Grzegorz Kapustka, Michal Kapustka
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 $\mathbb{P}^6$.
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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.
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 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.
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.
Geometric transitions between Calabi-Yau threefolds related to Kustin-Miller unprojections
我们研究Kustin-Miller非投影之间的卡-丘三倍或更精确的几何过渡，他们诱导。我们用它们来连接许多家庭的Calabi-Yau三倍皮卡德数1的网络的Calabi-Yau完全交叉。这使我们能够找到明确的描述几个已知的家庭卡-丘三倍的方程。此外，我们发现了两个新的例子，Calabi-Yau三重与皮卡德群的秩为一，描述的Pfidian方程的加权射影空间。
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 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。
Counting sheaves on Calabi-Yau and abelian threefolds
本文综述了具有平凡典范丛的代数三重层上的稳定层的Donaldson-Thomas不变量的基础，重点讨论了交换三重层的情形。
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.
Elliptic Calabi-Yau threefolds over a del Pezzo surface
我们考虑在射影平面上（更一般地在某些有理曲面上）具有维尔斯特拉斯标准形截面的某些椭圆三重曲面。特别地，在一个del Pezzo曲面的度为8，这些椭圆的三倍是Calabi-Yau三倍。我们将特别讨论Gromov-Witten和Gopakumar-Vafa不变量的生成函数。
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.
In this note, we consider crepant resolutions of the quotient varieties of smooth quintic threefolds by Gorenstein group actions. We compute their Hodge numbers via McKay correspondence. In this way, we find some new pairs $(h^{1,1},\;h^{2,1})$ of Hodge numbers of Calabi-Yau threefolds.
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-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。
Sections of Calabi-Yau threefolds with K3 fibration
利用K3曲面研究了曲线上的Calabi-Yau三重纤维截面。我们证明了在某些K3纤维的Calabi-Yau三重上存在无穷多个孤立的截面，并且由这些截面生成的Néron-Severi群的子群不是单生成的.本文还给出了复曲线的函数域F上的K3曲面的例子，该曲线的函数域F上具有Zebraki稠密的F-有理点，其几何模型为Calabi-Yau。
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。
Connecting certain rigid birational non-homeomorphic Calabi--Yau threefolds via Hilbert scheme
我们将给出一对双有理射影Calabi-Yau三重图，它们是刚性的，非同胚的，但在某些连通基方案上通过射影平坦变形连接。
We study the syzygetic structure of projections of del Pezzo surfaces in order to construct singular Calabi-Yau threefolds. By smoothing those threefolds, we obtain new examples of Calabi-Yau threefolds with Picard group of rank 1. We also give an example of type II primitive contraction whose exceptional divisor is the blow-up of the projective plane at a point.
We investigate the globally generated vector bundles on complete intersection Calabi-Yau threefolds with the first Chern class at most 2. We classify all the globally generated vector bundles of an arbitrary rank on quintic in $\mathbb{P}^4$ and investigate the globally generated vector bundles of rank 2 on complete intersection Calabi-Yau threefolds of codimension 2.
On Calabi-Yau threefolds with large nonabelian fundamental groups
本文用64阶非交换基本群构造了Calabi-Yau三重结构，作为M. Gross和S.波佩斯库
Monodromy of Picard-Fuchs differential equations for Calabi-Yau threefolds
本文研究了与单参数Calabi-Yau三重函数族有关的Picard-Fuch微分方程的单值性。我们的结果表明，在超几何情形下，单值性相对于Frobenius基的矩阵表示可以表示为基本的Calabi-Yau三重几何不变量.本文还对其它Calabi-Yau三重函数族的情况进行了数值验证。进一步，我们发现在适当的基基变换下，单值群包含在Sp（4，Z）的某些有限指数同余子群中，这些子群的水平与Calabi-Yau三重几何不变量有关
We study some conjectures about Chow groups of varieties of geometric genus one. Some examples are given of Calabi-Yau threefolds where these conjectures can be verified, using the theory of finite-dimensional motives.
We find a simple geometric construction of Tonoli's examples of Calabi--Yau threefolds of degree 17 in complex $\mathbb{P}^6$. We prove that the rank of the Picard group of elements of one of these families is at least $2$.
Kapustka and Rampazzo have exhibited pairs of Calabi-Yau threefolds $X$ and $Y$ that are L-equivalent and derived equivalent, without being birational. We complete the picture by showing that $X$ and $Y$ have isomorphic Chow motives.
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.
