Calabi-Yau complete intersections with infinitely many lines
Marcello Bernardara
本文给出了类元含有无穷多条直线的Calabi-Yau三重完全交族的两个新例子。我们得到了这些线的法丛和三重线的Hilbert格式的一些结果。
We give two new examples of families of Calabi-Yau complete intersection threefolds whose generic element contains infinitely many lines. We get some results about the normal bundles of these lines and the Hilbert scheme of lines on the threefolds.
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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.
Calabi--Yau threefolds with infinitely many divisorial contractions
我们研究了具有无穷多个整除收缩的Calabi-Yau 3-folds。我们还提出了用无限自同构群描述Calabi-Yau 3-折叠的方法。
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
我们给出了具有复乘的Calabi-Yau三维流形的一些具体例子。
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.
We classify all smooth Calabi-Yau threefolds of Picard number two that have a general hypersurface Cox ring.
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
研究了Calabi-Yau簇的正特征不变量，特别是Artin-Mazur形式群的高度不变量.我们用Fermat和Kummer型的Calabi-Yau簇来说明这些结果。
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 prove the non-vanishing conjecture for lc pairs $(X,Δ)$ when $X$ is of Calabi--Yau type.
Notes on Calabi-Yau ordinary differential equations
本文研究了Calabi-Yau微分方程的结构以及方程背后的Calabi-Yau簇的簇与算法的关系。这为Samol和van Straten最近的一篇论文中的一些观察和计算提供了解释。
We determine the Griffiths-Yukawa coupling length of the Calabi-Yau universal families coming from hyperplane arrangements.
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
本文描述了两种构造带截面有理椭圆曲面的纤维积与某些Calabi-Yau流形之间的有限有理态射的方法。我们用它们来构造这样的纤维积，承认最多五个奇异的纤维和刚性卡拉比-丘三倍之间的对应关系。
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 $\mathbb{P}^6$.
We show that there exists a non-Kähler Calabi-Yau fourfold, constructing an example by smoothing a normal crossing variety.
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
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.
A cascade of determinantal Calabi--Yau threefolds
我们研究Kustin-Miller的卡拉比-丘三重非投影。作为应用，我们给出了定义为行列式簇的线性截面的Calabi-Yau三重的几何性质。我们计算他们的Hodge数和描述的态射对应于他们的Kähler-Mori锥的脸。
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.
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 $\mathbb{P}^{n+1}$.
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。
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.
Orbifold Riemann-Roch for threefolds with an application to Calabi-Yau geometry
本文证明了一个关于极化三重（X，D）的轨道Riemann-Roch公式.作为应用，我们构造了新的投射Calabi-Yau三重簇.
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 $2p+1$.
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。
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
我们认为，存在的高程度的多节的某些卡-丘三倍的属1纤维化意味着存在对这样的品种，不是双理性的，但派生等价。它也（可能）意味着存在的非双有理反例的Torelli问题的卡-丘三倍。
Extending the notion of Frobenius-splitting, we prove that every finite height Calabi-Yau variety defined over an algebraically closed field of positive characteristic can be lifted to the ring of Witt vectors of length two.
We write out explicit proper Calabi-Yau structures, i. e. non-degenerate cyclic cocycles on the differential graded categories of matrix factorizations of regular functions with isolated critical points. The formulas involve the Kapustin-Li trace and its "higher corrections".
We show some inductive statements for the index conjecture for log canonical Calabi-Yau pairs. Using it, we show that boundedness of log canonical index for log canonical Calabi Yau pairs with rational DCC coefficients in dimension 3. We also show boundedness of log canonical index for klt log Calabi-Yau pairs in dimension 4 with $B\neq 0$.
We show an example of non-simply connected non-liftable Calabi-Yau threefold over an algebraically closed field of characteristic $3$. It is constructed from a simply connected example by S.~Schröer.
We introduce and we study a class of odd dimensional compact complex manifolds whose Hodge structure in middle dimension looks like that of a Calabi-Yau threefold. We construct several series of interesting examples from rational homogeneous spaces with special properties.
Toric Residue Mirror Conjecture for Calabi-Yau complete intersections
我们推广的环面剩余镜像猜想Batyrev和Materov不一定自反多面体。利用这一推广，我们证明了Gorenstein环面Fano簇的Calabi-Yau完全相交的环面剩余镜像猜想。
Calabi--Yau complete intersections in exceptional Grassmannians
本文分类了例外型李群格拉斯曼流形上能产生Calabi--Yau三维流形完全交的完全可约等变向量丛，并在$E_6$-格拉斯曼流形中发现了一族新的Calabi--Yau三维流形。
We consider fibrations by abelian surfaces and K3 surfaces over a one dimensional base that are Calabi-Yau and we obtain dual fibrations that are derived equivalent to the original fibration. Finally, we relate the problem to mirror symmetry.
5-th order differential equations related to Calabi-Yau differential equations
列出了极少数以四阶Calabi-Yau方程为拉回的五阶微分方程。我们使用杨一凡的回调，在大多数情况下，回调的程度比通常的回调要低得多。
Based on Cynk-Hulek method we construct complex Calabi-Yau varieties of arbitrary dimensions using elliptic curves with automorphism of order 6. Also we give formulas for Hodge numbers of varieties obtained from that construction. We shall generalize result of Katsura and Schütt to obtain arbitrarily dimensional Calabi-Yau manifolds which are Zariski in any characteristic $p\not\equiv 1\pmod{12}.$
Smooth finite abelian uniformizations of projective spaces and Calabi-Yau Orbifolds
给出了具有有限阿贝尔群作用的光滑复流形的一个分类，使得其商同构于一个射影空间。详细研究了流形是卡-丘流形的情形。
A Calabi-Yau threefold with Brauer group (Z/8Z)^2
本文计算了Mark Gross和Sorin Popescu发现的某个Calabi-Yau三重的Brauer群，它是由64个普通二重点构成的（2，2，2，2）型完全交的小分解.我们发现布劳尔群是（Z/8 Z）^2，这是已知的卡-丘三重布劳尔群的最大例子。
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.
How to tell you're hearing a Calabi-Yau: Universal variations of Hodge structure and local Schottky relations for Calabi-Yau manifolds
一个修订本，有一些更正和改进。
Calabi-Yau coverings over some singular varieties and new Calabi-Yau 3-folds with Picard number one
本文是关于某些奇异簇容许Calabi-Yau覆盖的观察的报告。作为应用，我们构造了18个新的Picard数为1的Calabi-Yau 3-折叠，它们具有一些有趣的性质.
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.
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