Hodge Representations of Calabi-Yau 3-fold Type

Calabi--Yau threefolds with infinitely many divisorial contractions

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.

How to tell you're hearing a Calabi-Yau: Universal variations of Hodge structure and local Schottky relations for Calabi-Yau manifolds

A revised version with a number of corrections and refinements.

Graded 3-Calabi-Yau algebras as Ore extensions of 2-Calabi-Yau algebras

We study a class of graded algebras obtained from Ore extensions of graded Calabi-Yau algebras of dimension 2. It is proved that these algebras are graded Calabi-Yau and graded coherent. The superpotentials associated to these graded Calabi-Yau algebras are also constructed.

A realization for a $\mathbb{Q}$-Hermitian variation of Hodge structure of Calabi-Yau type with real multiplication

We show that the $\mathbb{Q}$-descents of the canonical $\mathbb{R}$-variation of Hodge structure of Calabi-Yau type over a tube domain of type $A$ can be realized as sub-variations of Hodge structure of certain $\mathbb{Q}$-variations of Hodge structure which are naturally associated to abelian varieties of (generalized) Weil type.

Moduli of polarized Calabi-Yau pairs

We prove that the irreducible components of the moduli space of polarized Calabi-Yau pairs are projective.

Examples of Calabi-Yau 3-manifolds with complex multiplication

We give some concrete examples of Calabi-Yau 3-manifolds with complex multiplication.

Polarized Calabi-Yau 3-folds in codimension 4

We construct Calabi-Yau 3-folds as orbifolds embedded in weighted projective space in codimension 4. For each Hilbert series that is realised, there are at least two different components of Calabi-Yau 3-folds.

Hodge numbers for the cohomology of Calabi-Yau type local systems

We use Higgs cohomology to determine the Hodge numbers of the first intersection cohomology group of a local system V arising from the third direct image of a family of Calabi-Yau 3-folds over a smooth, quasi-projective curve. We give applications to Rhode's families of Calabi-Yau 3-folds without MUM.

Calabi-Yau coverings over some singular varieties and new Calabi-Yau 3-folds with Picard number one

This note is a report on the observation that some singular varieties admit Calabi--Yau coverings. As an application, we construct 18 new Calabi--Yau 3-folds with Picard number one that have some interesting properties.

On smooth Calabi-Yau threefolds of Picard number two

We classify all smooth Calabi-Yau threefolds of Picard number two that have a general hypersurface Cox ring.

Rigid realizations of modular forms in Calabi--Yau threefolds

We construct examples of modular rigid Calabi--Yau threefolds, which give a realization of some new weight 4 cusp forms.

On the height of Calabi-Yau varieties in positive characteristic

We study invariants of Calabi-Yau varieties in positive characteristic, especially the height of the Artin-Mazur formal group. We illustrate these results by Calabi-Yau varieties of Fermat and Kummer type.

Geometric transitions between Calabi-Yau threefolds related to Kustin-Miller unprojections

We study Kustin-Miller unprojections between Calabi-Yau threefolds or more precisely the geometric transitions they induce. We use them to connect many families of Calabi-Yau threefolds with Picard number one to the web of Calabi Yau complete intersections. This enables us to find explicit description of a few known families of Calabi-Yau threefolds in terms of equations. Moreover we find two new examples of Calabi-Yau threefolds with Picard group of rank one, described by Pfaffian equations in weighted projective spaces.

Characterization of Calabi--Yau variations of Hodge structure over tube domains by characteristic forms

Sheng and Zuo's characteristic forms are invariants of a variation of Hodge structure. We show that they characterize Gross's canonical variations of Hodge structure of Calabi-Yau type over (Hermitian symmetric) tube domains.

What is the monodromy property for degenerations of Calabi-Yau varieties?

In this survey, we discuss the state of art about the monodromy property for Calabi-Yau varieties. We explain what is the monodromy property for Calabi-Yau varieties and then discuss some examples of Calabi-Yau varieties that satisfy this property. After this recap, we discuss a possible approach to future research in this area.

$d$-semistable Calabi--Yau threefolds of Type III

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 several of them are new examples.

Non-vanishing theorem for lc pairs admitting a Calabi--Yau pair

We prove the non-vanishing conjecture for lc pairs $(X,Δ)$ when $X$ is of Calabi--Yau type.

