Calabi-Yau three-folds and moduli of abelian surfaces II

Abstract

Abstract. We give explicit descriptions of the moduli spaces of abelian surfaces with polarizations of type (1, d), for d = 12, 14, 16, 18 and 20. More precisely, in each case we show that a certain choice of moduli space of such abelian surfaces with a partial level structure can be described explicitly and is unirational, and in some cases rational. These moduli spaces with partial level structure are covers of the ordinary moduli spaces, so the Kodaira dimension of the ordinary moduli spaces in these cases is −∞. In addition, we give a few new examples of Calabi-Yau three-folds fibred in abelian surfaces. In the case of d = 20, such Calabi-Yau three-folds play a key role in the description of the abelian surfaces.

A cascade of determinantal Calabi–Yau threefolds

“…We shall describe a natural relation between the threefold X and a quintic in P 4 , that closes our cascade. As it was observed in the proof of [GP,Thm. 7.4], the smooth Calabi-Yau threefold X defined by the 3 × 3 minors of a symmetric 5 × 5 matrix in P 9 admits an unramified covering being a Calabi-Yau threefold.…”

Computations of Spaces of Paramodular Forms of General Level

“…Gritsenko [8] proved that S 3 (K(N )) = {0} for N > 36, so our Table 2 completely enumerates the twenty cases of dimension zero. For sixteen of these cases the rationality or unirationality of the moduli space is known, compare the work of Gross and Popescu [14]; our vanishing results for the four cases N = 15, 24, 30, 36 are new. For composite N > 4, all the nonzero dimensions are new.…”

A counterexample to the birational Torelli problem for Calabi-Yau threefolds

“…There are several counterexamples to (2) ⇒ (1): The Pfaffian-Grassmannian pairs [7,26,32], Hosono and Takagi's example [15], and the example studied by Gross-Popescu [13], Bak [6] and Schnell [33]. The example of this paper is the only one we know of for which the varieties are deformation equivalent.…”