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 $u$?

  • $\begingroup$ If an $n-q$ dimensional cohomology class could cupped with that class, giving a class in $H^n$, and if that space comes equipped with a norm, you're in business. $\endgroup$ – 54321user Jan 8 at 5:35

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