Combinatorial degenerations of surfaces and Calabi--Yau threefolds

Update on Modular Non-Rigid Calabi-Yau Threefolds

We review some recent results on the modularity of non-rigid Calabi-Yau threefolds.

Abelian Calabi-Yau threefolds: Néron models and rational points

We study Calabi-Yau threefolds fibered by abelian surfaces, in particular, their arithmetic properties, e.g., Néron models and Zariski density.

Small Resolutions and Non-Liftable Calabi-Yau threefolds

We use properties of small resolutions of the ordinary double point in dimension three to construct smooth non-liftable Calabi-Yau threefolds. In particular, we construct a smooth projective Calabi-Yau threefold over $\F_3$ that does not lift to characteristic zero and a smooth projective Calabi-Yau threefold over $\F_5$ having an obstructed deformation. We also construct many examples of smooth Calabi-Yau algebraic spaces over $\F_p$ that do not lift to algebraic spaces in characteristic zero.

On the modularity of Calabi-Yau threefolds containing elliptic ruled surfaces

We prove that (not necessarily rigid) Calabi-Yau threefolds defined over the rationals which contain sufficiently many elliptic ruled surfcaes are modular (under mild restrictions on the primes of bad reduction). Our proof uses the results of Dieulefait and Manoharmayum who proved modularity of rigid Calabi-Yau threefolds.

The modularity conjecture for rigid Calabi-Yau threefolds over Q

We formulate the modularity conjecture for rigid Calabi-Yau threefolds defined over the field Q of rational numbers. We establish the modularity for the rigid Calabi-Yau threefold arising from the root lattice A_3. Our proof is based on geometric analysis.

A modular quintic Calabi-Yau threefold of level 55

In this note we search the parameter space of Horrocks-Mumford quintic threefolds and locate a Calabi-Yau threefold which is modular, in the sense that the L-function of its middle-dimensional cohomology is associated to a classical modular form of weight 4 and level 55.

Rigid Calabi-Yau Threefolds over Q Are Modular

The proof of Serre's conjecture on Galois representations over finite fields allows us to show, using a method due to Serre himself, that all rigid Calabi-Yau threefolds defined over Q are modular.

普通卡拉比-丘三维簇的塔特猜想

The Tate Conjecture for Ordinary Calabi-Yau Threefolds

该研究利用典范提升理论，证明有限域上普通卡拉比-丘三维簇的积分几何塔特猜想及部分普通卡拉比-丘簇除子类的对应猜想。

Hilbert modularity of some double octic Calabi--Yau threefolds

We exhibit three double octic Calabi--Yau threefolds over the certain quadratic fields and prove their modularity. The non-rigid threefold has two conjugate Hilbert modular forms of weight [4,2] and [2,4] attached while the two rigid threefolds correspond to a Hilbert modular form of weight [4,4] and to the twist of the restriction of a classical modular form of weight 4.

Some Siegel threefolds with a Calabi-Yau model

In the following we describe some examples of Calabi-Yau manifolds, which arise as desingularizations of certain Siegel threefolds. 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.

Modularity of Maschke's octic and Calabi-Yau threefold

We prove the modularity of Maschke's octic and two Calabi-Yau threefolds derived from it as double octic and quotient thereof by the Heisenberg group, as conjectured by Bini and van Geemen. The proofs rely on automorphisms of the varieties and isogenies of K3 surfaces.

Calabi-Yau threefolds over finite fields and torsion in cohomologies

We study various examples of Calabi-Yau threefolds over finite fields. In particular, we provide a counterexample to a conjecture of K. Joshi on lifting Calabi-Yau threefolds to characteristic zero. We also compute the p-adic cohomologies of some Calabi-Yau threefolds constructed by Cynk-van Straten which have remarkable arithmetic properties, as well as those of the Hirokado threefold. These examples and computations answer some outstanding questions of B. Bhatt, T. Ekedahl, van der Geer-Katsura and Patakfalvi-Zdanowicz, and shed new light on the Beauville-Bogomolov decomposition in positive …

Quadratic twists of rigid Calabi-Yau threefolds over $\QQ$

We consider rigid Calabi--Yau threefolds defined over $\QQ$ and the question of whether they admit quadratic twists. We give a precise geometric definition of the notion of a quadratic twists in this setting. Every rigid Calabi--Yau threefold over $\QQ$ is modular so there is attached to it a certain newform of weight 4 on some $Γ_0(N)$. We show that quadratic twisting of a threefold corresponds to twisting the attached newform by quadratic characters and illustrate with a number of obvious and not so obvious examples. The question is motivated by the deeper question of which newforms of weigh…

Modularity of some non-rigid double octic Calabi-Yau threefolds

In this paper we discuss four methods of proving modularity of Calabi--Yau threefolds with $h^{12}=1$: existence of elliptic ruled surfaces inside (Hulek-Verrill), correspondence with a product of an elliptic curve and a K3 surface (Livné-Yui), correspondence with a (modular) rigid Calabi-Yau threefold, and existence of an involution splitting the fourdimensional representation into twodimensional subrepresentations. We apply these methods to prove modularity of 17 out of 18 double octic Calabi-Yau threefolds for which "numerical evidence of modularity" was found in the second Author's thesis.…

Some Siegel threefolds with a Calabi-Yau model II

In the paper [FSM] we described some Siegel modular threefolds which admit a Calabi-Yau model. Using a different method we give in this paper an enlarged list of such varieties that admits a Calabi-Yau model in the following weak sense: there exists a desingularization in the category of complex spaces of the Satake compactification which admits a holomorphic three-form without zeros and whose first Betti number vanishes Basic for our method is the paper [GN] of van Geemen and Nygaard.

A Modular Non-Rigid Calabi-Yau Threefold

We construct an algebraic variety by resolving singularities of a quintic Calabi-Yau threefold. The middle cohomology of the threefold is shown to contain a piece coming from a pair of elliptic surfaces. The resulting quotient is a two-dimensional Galois representation. By using the Lefschetz fixed-point theorem in étale cohomology and counting points on the variety over finite fields, this Galois representation is shown to be modular.