Rank 2 ACM bundles on complete intersection Calabi-Yau threefolds
本文的目的是对竞争交Calabi-Yau（CICY）三重上的不可分解秩2算术Cohen-Macaulay（ACM）丛进行分类，并证明其中一些不可分解秩2算术Cohen-Macaulay（ACM）丛的存在性.通过Serre对应，得到了秩为2的ACM丛对应的曲线的新的几何性质.这些都遵循最小的自由分解曲线在适当选择的四重（包含卡-丘三重作为超曲面）。证明了（2，4）型CICY三重图上存在一个不可分解的高秩向量丛.
We prove that a nef line bundle $\mathcal L$ with $c_1(\mathcal L)^2 \ne 0$ on a Calabi-Yau threefold $X$ with Picard number $2$ and with $c_3(X) \ne 0$ is semiample, that is, some multiple of $\mathcal L$ is generated by global sections.
In this paper, the numbers of rational curves on general complete intersection Calabi-Yau threefolds in complex projective spaces are computed up to degree six. The results are all in agreement with the predictions made from mirror symmetry.
New examples of modular rigid Calabi-Yau threefolds
本文给出了5个新的模刚性Calabi-Yau三重性的例子，这些例子产生于6阶模椭圆曲面。给出了新形式的权重4和水平10、17、21和73的显式对应关系。
The main purpose of this article is to discuss a project relating Gopakumar-Vafa invariants to quantum K-invariants on Calabi-Yau threefolds. Results in genus zero, including recent and forthcoming works, are reported.
Stability conditions on Calabi-Yau threefolds via Brill-Noether theory of curves
通过Brill-Noether理论研究Calabi-Yau三重曲面的稳定性条件，证明了相关不等式并确保稳定性条件的存在。
Ito-Miura-Okawa-Ueda have constructed a pair of Calabi-Yau threefolds $X$ and $Y$ that are L-equivalent and derived equivalent, but not stably birational. We complete the picture by showing that $X$ and $Y$ have isomorphic Chow motives.
Primitive contractions of Calabi-Yau threefolds II
我们构造了16个新的Calabi-Yau三重群的例子，它们都是秩为1的Picard群.这些例子中的每一个都是通过平滑原始收缩的图像来获得的，其中异常因子是6、7或$\mathbb{P}^1\times \mathbb{P}^1$度的del Pezzo曲面。
Primitive contractions of Calabi-Yau threefolds I
我们构造了Calabi-Yau三折叠的原始收缩的例子，其例外轨迹是$ \mathbb{P}^1 \times \mathbb{P}^1$，$\mathbb{P}^2$，以及$\leq 5$度的光滑del Pezzo曲面。我们描述了这些原始压缩的图像，并找到他们的光滑家庭。特别是，我们给出了一种方法来计算的一个通用的纤维的光滑家庭的每个Calabi-丘三倍与一个孤立奇点后得到的原始收缩的II型。作为应用，我们给出了Calabi-Yau三重函数族之间的自然锥变换的例子.
Some Calabi-Yau threefolds with obstructed deformations over the Witt vectors
我构造了一些光滑的卡-丘三重特征2和3，不提升到特征零。这三重是超奇异K3-曲面的束。该构造依赖于Moret-Bailly的交换曲面束和Katalina的广义Kummer曲面分析。特征2中的三重性是非刚性的。
We extend the known classification of threefolds of general type that are complete intersections to various classes of non-complete intersections, and find other classes of polarised varieties, including Calabi-Yau threefolds with canonical singularities, that are not complete intersections. Our methods apply more generally to construct orbifolds described by equations in given Gorenstein formats.
Calabi-Yau-threefolds with Picard number $ρ(X)=2$ and their Kaehler cone I
用Picard数$ρ(X)=2$刻画了一般Calabi-Yau-三重Kähler锥的性质，证明了当$X$是${\mathbb P}^2$和$c_3(X)\le -54$上的${\mathbb P}^2$ -丛中的Calabi-Yau-超曲面时，Kähler锥的合理性.没有后者的假设，我们证明了积极的$c_2(X)$。
We prove that the Borcea-Voisin mirror pairs of Calabi-Yau threefolds admit projective birational models that satisfy the Berglund-Hübsch-Chiodo-Ruan transposition rule. This shows that the two mirror constructions provide the same mirror pairs, as soon as both can be defined.
Number of triple points on complete intersection Calabi-Yau threefolds
本文研究射影及加权射影空间中 Calabi–Yau 三维簇的普通三重点数量，证明二次–四次完全交至多有10个，并给出六次超曲面的精确界与实例。
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。
Automorphisms of Calabi-Yau threefolds with Picard number three
证明了皮卡德数为3的卡-丘三重群的自同构群要么是有限的，要么同构于无限循环群，直到有限核和上核。
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.
Raynaud-Mukai construction and Calabi-Yau Threefolds in Positive Characteristic
在这篇文章中，我们研究的可能性，产生一个卡拉比-丘三倍的积极特点，这是一个反例小平消失。唯一已知的构造反例的方法是所谓的归纳法，例如Raynaud-Mukai构造或Russel构造。我们考虑Mukai的方法及其修改。最后，作为Fano三重格的Shepherd-Barron消灭定理的应用，我们计算了Kodaira消失不成立的Calabi-Yau三重格$X$上任意样本线丛$H$的$H^1(X, H^{-1})$.
Semistability of certain bundles on a quintic Calabi-Yau threefold
在Douglas和Zhou的论文《弦理论中的手征变化》中，作者给出了五次Calabi-Yau三重上的丛列表。在这里，我们证明了这些丛的半稳定性。这提供了弦理论紧化的例子，它们具有不同的代数，并且可以连接。
We describe explicitly the chamber structure of the movable cone for a general smooth complete intersection Calabi-Yau threefold $X$ of Picard number two in certain Pr-ruled Fano manifold and hence verify the Morrison-Kawamata cone conjecture for such $X$. Moreover, all birational minimal models of such Calabi-Yau threefolds are found, whose number is finite up to isomorphism.
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。
Stability conditions on some families of Calabi-Yau threefolds via orbifolding
该研究通过轨形构造方法，证明了源自具有Bridgeland稳定性条件的Calabi-Yau三维流形族的衍生族也满足该条件，并验证了五次镜像簇的相关性质，推进了代数几何中稳定性条件的研究。
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.
In the present paper we propose a combinatorial approach to study the so called double octic Clabi--Yau threefolds. We use this description to give a complete classification of double octics with $h^{1,2}\le1$ and to derive their geometric properties (Kummer surface fibrations, automorphisms, special elements in families).
Calabi-Yau threefolds of quotient type
首先，利用Beauville引入的具有无限基本群的Calabi-Yau三重的极小分裂覆盖，对它们进行了分类，并证明了当第二类恒为零时，nef锥是有理单纯锥，且任何有理nef因子都是半充分的.我们还得到了一个基本群用Picard数表示为有限的最佳形式的充分条件。其次，我们给出了关于Calabi-Yau三重的$c_{2}$ -压缩的一个具体结构定理，作为对我们以前关于单连通的工作的推广和修正。最后，作为应用，我们证明了每个Calabi-Yau三重的$c_{2}$ -压缩的同构类的有限性。
We prove the non-vanishing conjecture for lc pairs $(X,Δ)$ when $X$ is of Calabi--Yau type.
Automorphisms of Calabi-Yau threefolds from algebraic dynamics and the second Chern class
本文是作者在MFO的报告记录，基于代数动力学视角，对严格Calabi-Yau三维流形的自同构及第二陈类问题进行了推测性探讨。
Fibered Calabi-Yau threefolds with relative automorphisms of positive entropy and $c_2$-contractions
本文研究纤维化Calabi-Yau三维流形上保持纤维化的正熵自同构，证明阿贝尔纤维化情形在同构意义下唯一，并给出椭圆纤维化情形的显式结构定理。
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.
On isolated smooth curves of low genera in Calabi-Yau complete intersection threefolds
克莱门斯和Kley的结果的基础上，我们发现标准的连续家庭的曲线在一个节点$K$ -平凡的三倍$Y_0$变形的计划，许多光滑的孤立曲线在一般变形$Y_t$的$Y_0$。作为应用，我们三次证明了在Calabi-Yau完全交中有界亏格和无界度的光滑孤立曲线的存在性。
We study threefolds fibred by Kummer surfaces associated to products of elliptic curves, that arise as resolved quotients of threefolds fibred by certain lattice polarized K3 surfaces under a fibrewise Nikulin involution. We present a general construction for such surfaces, before specializing our results to study Calabi-Yau threefolds arising as resolved quotients of threefolds fibred by mirror quartic K3 surfaces. Finally, we give some geometric properties of the Calabi-Yau threefolds that we have constructed, including expressions for Hodge numbers.
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