Related papers: Examples of Calabi-Yau 3-manifolds with complex mu…
We construct several examples of higher-dimensional Calabi-Yau manifolds and prove their modularity.
We construct families of Calabi-Yau manifolds with dense set of complex multiplication fibers in an arbitrary dimension. We will also give explicite examples of complex multiplication fibers. For this construction we use families of curves…
We construct examples of modular rigid Calabi--Yau threefolds, which give a realization of some new weight 4 cusp forms.
We develop some methods to construct normal crossing varieties whose dual complexes are two-dimensional, which are smoothable to Calabi--Yau threefolds. We calculate topological invariants of smoothed Calabi--Yau threefolds and show that…
In this note we provide examples of families of Calabi-Yau 3-manifolds over Shimura varieties, whose mirror families contain subfamilies over Shimura varieties. Therefore these original families and subfamilies on the mirror side contain…
We study Calabi--Yau 3-folds with infinitely many divisorial contractions. We also suggest a method to describe Calabi--Yau 3-folds with the infinite automorphism group.
This paper presents five new examples of modular rigid Calabi-Yau threefolds arising from the modular elliptic surface of level 6. Explicit correspondences to newforms of weight 4 and level 10, 17, 21, and 73 are given.
I point out some very elementary examples of special Lagrangian tori in certain Calabi-Yau manifolds that occur as hypersurfaces in complex projective space. All of these are constructed as real slices of smooth hypersurfaces defined over…
Birational Calabi-Yau threefolds in the same deformation family provide a `weak' counterexample to the global Torelli problem, as long as they are not isomorphic. In this paper, it is shown that deformations of certain desingularized…
We give some examples of Calabi-Yau 3-folds with $\rho=1$, defined over $\mathbb{Q}$ and constructed as 4-codimensional subvarieties of $\mathbb{P}^7$ via commutative algebra methods. We explain how to deduce their Hodge diamond and top…
We construct examples of complete Calabi-Yau metrics on smoothings of 3-dimensional Calabi-Yau cones that are not products of lower-dimensional Calabi-Yau cones and that have orbifold singularities away from the vertex.
We construct non-K\"ahler Calabi-Yau manifolds of dimension $\ge$ 4 with arbitrarily large 2nd Betti numbers by smoothing normal crossing varieties. The examples have K3 fibrations over smooth projective varieties and their algebraic…
We study Calabi-Yau manifolds constructed as double covers of ${\mathbb P}^3$ branched along an octic surface. We give a list of 85 examples corresponding to arrangements of eight planes defined over ${\mathbb Q}$. The Hodge numbers are…
We give a counterexample of Morrison's cone conjecture for a strict Calabi-Yau threefold.
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…
In this expository paper, we illustrate two explicit methods which lead to special $L$-values of certain modular forms admitting complex multiplication (CM), motivated in part by properties of $L$-functions obtained from Calabi-Yau…
In this note, an overview of Calabi-Yau varieties in positive characteristic is presented. Although Calabi-Yau varieties in characteristic zero are unobstructed, there are examples of Calabi-Yau threefolds in positive characteristic which…
We propose a conceptual framework that leads to an abstract characterization for the exact solvability of Calabi-Yau varieties in terms of abelian varieties with complex multiplication. The abelian manifolds are derived from the cohomology…
We present the first explicit examples of a rational threefold and a Calabi-Yau threefold, admitting biregular automorphisms of positive entropy not preserving any dominant rational maps to lower positive dimensional varieties. The most…
We prove modularity for a huge class of rigid Calabi-Yau threefolds over $\Q$. In particular we prove that every rigid Calabi-Yau threefold with good reduction at 3 and 7 is modular.