Notes on Calabi-Yau ordinary differential equations

We investigate the structures of Calabi-Yau differential equations and the relations to the arithmetic of the pencils of Calabi-Yau varieties behind the equations. This provides explanations of some observations and computations in a recent paper by Samol and van Straten.

On the Griffiths-Yukawa coupling length of some Calabi-Yau families

We determine the Griffiths-Yukawa coupling length of the Calabi-Yau universal families coming from hyperplane arrangements.

Calabi-Yau threefolds in positive characteristic

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 cannot be lifted to characteristic zero, although one-dimensional and two-dimensional Calabi-Yau varieties, i.e., elliptic curves and K3 surfaces, are all liftable to characteristic zero. In this respect, Calabi-Yau threefolds in positive characteristic are interesting in view of deformation theory and they are still very mysterious.

Correspondences between modular Calabi--Yau fiber products

We describe two ways to construct finite rational morphisms between fiber products of rational elliptic surfaces with section and some Calabi--Yau manifolds. We use them to construct correspondences between such fiber products that admit at most five singular fibers and rigid Calabi--Yau threefolds.

Calabi-Yau manifolds from pairs of non-compact Calabi-Yau manifolds

Most of Calabi-Yau manifolds that have been considered by physicists are complete intersection Calabi-Yau manifolds of toric varieties or some quotients of product types. Purpose of this paper is to introduce a different and rather new kind of construction method of Calabi-Yau manifolds by pasting two non-compact Calabi-Yau manifolds. We will also in some details explain a curious and mysterious similarity with construction of some $G_2$-manifolds (also called Joyce manifolds), which are base spaces for M-theory.

Boundedness of log pluricanonical representations of log Calabi-Yau pairs in dimension 2

We show the boundedness of B-pluricanonical representations of lc log Calabi-Yau pairs in dimension $2$. As applications, we prove the boundedness of indices of slc log Calabi-Yau pairs up to dimension $3$ and that of non-klt lc log Calabi-Yau pairs in dimension $4$.

Some Calabi-Yau fourfolds verifying Voisin's conjecture

Motivated by the Bloch-Beilinson conjectures, Voisin has made a conjecture concerning the behaviour of zero-cycles on self-products of Calabi-Yau varieties. This note contains some examples of Calabi-Yau fourfolds verifying Voisin's conjecture.

Degenerations and mirror contractions of Calabi-Yau complete intersections via Batyrev-Borisov Mirror symmetry

We show that the dual of the Cayley cone, associated to a Minkowski sum decomposition of a reflexive polytope, contains a reflexive polytope admitting a nef-partition. This nef-partition corresponds to a Calabi-Yau complete intersection in a Gorenstein Fano toric variety degenerating to an ample Calabi-Yau hypersurface in another Fano toric variety. Using the Batyrev-Borisov mirror symmetry construction, we found the mirror contraction of a Calabi-Yau complete intersection to the mirror of the ample Calabi-Yau hypersurface.

Construction of non-Kähler Calabi-Yau manifolds by log deformations

We survey recent construction of non-Kähler Calabi-Yau manifolds by smoothing SNC varieties obtained by non-trivial isomorphisms of strict Calabi-Yau manifolds. We also give a new example by smoothing an SNC 3-fold which are constructed from automorphisms of an abelian surface.

Some examples of Calabi-Yau pairs with maximal intersection and no toric model

It is known that a maximal intersection log canonical Calabi-Yau surface pair is crepant birational to a toric pair. This does not hold in higher dimension: this paper presents some examples of maximal intersection Calabi-Yau pairs that admit no toric model.

Calabi--Yau 3-folds from projective joins of del Pezzo manifolds

In this paper, we will construct new examples of derived equivalent Calabi--Yau 3-folds with Picard number greater than one. We also study their mirror Calabi--Yau manifolds and find that they are given by Schoen's fiber products of suitable rational elliptic modular surfaces.

Derived equivalence of Ito-Miura-Okawa-Ueda Calabi-Yau 3-folds

We prove derived equivalence of Calabi-Yau threefolds constructed by Ito-Miura-Okawa-Ueda as an example of non-birational Calabi-Yau varieties whose difference in the Grothendieck ring of varieties is annihilated by the affine line.

