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2019

The area is a good enough metric

Costantini, Matteo, Möller, Martin, Zachhuber, Jonathan

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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.

The area is a good enough metric · Around

Built on

“The moduli space of multi-scale differentials”, (2019)
Original
M. Bainbridge et al · 1910
Earlier work this paper cites.
“Masur-Veech volumes and intersection theory: quadratic differentials” With an appendix by G. Borot, A. Giacchetto and D. Lewanski), (2019)
Original
D. Chen, M. M“”oller and A. Sauvaget · 1912
Earlier work this paper cites.
“Hirzebruch’s proportionality theorem in the noncompact case”
D. Mumford · 1977
Earlier work this paper cites.
“Flat surfaces”
W. Veech · 1993
Earlier work this paper cites.

Similar

“Volumes of moduli spaces of flat surfaces”, (2020)
Original
A. Sauvaget · 2004
Cited alongside, same era.
“The Chern classes and the Euler characteristic of the moduli spaces of Abelian differentials”, (2020)
Original
M. Costantini, M. M“”oller and J. Zachhuber · 2006
Cited alongside, same era.
“Geometry of algebraic curves. Volume II” With a contribution by J. Harris 268
E. Arbarello, M. Cornalba and P. Griffiths · 2011
Cited alongside, same era.
“Compactification of strata of abelian differentials”
M. Bainbridge et al · 2018
Cited alongside, same era.

Then

“Volumes and Siegel–Veech constants of ℋ \mathcal{H} (2g – 2) and Hodge integrals”
A. Sauvaget · 2018
Later among the works it cites.
“Strata of k k -differentials”
M. Bainbridge et al · 2019
Closest in time.
“Masur-Veech volumes and intersection theory on moduli spaces of Abelian differentials”
D. Chen, M. M“”oller, Adrien Sauvaget and D. Zagier · 2020
Closest in time.

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