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 . 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 and 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 (we need ). For a particular case we produce the equations for the corresponding Siegel threefold.
@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}, pages = {833--850}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 9}, number = {4}, year = {2010}, zbl = {1232.11058}, language = {en}, url = {http://www.numdam.org/item/ASNSP_2010_5_9_4_833_0/} }
TY - JOUR AU - Freitag, Eberhard AU - Manni, Riccardo Salvati TI - Some Siegel threefolds with a Calabi-Yau model JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2010 SP - 833 EP - 850 VL - 9 IS - 4 PB - Scuola Normale Superiore, Pisa UR - http://www.numdam.org/item/ASNSP_2010_5_9_4_833_0/ LA - en ID - ASNSP_2010_5_9_4_833_0 ER -
%0 Journal Article %A Freitag, Eberhard %A Manni, Riccardo Salvati %T Some Siegel threefolds with a Calabi-Yau model %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2010 %P 833-850 %V 9 %N 4 %I Scuola Normale Superiore, Pisa %U http://www.numdam.org/item/ASNSP_2010_5_9_4_833_0/ %G en %F 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, Tome 9 (2010) no. 4, pp. 833-850. http://www.numdam.org/item/ASNSP_2010_5_9_4_833_0/
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