Calabi--Yau threefolds in $\mathbb{P}^6$

We study the geometry of $3$-codimensional smooth subvarieties of the complex projective space. In particular, we classify all quasi-Buchsbaum Calabi--Yau threefolds in projective $6$-space. Moreover, we prove that this classification includes all Calabi--Yau threefolds contained in a possibly singular 5-dimensional quadric as well as all Calabi--Yau threefolds of degree at most $14$ in $\mathbb{P}^6$.

An example of non-Kähler Calabi-Yau fourfold

We show that there exists a non-Kähler Calabi-Yau fourfold, constructing an example by smoothing a normal crossing variety.

Some ball quotients with a Calabi--Yau model

The aim of this note is to treat one distinguished example of a Calabi--Yau variety that appears as a small resolution of a Picard modular variety

Mirror Symmetry for Calabi-Yau complete intersections in Fano toric varieties

Generalizing the notions of reflexive polytopes and nef-partitions of Batyrev and Borisov, we propose a mirror symmetry construction for Calabi-Yau complete intersections in Fano toric varieties.

Calabi-Yau double coverings of Fano-Enriques threefolds

This note is a report on the observation that the Enriques-Fano threefolds with terminal cyclic quotient singularities admit Calabi-Yau threefolds as their double coverings. We calculate the invariants of those Calabi-Yau threefolds when the Picard number is one. It turns out that all of them are new examples.

Arithmetically Gorenstein Calabi-Yau threefolds in $\mathbb{P}^7$

We present a list of arithmetically Gorenstein Calabi-Yau threefolds in $\mathbb{P}^7$ and give evidence that this is a complete list. In particular we construct three new families of arithmetically Gorenstein Calabi-Yau threefolds in $\mathbb{P}^7$ for which no mirror construction is known.

A cascade of determinantal Calabi--Yau threefolds

We study Kustin--Miller unprojections of Calabi--Yau threefolds. As an application we work out the geometric properties of Calabi--Yau threefolds defined as linear sections of determinantal varieties. We compute their Hodge numbers and describe the morphisms corresponding to the faces of their Kähler--Mori cone.

Sections of Calabi-Yau threefolds with K3 fibration

We study sections of a Calabi-Yau threefold fibered over a curve by K3 surfaces. We show that there exist infinitely many isolated sections on certain K3 fibered Calabi-Yau threefolds and the subgroup of the Néron-Severi group generated by these sections is not finitely generated. This also gives examples of K3 surfaces over the function field F of a complex curve with Zariski dense F-rational points, whose geometric model is Calabi-Yau.

A global Torelli theorem for certain Calabi-Yau threefolds

We establish a global Torelli theorem for the complete family of Calabi-Yau threefolds arising from cyclic triple covers of $\mathbb P^3$ branched along stable hyperplane arrangements.

Primitive Calabi-Yau Threefolds

A primitive Calabi-Yau threefold is a non-singular Calabi-Yau threefold which cannot be written as a crepant resolution of a singular fibre of a degeneration of Calabi-Yau threefolds. These should be thought as the most basic Calabi-Yau manifolds; all others should arise through degenerations of these. This paper first continues the study of smoothability of Calabi-Yau threefolds with canonical singularities begun in the author's previous paper, ``Deforming Calabi-Yau Threefolds.'' (alg-geom 9506022). If X' is a non-singular Calabi-Yau threefold, and f:X'->X is a contraction of a divisor to a …

Algebraic characterization of differential operators of Calabi-Yau type

We give an algebraic characterization of Picard-Fuchs operators attached to families of Calabi-Yau manifolds with a point of maximally unipotent monodromy and discuss possibilities for their differential Galois groups.

On the Hodge Metric of the Universal Deformation Space of Calabi-Yau Threefolds

In this paper, we represent the Hodge metric in terms of the Weil-Petersson metric and its Ricci curvature on the moduli spaces of polarized Calabi-Yau threefolds.

Mirror symmetry for Pfaffian Calabi-Yau 3-folds via conifold transitions

In this note we construct conifold transitions between several Calabi-Yau threefolds given by Pfaffians in weighted projective spaces and Calabi-Yau threefolds appearing as complete intersections in toric varieties. We use the obtained results to predict mirrors following ideas of \cite{BCKS, Batsmalltoricdegen}. In particular we consider the family of Calabi--Yau threefolds of degree 25 in $\mathbb{P}^9$ obtained as a transverse intersection of two Grassmannians in their Plucker embeddings.

