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.

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