In this article we introduce an ordinary differential equation associated to the one parameter family of Calabi-Yau varieties which is mirror dual to the universal family of smooth quintic three folds. It is satisfied by seven functions written in the $q$-expansion form and the Yukawa coupling turns out to be rational in these functions. This is a generalization of the Ramanujan differential equation satisfied by three Eisenstein series.
This paper presents the current status on modularity of Calabi-Yau varieties since the last update in 2003. We will focus on Calabi-Yau varieties of dimension at most three. Here modularity refers to at least two different types: arithmetic modularity and geometric modularity. These will include: (1) the modularity (automorphy) of Galois representations of Calabi-Yau varieties (or motives) defined over Q or number fields, (2) the modularity of solutions of Picard--Fuchs differential equations of families of Calabi-Yau varieties, and mirror maps (mirror moonshine), (3) the modularity of generat…
We consider certain $K3$-fibered Calabi--Yau threefolds. One class of such Calabi--Yau threefolds are constructed by Hunt and Schimmrigk using twist maps. They are realized in weighted projective spaces as orbifolds of hypersurfaces. Our main goal of this paper is to investigate arithmetic properties of these Calabi--Yau threefolds. We also consider deformations of our Calabi--Yau threefolds, and we study the variation of the zeta-functions using $p$-adic rigid cohomology theory.
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.
We consider Calabi-Yau threefolds of Borcea-Voisin type over Q. They are constructed from products of K3 surfaces and elliptic curves. We use concrete K3 surfaces and discuss the automorphy of the Galois representations associated to the Calabi-Yau threefolds. The moduli spaces of these Calabi-Yau threefolds are Shimura varieties. Our result shows the existence of a CM point in the moduli space. We also consider mirror symmetry of Calabi-Yau threefolds.
We construct many new non-liftable three-dimensional Calabi-Yau spaces in positive characteristic. The technique relies on lifting a nodal model to a smooth rigid Calabi-Yau space over some number field as introduced by the first author and D. van Straten.
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.
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.
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.
We discuss Calabi-Yau and fractional Calabi-Yau semiorthogonal components of derived categories of coherent sheaves on smooth projective varieties. The main result is a general construction of a fractional Calabi-Yau category from a rectangular Lefschetz decomposition and a spherical functor. We give many examples of application of this construction and discuss some general properties of Calabi-Yau categories.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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 $4$.
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.
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.
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.
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.
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.
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.
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 $\mathbb{P}^6$.
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.
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.
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.
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.
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.
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.
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 …
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.
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.
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 $\mathbb{P}^{n+1}$.
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.
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.
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.
School of Mathematics and Physics, University of Science and Technology Beijing（北京科技大学数理学院）；Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）；Instituto de Matemática Pura e Aplicada, IMPA（巴西纯粹与应用数学研究所）
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.
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 $2p+1$.
