# (Singular) metric associated to the higher cohomology

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 $$u$$?

• 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. – 54321user Jan 8 at 5:35