I am trying to understand a detail in the following paper:
Bott-Chern currents and complex immersions
J.-M. Bismut, H. Gillet, and C. Soulé
https://projecteuclid.org/euclid.dmj/1077297147
On page 262, they write:
Let $C^1(M)$ be the set of continuous differential forms on $M$ which have continuous first derivatives. Let $|| \, ||_{C^1(M)}$ be a norm on $C^1(M)$ such that $||\mu_n ||_{C^1(M)} \to 0$ if and only if $\mu_n$ tends to $0$ uniformly on $M$ together with its first derivatives.
Earlier, $M$ is defined to be a compact complex manifold.
- What does it mean for a differential form to tend to $0$ uniformly on $M$?
- What is the norm they describe on $C^1(M)$?
The usual notion of uniform convergence of real functions $f_n$ on a compact set $K$ to a limit function $f$ makes sense, because we can take the sup norm $\sup_{x \in K} |f_n(x) - f(x)|$. However, here, differential forms are sections of the cotangent bundle, and the fibers $\bigwedge T^*_xM$ do not have a natural norm so I'm not sure how to measure how ``far'' away a differential form is from 0.
Thank you.
