Some Siegel threefolds with a Calabi-Yau model
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Volume 9 (2010) no. 4, p. 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 Γ 2,0 [2]. 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 Γ 2 [2] and Γ 2,0 [2] 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
@article{ASNSP_2010_5_9_4_833_0,
     author = {Freitag, Eberhard and Manni, Riccardo Salvati},
     title = {Some Siegel threefolds with a Calabi-Yau model},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 9},
     number = {4},
     year = {2010},
     pages = {833-850},
     zbl = {1232.11058},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2010_5_9_4_833_0}
}
Freitag, Eberhard; Manni, Riccardo Salvati. Some Siegel threefolds with a Calabi-Yau model. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Volume 9 (2010) no. 4, pp. 833-850. http://www.numdam.org/item/ASNSP_2010_5_9_4_833_0/

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