On Siegel threefolds with a projective Calabi–Yau model

In the papers [FS1], [FS2] 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. His comment was a starting motivation for this paper. We mention that 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. For example, we obtain a rigid Calabi–Yau manifold with Euler number 4. Basic for our examples is a certain complete intersection X of four quadrics introduced the paper [GN] of van Geemen and Nygaard: