $(1,1)$-dimensional cohomology of a compact Riemann surface $X$ is one-dimensional, so any two Kähler classes are positively proportional: If $\Omega$ and $\Omega'$ are smooth Kähler forms on $X$, there exists a unique positive real number $c$ such that $\Omega$ and $c\Omega'$ are cohomologous. The $\dd\bar{\dd}$-lemma then guarantees there exists a smooth, real-valued function $\phi$ satisfying $\Omega = c\Omega'+ i\dd\bar{\dd}\phi$.