Geodesic
Kähler Geometry on Hurwitz Spaces
Documenta mathematica, Tome 23 (2018), pp. 1829-1861.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

The classical Hurwitz space $\mathcal{H}^{n,b}$ is a fine moduli space for simple branched coverings of the Riemann sphere $\mathbb{P}^1$ 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 $\mathcal{H}^{n,b} \to \mathcal{M}_g$.
Classification : 32G05, 32G15, 14H15
Keywords: deformations of holomorphic maps, Hurwitz spaces, Weil-Petersson metric 
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     author = {Naumann, Philipp},
     title = {K\"ahler {Geometry} on {Hurwitz} {Spaces}},
     journal = {Documenta mathematica},
     pages = {1829--1861},
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     volume = {23},
     year = {2018},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DOCMA_2018__23__a11/}
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Naumann, Philipp. Kähler Geometry on Hurwitz Spaces. Documenta mathematica, Tome 23 (2018), pp. 1829-1861. http://geodesic.mathdoc.fr/item/DOCMA_2018__23__a11/