Geodesic
Kähler Geometry on Hurwitz Spaces
Documenta mathematica, Tome 23 (2018), pp. 1829-1861
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The classical Hurwitz space Hn,b is a fine moduli space for simple branched coverings of the Riemann sphere P1 by compact hyperbolic Riemann surfaces. In the article we study a generalized Weil-Petersson metric on the Hurwitz space, which was introduced in [R. Axelsson et al., Manuscr. Math. 147, No. 1–2, 63–79 (2015; Zbl 1319.32012)]. For this purpose, Horikawa's deformation theory of holomorphic maps is refined in the presence of hermitian metrics in order to single out distinguished representatives. Our main result is a curvature formula for a subbundle of the tangent bundle on the Hurwitz space obtained as a direct image. This covers the case of the curvature of the fibers of the natural map Hn,b→Mg​.
DOI : 10.4171/dm/661
Classification : 14H15, 32G05, 32G15
Mots-clés : Weil-Petersson metric, deformations of holomorphic maps, Hurwitz spaces 
@article{10_4171_dm_661,
     author = {Philipp Naumann},
     title = {K\"ahler {Geometry} on {Hurwitz} {Spaces}},
     journal = {Documenta mathematica},
     pages = {1829--1861},
     year = {2018},
     volume = {23},
     doi = {10.4171/dm/661},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/661/}
}
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Philipp Naumann. Kähler Geometry on Hurwitz Spaces. Documenta mathematica, Tome 23 (2018), pp. 1829-1861. doi: 10.4171/dm/661

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