Abstract
In this paper, we investigate the geometry of the base complex manifold of an effectively parametrized holomorphic family of stable Higgs bundles over a fixed compact Kähler manifold. The starting point of our study is Schumacher–Toma/Biswas–Schumacher’s curvature formulas for Weil–Petersson-type metrics, in Sect. 2, we give some applications of their formulas on the geometric properties of the base manifold. In Sect. 3, we calculate the curvature on the higher direct image bundle, which recovers Biswas–Schumacher’s curvature formula. In Sect. 4, we construct a smooth and strongly pseudo-convex complex Finsler metric for the base manifold, the corresponding holomorphic sectional curvature is calculated explicitly.
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Notes
Throughout this paper, we use the lowercase Greek letters \(\alpha ,\beta ,\gamma ,\cdots \) and Roman letters \(i,j,k,\cdots \) for the coordinates on X and S, respectively.
References
Ahlfors, L.: Some remarks on Teichmüller’s space of Riemann surfaces. Ann. Math. 74, 171–191 (1961)
Ahlfors, L.: Curvature properties of Teichmüller’s space. J. Anal. Math. 9, 161–176 (1961)
Aubin, T.: Équations du type Monge-Ampère sur les variétés kähleriennes compactes. C. R. Acad. Sci. Paris Sér. A-B 283 (1976), Aiii, A119–A121
Biswas, I., Schumacher, G.: Geometry of moduli spaces of Higgs bundles. Comm. Anal. Geom. 14, 765–793 (2006)
Deng, Y.: On the hyperbolicity of base spaces for maximally variational families of smooth projective varieties. arXiv:1806.01666 to appear in Jour. Eur. Math. Soc
Donagi, R., Pantev, T., Simpson, C.: Direct image in non abelian Hodge theory. arXiv:1612.06388
Donaldson, S.: Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles. Proc. London Math. Soc. 50, 1–26 (1985)
Drezet, J., Narasimhan, M.: Groupe de Picard des variétés de modules de fibrés semi-stables sur les courbes algébriques. Invent. Math. 97, 53–94 (1989)
Fischer, A., Tromba, A.: On the Weil-Petersson metric on Teichmüller space. Trans. Am. Math. Soc. 284, 319–335 (1984)
Geiger, T., Schumacher, G.: Curvature of higher direct image sheaves. Adv. Stud. Pure Math. 74, 171–184 (2017)
Griths, P. (ed.): Topics in transcendental algebraic geometry. Ann. of Math. Studies 106, Princeton, NJ, Princeton Univ. Press (1984)
Heier, G., Wong, B.: Scalar curvature and uniruledness on projective manifolds. Comm. Anal. Geom. 20, 751–764 (2012)
Hitchin, N.: The self-duality equations on a Riemann surface. Proc. London Math. Soc. 55, 59–126 (1987)
Kobayashi, S.: Hyperbolic complex spaces, Grundlehren der Mathematischen Wissenschaften 318. Springer, Berlin (1998)
Mochizuki, T.: Asymptotic behaviour of certain families of harmonic bundles on Riemann surfaces. J. Topol. 9, 1021–1073 (2016)
Narasimhan, M., Sechadri, C.: Stable and unitary vector bundles on a compact Riemann surafce. Ann. Math. 82, 540–564 (1965)
Royden, H.: Intrinsic metrics on Teichmüller space, in Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974, Vol. 2) Canad. Math. Congress, Montreal, QC, pp. 217–221 (1975)
Schumacher, G.: Positivity of relative canonical bundles and applications. Invent. Math. 190, 1–56 (2012)
Schumacher, G., Toma, M.: On the Petersson-Weil metric for the moduli space of Hermitian-Einstein bundles and its curvature. Math. Ann. 293, 101–107 (1992)
Simpson, C.: Constructing of variations of Hodge structure using Yang-Mills theory and applications to uniformization. J. Am. Math. Soc. 1, 867–918 (1988)
Siu, Y.-T.: Curvature of the Weil-Petersson metric in the moduli spaces of compact Kähler-Einstein manifolds of negative first Chern class, In: Contributions to several complex variables, 261–298, ed. P.-M. Wong and A. Howard, Vieweg (1986)
Strominger, A.: Special geometry. Comm. Math. Phys. 133, 163–180 (1990)
To, W.-K., Weng, L.: Curvature of the \(L^2\)-metric on the direct image of a family of Hermitian-Einstein vector bundles. Am. J. Math. 120, 649–661 (1998)
To, W.-K., Yeung, S.-K.: Finsler metrics and Kobayashi hyperbolicity of the moduli spaces of canonically polarized manifolds. Ann. Math. 181, 547–586 (2015)
Uhlenbeck, K., Yau, S.-T.: On the existence of Hermitian-Yang-Mills connections on stable vector bundles. Comm. Pure Appl. Math. 39, 5257–5293 (1986)
Wan, X.: Holomorphic sectional curvature of complex Finsler manifolds. J. Geom. Anal. 29, 194–216 (2019)
Wolpert, S.: Chern forms and the Riemann tensor for the moduli space of curves. Invent. Math. 85, 119–145 (1986)
Yang, X.: Hermitian manifolds with semi-positive holomorphic sectional curvature. Math. Res. Lett. 23, 939–952 (2016)
Yau, S.-T.: On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation I. Comm. Pure Appl. Math. 31, 339–411 (1978)
Acknowledgements
The author P. Huang acknowledges funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC 2181/1 - 390900948 (the Heidelberg STRUCTURES Excellence Cluster), and by Collaborative Research Center/Transregio (CRC/TRR 191; 281071066-TRR 191). The authors would like to express their deep gratitude to the anonymous referee for many valuable suggestions.
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Hu, Z., Huang, P. Curvature Formulas Related to a Family of Stable Higgs Bundles. Commun. Math. Phys. 387, 1491–1514 (2021). https://doi.org/10.1007/s00220-021-04132-9
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DOI: https://doi.org/10.1007/s00220-021-04132-9