We identify the weight four newform of a modular Calabi-Yau manifold studied by Hulek and Verrill. The main obstacle is that this Calabi-Yau manifold is not rigid and has bad reduction at prime 13. Replacing the original fiber product of elliptic fibrations with a fiberwise Kummer construction we reduce the problem to studying the modularity of a rigid Calabi-Yau manifold with good reduction at primes p ≥ 5.
In this brief note we give a proof that a certain family of Fano 4-folds, described below, is complex (locally) complete and effectively parametrized in the sense of Kodaira-Spencer [Ko-Sp].
We show that the moduli space of polarized irreducible symplectic manifolds of -type, of fixed polarization type, is not always connected. This can be derived as a consequence of Eyal Markman’s characterization of polarized parallel-transport operators of -type.
We describe explicitly the moduli spaces of polystable holomorphic structures with on a rank two vector bundle with and for all minimal class VII surfaces with and with respect to all possible Gauduchon metrics . These surfaces are non-elliptic and non-Kähler complex surfaces and have recently been completely classified. When is a half or parabolic Inoue surface, is always a compact one-dimensional complex disc. When is an Enoki surface, one obtains a complex disc with finitely...