Some Siegel threefolds with a Calabi-Yau model
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 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 ${\Gamma }_{2,0}\left[2\right]$. 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
@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, Serie 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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