The area is a good enough metric
Abstract
In the first part we extend the construction of the smooth normal-crossing divisors compactification of projectivized strata of abelian differentials given by Bainbridge, Chen, Gendron, Grushevsky and Moeller to the case of k-differentials. Since the generalized construction is closely related to the original one, we mainly survey their results and justify the details that need to be adapted in the more general context. In the second part we show that the flat area provides a canonical hermitian metric on the tautological bundle over the projectivized strata of finite area k-differentials whose curvature form represents the first Chern class. This result is useful in order to apply Chern-Weyl theory tools. It has already been used as an assumption in the work of Sauvaget for abelian differentials and is also used in a paper of Chen, M\"oller and Sauvaget for quadratic differentials.
Cite
@article{arxiv.1910.14151,
title = {The area is a good enough metric},
author = {Matteo Costantini and Martin Möller and Jonathan Zachhuber},
journal= {arXiv preprint arXiv:1910.14151},
year = {2022}
}
Comments
Integrability of the curvature form and its wedge powers clarified in Section 4. All applications unchanged