Calabi-Yau hypersurfaces in the direct product of $\mathbb{P}^{1}$ and inertia groups

We produce the family of Calabi-Yau hypersurfaces $X_{n}$ of $(\mathbb{P}^{1})^{n+1}$ in higher dimension whose inertia group contains non commutative free groups. This is completely different from Takahashi's result \cite{ta98} for Calabi-Yau hypersurfaces $M_{n}$ of $\mathbb{P}^{n+1}$.

带极化的对数Calabi-Yau纤维化的有界性

Boundedness of polarized log Calabi-Yau fibrations

该文研究带对数Calabi-Yau纤维化结构的对数对的有界性，证明在对数典范除子Iitaka体积有界、一般纤维属有界极化对数Calabi-Yau对族的条件下，纤维化全空间在crepant双有理等价下有界。

On autoequivalences of some Calabi--Yau and hyperkähler varieties

We show how one can construct new autoequivalences of some Calabi-Yau and hyperk"ahler varieties by passing from a smooth projective surface to the associated Hilbert scheme of points.

Locally stable degenerations of log Calabi-Yau pairs

We study the birational boundedness of special fibers of log Calabi-Yau fibrations and Fano fibrations. We show that for a locally stable family of Fano varieties or polarised log Calabi-Yau pairs over a curve, if the general fiber satisfies some natural boundedness conditions, then every irreducible component of the special fiber is birationally bounded.

The integral Hodge conjecture for two-dimensional Calabi-Yau categories

We formulate a version of the integral Hodge conjecture for categories, prove the conjecture for two-dimensional Calabi-Yau categories which are suitably deformation equivalent to the derived category of a K3 or abelian surface, and use this to deduce cases of the usual integral Hodge conjecture for varieties. Along the way, we prove a version of the variational integral Hodge conjecture for families of two-dimensional Calabi-Yau categories, as well as a general smoothness result for relative moduli spaces of objects in such families. Our machinery also has applications to the structure of int…

Orbifold Riemann-Roch for threefolds with an application to Calabi-Yau geometry

We prove an orbifold Riemann--Roch formula for a polarized 3--fold (X,D). As an application, we construct new families of projective Calabi--Yau threefolds.

Infinite dimensional families of Calabi-Yau threefolds and moduli of vector bundles

We study noncompact Calabi-Yau threefolds, their moduli spaces of vector bundles and deformation theory. We present Calabi-Yau threefolds that have infinitely many distinct deformations, constructing them explicitily, and describe the effect that such deformations produce on moduli spaces of vector bundles.

Non-liftable Calabi-Yau varieties in characteristic $p \geq 5$

Based on recent work of Totaro, we provide two simple constructions of non-liftable Calabi-Yau varieties in characteristic $p \geq 5$. Our examples are of dimension $2p$ and $2p+1$.

Calabi-Yau流形族与镜像对称

Families of Calabi-Yau manifolds and mirror symmetry

本文从Calabi-Yau流形族与周期积分视角综述镜像对称，强调GKZ超几何级数的性质，并基于中心荷公式证明自反多胞形相关流形对的镜像对称，展示模空间全局描述带来的双有理几何与Fourier-Mukai配对例子。

Semiampleness for Calabi--Yau surfaces in positive and mixed characteristic

In this note we prove the semiampleness conjecture for klt Calabi--Yau surface pairs over an excellent base ring. As applications we deduce that generalised abundance and Serrano's conjecture hold for surfaces. Finally, we study the semiampleness conjecture for Calabi--Yau threefolds over a mixed characteristic DVR.

Non-birational Calabi-Yau threefolds that are derived equivalent

We argue that the existence of genus one fibrations with multisections of high degree on certain Calabi-Yau threefolds implies the existence of pairs of such varieties that are not birational, but are derived equivalent. It also (likely) implies the existence of non-birational counterexamples to the Torelli problem for Calabi-Yau threefolds.

Orbifold Hodge Numbers of Calabi-Yau Hypersurfaces

We identify the twisted sectors of a compact simplicial toric variety. We do the same for a generic nondegenerate Calabi-Yau hypersurface of an $n$-dimensional simplicial Fano toric variety and then explicitly compute $h^{1,1}_{orb}$ and $h^{n-2,1}_{orb}$ for the hypersurface. We give applications to the orbifold string theory conjecture and orbifold mirror symmetry.