Some Siegel threefolds with a Calabi-Yau model

2009 Introduction In the following we describe some examples of Calabi-Yau manifolds that arise as desingularizations of certain Siegel threefolds. Here by a Calabi-Yau mani-fold we understand a smooth complex projective variety which admits a holo-morphic differential form of degree three without zeros and such that the first Betti number is zero. This differential form is unique up to a constant factor, and we call it the Calabi-Yau form. Our interest in this subject is influenced by work of Gritsenko and many discussions with him. The first Siegel modular variety with a Calabi-Yau model and the essentially only one up to now has been discovered by Barth and Nieto. They showed that the " Nieto quintic " {x ∈ P 5 (C), σ 1 (x) = σ 5 (x) = 0}, where σ i denote the elementary symmetric polynomials, has a Calabi-Yau model and they derived that the Siegel modular variety A 1,3 (2) of polarization type (1, 3) and a certain level two structure has a Calabi-Yau model. Since the Jacobian of a symplectic substitution is det(CZ + D) −3 , the Calabi-Yau three-form produces a modular form of weight three and this must be a cusp form, since it survives on a non-singular model as a holomorphic differential form ([Fr], III.2.6). In the paper [GH] Gritsenko and Hulek gave a direct construction of this modular form and obtained a new proof for the fact that A 1,3 (2) has a Calabi-Yau model. We also refer to [GHSS] for further investigations. Besides this example and some small extensions of this group with the same three-form no other examples of Siegel threefolds with Calabi-Yau model seem to be known. Gritsenko raised the problem of determing all Siegel threefolds which admit a Calabi-Yau model. As we mentioned already, such a threefold will produce a certain cusp form of weight three for the considered modular group Γ. This cusp form has very restrictive properties. Since the induced differential form should have no zero at least at the regular locus of the quotient H 2 /Γ, all zeros of the form must be contained in the ramification of H 2 → H 2 /Γ. Gritsenko gave examples of such modular forms: we refer to the paper [GC] which contains some systematic study of them. One example that Gritsenko and Cléry describe, is the form ∇ 3 , which 2 Some Siegel threefolds with …