Hostname: page-component-848d4c4894-sjtt6 Total loading time: 0 Render date: 2024-06-28T11:55:19.934Z Has data issue: false hasContentIssue false
  • English
  • Français

Geometry and Arithmetic of Certain Double Octic Calabi–Yau Manifolds

Published online by Cambridge University Press:  20 November 2018

Sławomir Cynk  and
Christian Meyer
Sławomir Cynk
Affiliation:
Instytut Matematyki, Uniwersytetu Jagiellońskiego, ul. Reymonta 4, 30–059 Kraków, Poland e-mail: cynk@im.uj.edu.pl
Christian Meyer
Affiliation:
Fachbereich Mathematik und Informatik, Johannes Gutenberg-Universität, Staudingerweg 9, D–55099 Mainz, Germany e-mail: cm@mathematik.uni-mainz.de
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study Calabi–Yau manifolds constructed as double coverings of ${{\mathbb{P}}^{3}}$ branched along an octic surface. We give a list of 87 examples corresponding to arrangements of eight planes defined over $\mathbb{Q}$. The Hodge numbers are computed for all examples. There are 10 rigid Calabi–Yau manifolds and 14 families with ${{h}^{1,2}}\,=\,1.$ The modularity conjecture is verified for all the rigid examples.

Keywords

14G1014J32Calabi–Yaudouble coveringsmodular forms
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 48 , Issue 2 , 01 June 2005 , pp. 180 - 194
Copyright
Copyright © Canadian Mathematical Society 2005

References

[1] Cynk, S., Double coverings of octic arrangements with isolated singularities. Adv. Theor. Math. Phys. 3(1999), 217–225.Google Scholar
[2] Cynk, S., Cohomologies of a double covering of a non–singular algebraic 3-fold. Math. Z. 240 (2002), 731–743.Google Scholar
[3] Cynk, S. and van Straten, D., Infinitesimal deformations of smooth algebraic varieties. To appear in Math. Nachrichten. Preprint (2003), AG/0303329.Google Scholar
[4] Cynk, S. and Szemberg, T., Double covers and Calabi–Yau varieties. Banach Center Publ. 44, Polish Acad. Sci., Warsaw, 1998, pp. 93–101.Google Scholar
[5] Dieulefait, L. and Manoharmayum, J., Modularity of rigid Calabi-Yau threefolds over ℚ. In: Calabi-Yau varieties and mirror symmetry. Proceedings of the Workshop on Arithmetic, Geometry and Physics around Calabi-Yau Varieties andMirror Symmetry (Toronto, 2001). Yui, Noriko and Lewis, James D., eds. Fields Institute Communications, 38, American Mathematical Society, Providence, RI, 2003, pp. 159–166.Google Scholar
[6] Esnault, H. and Viehweg, E., Lectures on vanishing theorems. DMV Seminar 20, Birkhäuser Verlag, Basel, 1992.Google Scholar
[7] Greuel, G.-M., Pfister, G., and Schönemann, H.. Singular 2.0. A Computer Algebra System for Polynomial Computations. Centre for Computer Algebra, University of Kaiserslautern (2001). http://www.singular.uni-kl.de.Google Scholar
[8] Hulek, K., Spandaw, J., van Geemen, B., and van Straten, D., The modularity of the Barth–Nieto quintic and its relatives. Adv. Geom. 1(2001), 263–289.Google Scholar
[9] Meyer, C., A Dictionary of Modular Threefolds. Thesis, Mainz, 2005.Google Scholar
[10] Saito, M. and Yui, N., The modularity conjecture for rigid Calabi–Yau threefolds over Q. J. Math. Kyoto Univ. 41(2001), 403–419.Google Scholar
[11] Stein, W. A., Modular forms database. http://modular.fas.harvard.edu/.Google Scholar
[12] Szendröi, B., Calabi–Yau threefolds with a curve of singularities and counterexamples to the Torelli problem. Internat. J. Math. 11(2000), 449–459.Google Scholar
[13] Szendröi, B., Calabi–Yau threefolds with a curve of singularities and counterexamples to the Torelli problem. II. Math. Proc. Cambridge Philos. Soc. 129(2000), 193–204.Google Scholar
[14] Verrill, H. A., The L-series of certain rigid Calabi–Yau threefolds. J. Number Theory 81(2000), 310–334.Google Scholar
[15] Wilson, P. M. H., The Kähler cone on Calabi–Yau threefolds. Invent. Math. 107(1992), 561–583.Google Scholar