Some Siegel threefolds with a Calabi-Yau model
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 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
Freitag, Eberhard 1 ; Manni, Riccardo Salvati 2

1 Mathematisches Institut, Universität Heidelberg, Im Neuenheimer Feld 288, 69120 Heidelberg
2 Dipartimento di Matematica, Sapoienza Università di Roma, Piazzale Aldo Moro, 5, 00185 Roma, Italia
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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, Tome 9 (2010) no. 4, pp. 833-850. http://www.numdam.org/item/ASNSP_2010_5_9_4_833_0/

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