Some Siegel threefolds with a Calabi-Yau model

Freitag, Eberhard; Manni, Riccardo Salvati

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 9 (2010) no. 4, pp. 833-850.

We describe some examples of projective Calabi-Yau manifolds which arise as desingularizations of Siegel threefolds. There is a certain explicit product of six theta constants which defines a cusp form of weight three for a certain subgroup of index two of the Hecke group ${\Gamma }_{2,0}\left[2\right]$. This form defines an invariant differential form for this group and for any subgroup of it. We study the question whether the Satake compactification for such a subgroup admits a projective desingularization on which this differential form is holomorphic and without zeros. Then this desingularization is a Calabi-Yau manifold. We shall prove: For any group between ${\Gamma }_2\left[2\right]$ and ${\Gamma }_{2,0}\left[2\right]$ there exists a subgroup of index two which produces a (projective) Calabi-Yau manifold. The proof rests on a detailed study of this cusp form and on Igusa’s explicit desingularization of the Siegel threefolds with respect to the principal congruence subgroup of level $q>2$ (we need $q=4$). For a particular case we produce the equations for the corresponding Siegel threefold.

Classification : 11F46, 14J32

[1] W. Barth and I. Nieto, Abelian surfaces of type $(1,3)$ and quartic surfaces with 16 skew lines, J. Algebraic Geom. 3 (1994), 173–222. | Zbl

[2] T. Bridgeland, A. King and M. Reid, The McKay correspondence as an equivalence of derived categories, J. Amer. Math. Soc. 14 (2001), 535–554. | Zbl

[3] E. Freitag, “Siegelsche Modulfunktionen”, Grundlehren der mathematischen Wissenschaften, Bd., Vol. 254, Berlin Heidelberg New York: Springer, 1983. | Zbl

[4] V. Gritsenko and F. Cléry, The Siegel modular form of genus 2 with the simplest divisor, arXiv: 0812.3962 (2008). | Zbl

[5] V. Gritsenko and K. Hulek, The modular form of the Barth-Nieto quintic, Internat. Math. Res. Notices 17 (1999). | Zbl

[6] B. Van Geemen and N. O. Nygaard, On the geometry and arithmetic of some Siegel modular threefolds, J. Number Theory 53 (1995), 45–87. | Zbl

[7] B. Van Geemen and D. Van Straten, The cusp form of weight 3 on ${\Gamma }_2(2,4,8)$, Math. Comp. 61 (1993), 849–872. | Zbl

[8] B. Van Geemen, K. Hulek, J. Spandau and D. Van Straten, The modularity of the Barth-Nieto quintic and its relatives, Adv. Geom. 1 (2001), 263–289. | EuDML | Zbl

[9] I. Igusa, On Siegel modular forms of genus two II, Amer. J. Math. 86 (1964), 392–412. | Zbl

[10] I. Igusa, A desingularization problem in the theory of Siegel modular functions, Math. Ann. 168 (1967), 228–260. | EuDML | Zbl

[11] Y. Ito, Gorenstein quotient singularities of monomial type in dimension three, J. Math. Sci. Univ. Tokyo 2 (1995), 419–440. | Zbl

[12] R. Lee, and S. H. Weintraub, The Siegel modular variety of degree two and level four, Mem. Amer. Math. Soc. 631 (1998), 3–58. | Zbl

[13] M. Reid, Minimal models of canonical 3-folds, In: “Algebraic Varieties and Analytic Varieties” (Tokyo, 1981), Adv. Stud. Pure Math., Vol. 1, Amsterdam: North-Holland, 131–180. | Zbl

[14] M. Reid, “La correspondence de McKay”, Séminaire Bourbaki 1999/2000, Astérisque, Vol. 276, 2002, 53–72. | EuDML

[15] B. Runge, On Siegel modular form, part I, J. Reine Angew. Math. 436 (1993), 57–85. | EuDML | Zbl

[16] B. Runge, On Siegel modular forms, part II, Nagoya Math. J. 138 (1995), 179–197. | Zbl

[17] R. Salvati Manni, Thetanullwerte and stable modular forms, Amer. J. Math. 111 (1989), 435–455. | Zbl