Communications in Number Theory and Physics

Volume 5 (2011)

Number 3

On Siegel three-folds with a projective Calabi–Yau model

Pages: 713 – 750



Eberhard Freitag (Mathematisches Institut, Ruprecht-Karls-Universität, Heidelberg, Germany)

Riccardo Salvati Manni (Dipartimento di Matematica, Sapienza Università di Roma, Italy)


In two recent papers we described some Siegel modular threefolds which admit a weak Calabi–Yau model. Not all of them admit a projective model. In fact, Bert van Geemen, in a private communication, pointed out a significative example which cannot admit a projective model. A weak Calabi–Yau threefold is projective if, and only if, it admits a Kaehler metric. The purpose of this paper is to exhibit criteria for the projectivity, to treat several examples, and to compute their Hodge numbers. Some of these Calabi–Yau manifolds seem to be new. We obtain a rigid Calabi–Yau manifold with Euler number 4.

Published 16 December 2011