A Global Torelli Theorem for Certain Calabi-Yau Threefolds

Abstract

We establish a global Torelli theorem for the complete family of Calabi-Yau threefolds arising from cyclic triple covers of

{\mathbb{P}}^3

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb {P}}}^3$$\end{document}

branched along six stable hyperplanes.

Keywords

Global Torelli theorem / Calabi-Yau threefolds / Hyperplane arrangements

References

[1.] Allcock D, Carlson JA, Toledo D. The complex hyperbolic geometry of the moduli space of cubic surfaces. J. Algebraic Geom., 2002, 11(4): 659-724,

[2.] Andreotti A. On a theorem of Torelli. Am. J. Math., 1958, 80: 801-828,

[3.] Burns D, Rapoport M. On the Torelli problem for Kählerian K3 surfaces. Ann. Sc. ENS., 1975, 8: 235-274

[4.] Deligne P, Mostow GD. Monodromy of hypergeometric functions and non-lattice integral monodromy. Publ. Math. Inst. Hautes Études Sci., 1986, 63: 5-89,

[5.] Dolgachev, I., Ortland, D.: Point sets in projective spaces and theta functions, Astérisque 165 (1988)

[6.] Fox, R.H.: Covering spaces with singularities, in Lefschetz Symposium, Priceton University Press (1957), pp. 243-262

[7.] Kennichiro S. Generic Torelli theorem for one-parameter mirror families to weighted hypersurfaces. Proc. Japan Acad. Ser. A Math. Sci., 2009, 85(10): 167-170

[8.] Laza R. The moduli space of cubic fourfolds via the period map. Ann. of Math., 2010, 172(2): 673-711,

[9.] Looijenga E. The period map for cubic fourfolds. Invent. Math., 2009, 177: 213-233,

[10.] Looijenga E, Peters C. Torelli theorems for Kähler K3-surfaces. Compos. Math., 1981, 42: 145-186

[11.] McMullen CT. Braid groups and Hodge theory. Math. Ann., 2013, 355: 893-946,

[12.] Pham F. Formules de Picard-Lefschetz généralisées et ramification des intégrales Bull. Soc. Math. France., 1965, 93: 333-367,

[13.] Piatečkii-Shapiro II, Šafarevič IR. A Torelli theorem for algebraic surfaces of type K3. Math. USSR, Izv., 1971, 5: 547-588,

[14.] Rohde, J.: Cyclic coverings, Calabi-Yau manifolds and complex multiplication. Lecture Notes in Math, vol. 1975. Springer, Berlin (2009)

[15.] Sheng M, Xu J, Zuo K. Maximal families of Calabi-Yau manifolds with minimal length Yukawa coupling. Comm. Math. Statist., 2013, 1(1): 73-92,

[16.] Sheng, M., Xu, J., Zuo, K.: The monodromy groups of Dolgachev’s CY moduli spaces are Zariski dense. Adv. Math. 272, 699–742 (2015)

[17.] Usui S. Generic Torelli theorem for quintic-mirror family. Proc. Japan Acad. Ser. A Math. Sci., 2008, 84(8): 143-146,

[18.] Verbitsky M. A global Torelli theorem for hyperkahler manifolds. Duke Math. J., 2013, 162: 2929-2986,

[19.] Voisin, C.: Théorème de Torelli pour les cubiques de ${\mathbb{P}}^5$, Invent. Math. 86, 577-601 (1986). Erratum in Invent. Math. 172, pp. 455-458 (2008)

[20.] Voisin, C.: A generic Torelli theorem for the quintic threefold, in New trends in algebraic geometry (Warwick, 1996), pp. 425-463, Cambridge Univ. Press, Cambridge

[21.] Wang C-L. On the incompleteness of the Weil-Petersson metric along degenerations of Calabi-Yau mainfolds. Math. Res. Lett., 1997, 4: 157-171,

Funding

National Key Research and Development Project(2020YFA0713100); Fundamental Research Funds for the Central Universities(WK3470000018); National Natural Science Foundation of China(11721101); Youth Innovation Promotion Association of the Chinese Academy of Sciences; Anhui Provincial Natural Science Foundation(2008085MA04); Anhui Initiative in Quantum Information Technologies(AHY150200)