Abstract
The classical Hurwitz spaces, that parameterize compact Riemann surfaces equipped with covering maps to \({\mathbb{P}_1}\) of fixed numerical type with simple branch points, are extensively studied in the literature.We apply deformation theory, and present a study of the Kähler structure of the Hurwitz spaces, which reflects the variation of the complex structure of the Riemann surface as well as the variation of the meromorphic map. We introduce a generalized Weil–Petersson Kähler form on the Hurwitz space. This form turns out to be the curvature of a Quillen metric on a determinant line bundle. Alternatively, the generalized Weil–Petersson Kähler form can be characterized as the curvature form of the hermitian metric on the Deligne pairing of the relative canonical line bundle and the pull back of the anti-canonical line bundle on \({\mathbb{P}_1}\). Replacing the projective line by an arbitrary but fixed curve Y, we arrive at a generalized Hurwitz space with similar properties. The determinant line bundle extends to a compactification of the (generalized) Hurwitz space as a line bundle, and the Quillen metric yields a singular hermitian metric on the compactification so that a power of the determinant line bundle provides an embedding of the Hurwitz space in a projective space.
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References
Artin M.: Algebraization of formal moduli. Global Analysis. Papers in Honor of K. Kodaira 21–71 (1969), and Algebraization of formal moduli II. Existence of modifications. Ann. Math. 91, 88–135 (1970)
Axelsson R., Schumacher G.: Geometric approach to the Weil–Petersson symplectic form. Comment. Math. Helv. 85, 243–257 (2010)
Axelsson R., Schumacher G.: Variation of geodesic length functions in families of Kähler–Einstein manifolds and applications to Teichmüller space. Ann. Acad. Sci. Fenn. Math. 37, 91–106 (2012)
Bismut, J.-M., Gillet, H., Soulé Ch.: Analytic torsion and holomorphic determinant bundles I, II, III. Commun. Math. Phys. 115, 49–78, 79–126, 301–351 (1988)
Biswas, I., Schumacher, G.: Deligne pairing and Quillen metric. Internat. J. Math. 25(14), Art. ID 1450122 (2014)
Biswas I., Schumacher G., Weng L.: Deligne pairing and determinant bundle. Electron. Res. Announc. Math. Sci. 18, 91–96 (2011)
Deligne, P.: Le déterminant de la cohomologie. In: Current trends in arithmetical algebraic geometry, Proc. Summer Res. Conf., Arcata/Calif. 1985, Contemp. Math., vol. 67, pp. 93–177 (1987)
Fujiki A., Schumacher G.: The moduli space of extremal compact Kähler manifolds and generalized Weil–Petersson metrics. Publ. Res. Inst. Math. Sci. 26, 101–183 (1990)
Fulton W.: Hurwitz schemes and irreducibility of moduli of algebraic curves. Ann. Math. 90, 542–575 (1969)
Harris, J., Graber, T., Starr, J.: A note on Hurwitz schemes of covers of a positive genus curve. arxiv:math/0205056v1
Hurwitz A.: Über Riemann’sche Flächen mit gegebenen Verzweigungspunkten. Math. Ann. 39, 1–61 (1891)
Kani E.: Hurwitz spaces of genus 2 covers of an elliptic curve. Collect. Math. 54, 1–51 (2003)
Looijenga E.: Smooth Deligne–Mumford compactifications by means of Prym level structures. J. Algbr. Geom. 3, 283–293 (1994)
Palamodov V.P.: Deformations of complex spaces. Russ. Math. Surv. 31, 129–197 (1976)
Palamodov V.P.: Deformations of complex spaces. Uspekhi Mat. Nauk 31, 129–194 (1976)
Pikaart, M., de Jong, A.J.: Moduli of curves with non-abelian level structure. In: The Moduli Space of Curves (Texel Island, 1994). In: Progr. Math., vol. 129, pp. 483–509, Birkhäuser Boston, Boston, MA (1995)
Romagny, M., Wewers, S.: Hurwitz spaces. In: Groupes de Galois arithmétiques et différentiels. In: Sémin. Congr., vol. 13, pp. 313–341, Soc. Math. France, Paris (2006)
Schumacher, G.: Positivity of relative canonical bundles and applications. Invent. Math. 190, 1–56 (2012); and Erratum to: Positivity of relative canonical bundles and applications. Invent. Math. 192, 253–255 (2013)
Siu, Y.-T.: Curvature of the Weil–Petersson metric in the moduli space of compact Kähler–Einstein manifolds of negative first Chern class. Contributions to several complex variables. In: Stoll, H.W. (eds.) Proceedings of the Conference Complex Analysis, Notre Dame/Indiana 1984. Aspects Math. vol. E9, 261–298 (1986)
Wolpert S.P.: Chern forms and the Riemann tensor for the moduli space of curves. Invent. Math. 85, 119–145 (1986)
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Axelsson, R., Biswas, I. & Schumacher, G. Kähler structure on Hurwitz spaces. manuscripta math. 147, 63–79 (2015). https://doi.org/10.1007/s00229-015-0738-6
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DOI: https://doi.org/10.1007/s00229-015-0738-6