The area is a good enough metric (Q6567973)

The study of compactifications of spaces of abelian, quadratic, and higher differentials on Riemann surfaces has been an area of significant recent activity. Pioneering work of Bainbridge, Chen, Gendron, Grushevsky and Moeller introduced a compactification of the space of abelian differentials. The paper under review generalizes this construction to spaces of \(k\)-differentials, and in addition generalizes (implicit) results of Sauvaget, and Chen-Moeller-Sauvaget (in the \(k=1\) and \(k=2\) cases) 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.