Suppose $X$ is a smooth complex variety and $L$ is a line bundle with a metric $h_L$, then a section $s \in H^0(X, L)$ gives another metric $\tilde h_L:= e^{-\phi}h_L$ where $\phi=\log \|s\|^2_{h_L}$.
If $u \in H^q(X,L)$ is a section (or just $u\in H^q(X,K_X)$), is there a way to construct a metric on $L$ with some relation to $u$?
