Abstract
We prove the existence of global sections trivializing the Hodge bundles on the Hodge metric completion space of the Torelli space of Calabi–Yau manifolds, a global splitting property of these Hodge bundles. We also prove that a compact Calabi–Yau manifold can not be deformed to its complex conjugate. These results answer certain open questions in the subject. A general result about certain period map to be bi-holomorphic from the Hodge metric completion space of the Torelli space of Calabi–Yau type manifolds to their period domains is proved and applied to the cases of K3 surfaces, cubic fourfolds, and hyperkähler manifolds.
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References
Chen, X., Liu, K., Shen, Y.: Hodge metric completion of the Teichmüller space of Calabi-Yau manifolds. arXiv:1305.0231
Chen, X., Liu, K., Shen, Y.: Global Torelli theorem for projective manifolds of Calabi-Yau type. arXiv:1205.4207
Chen, X., Liu, K., Shen, Y.: Applications of the affine structures on the Teichmüller spaces. arXiv:1402.5570
Griffiths, P., Schmid, W.: Locally homogeneous complex manifolds. Acta Math. 123, 253–302 (1969)
Griffiths, P., Wolf, J.: Complete maps and differentiable coverings. Michigan Math. J. 10(3), 253–255 (1963)
Harish-Chandra.: Representation of semisimple Lie groups VI. Am. J. Math. 78, 564–628 (1956)
Helgason, S.: Differential Geometry, Lie Groups, and Symmetric Spaces. Academic Press, New York (1978)
Hörmander, L.: An Introduction to Complex Analysis in Several Variables. Van Nostrand, Princeton, NJ (1973)
Huybrechts, D.: Compact hyperkähler manifolds: basic results. Invent. Math. 135, 63–113 (1999)
Kharlamov, V., Kulikov, V.: Deformation inequivalent complex conjugated complex structures and applications. Turk. J. Math. 26, 1–25 (2002)
Liu, K., Rao, S., Yang, X.: Quasi-isometry and deformations of Calabi-Yau manifolds. arxiv:1207.1182
Matsushima, Y.: Affine structure on complex manifold. Osaka J. Math. 5, 215–222 (1968)
Mok, N.: Metric Rigidity Theorems on Hermitian Locally Symmetric Manifolds. Series in Pure Mathematics, vol. 6. World Scientific, Singapore (1989)
Mok, N., Yau, S.T.: Completeness of the Kähler-Einstein metric on bounded domains and the characterization of domain of holomorphi by curvature condition. Symposia in Pure Math. The Mathematical Heritage of Henri Poincaré, vol. 39. AMS (1983), pp. 41–60
Popp, H.: Moduli theory and classification theory of algebraic varieties. Lecture Notes in Mathemathics, vol. 620. Springer, Berlin (1977)
Piatetski-Shapiro, I.I., Shafarevich, I.R.: A Torelli theorem for algebraic surfaces of type K\(3\). USSR Izv. Ser. Math. 5, 547–588 (1971)
Schmid, W.: Variation of Hodge structure: the singularities of the period mapping. Invent. Math. 22, 211–319 (1973)
Sugiura, M.: Conjugate classes of Cartan subalgebra in real semi-simple Lie algebras. J. Math. Soc. Jpn 11, 374–434 (1959)
Sugiura, M.: Correction to my paper: Conjugate classes of Cartan subalgebra in real semi-simple Lie algebras. J. Math. Soc. Jpn 23, 374–383 (1971)
Szendröi, B.: Some finiteness results for Calabi-Yau threefolds. J. Lond. Math. Soc. Sec. Ser. 60, 689–699 (1999)
Tian, G.: Smoothness of the universal deformation space of compact Calabi-Yau manifolds and its Petersson-Weil metric. Mathematical aspects of string theory(San Diego, Calif.), (1986), World Sci. Publishing, Singapore. Adv. Ser. Math. Phys. 1, 629–646 (1987)
Todorov, A.N.: The Weil-Petersson geometry of the moduli space of \({\rm { SU}}(n\ge 3)\) (Calabi-Yau) manifolds. I. Commun. Math. Phys. 126, 325–346 (1989)
Verbitsky, M.: A global Torelli theorem for hyperkähler Manifolds. arXiv:0908.4121
Viehweg, E.: Quasi-projective moduli for polarized manifolds. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 30. Springer, Berlin (1995)
Voisin, C.: Hodge Theory and Complex Algebraic Geometry I. Cambridge Universigy Press, New York (2002)
Xu, Y.: Lie Groups and Hermitian Symmetric Spaces. Science Press in China (2001) (in Chinese)
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Liu, K., Shen, Y., Chen, X. (2016). Applications of the Affine Structures on the Teichmüller Spaces. In: Futaki, A., Miyaoka, R., Tang, Z., Zhang, W. (eds) Geometry and Topology of Manifolds. Springer Proceedings in Mathematics & Statistics, vol 154. Springer, Tokyo. https://doi.org/10.1007/978-4-431-56021-0_3
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DOI: https://doi.org/10.1007/978-4-431-56021-0_3
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