# Some Siegel Threefolds with a Calabi-Yau Model II

Freitag, Eberhard;Manni, Riccardo Salvati

• Accepted : 2011.07.05
• Published : 2013.06.23
• 16 11

#### Abstract

In a previous paper, we described some Siegel modular threefolds which admit a Calabi-Yau model. Using a different method we give in this paper an enlarged list of such varieties. Basic for this method is a paper of van Geemen and Nygaard. They study a variety $\mathcal{X}$ that is the complete intersection of four quadrics in $\mathbb{P}^7(\mathbb{C})$. This is biholomorphic equivalent to the Satake compactification of $\mathcal{H}_2/{\Gamma}^{\prime}$ for a certain subgroup ${\Gamma}^{\prime}{\subset}Sp(2,\mathbb{Z})$ and it will be the starting point of our investigation. It has been pointed out that a (projective) small resolution of this variety is a rigid Calabi-Yau manifold $\tilde{\mathcal{X}}$. Then we will consider the action of quotients of modular groups on $\mathcal{X}$ and study possible resolutions that admits a Calabi-Yau model in the category of complex spaces.

#### Keywords

Calabi-Yau;Siegel modular varieties

#### References

1. L. Borisov, Z. Hua, On Calabi-Yau threefolds with large nonabelian fundamental groups, Proc. Amer. Math. Soc. 136, (2008), 1549-1551
2. W. Bosma, J. Cannon, C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (3-4), (1997), 235-265 https://doi.org/10.1006/jsco.1996.0125
3. T. Bridgeland, A. King, M. Reid, The McKay correspondence as an equivalence of derived categories, J. Amer. Math. Soc. 14, (2001), 535-554 https://doi.org/10.1090/S0894-0347-01-00368-X
4. S. Cynk, C. Meyer, Modular Calabi-Yau Threefolds of level eight, Internat. J. Math. 18, no. 3, (2007), 331-347 https://doi.org/10.1142/S0129167X07004126
5. S. Cynk, E. Freitag, R.Salvati Manni, The geometry and arithmetic of a Calabi-Yau Siegel threefold, Int. Jour. Math. 29 (2011), 1561-1583
6. R. Davis, Quotients of the conifold in compact Calabi-Yau threefolds, and new topo- logical transitions, eprint arXiv: 0911.0708 [hep-th] (2009)
7. E. Freitag, R.Salvati Manni, Some Siegel threefolds with a Calabi-Yau model, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) Vol IX (2010), 833-850
8. E. Freitag, Siegelsche Modulfunktionen, Grundlehren der mathematischen Wissenschaften, Bd. 254. Berlin Heidelberg New York: Springer (1983)
9. E. Freitag, Singular modular forms and theta relations, Lecture notes in Math. 1487, Springer-Verlag, Berlin Heidelberg New York (1991)
10. J. Kollar, Flops, Nagoya Math. J. 113, (1989), 15-36 https://doi.org/10.1017/S0027763000001240
11. B. van Geemen, N. O. Nygaard, On the geometry and arithmetic of some Siegel modular threefolds, Journal of Number Theory 53, (1995), 45-87 https://doi.org/10.1006/jnth.1995.1078
12. M. Reid, La correspondence de McKay, Seminaire Bourbaki 1999/2000. Asterisque No. 276,(2002), 53-72
13. S. Roan, Minimal resolutions of Gorenstein orbifolds in dimension three. Topology 35,(1996), 489-508 https://doi.org/10.1016/0040-9383(95)00018-6
14. R. Friedmann, Simultaneous resolution of threefold double points, Mathematische Ann. 274, (1986), 671-689 https://doi.org/10.1007/BF01458602

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