Slawomir Cynk, Matthias Schütt, Duco van Straten
We exhibit three double octic Calabi--Yau threefolds over the certain quadratic fields and prove their modularity. The non-rigid threefold has two conjugate Hilbert modular forms of weight [4,2] and [2,4] attached while the two rigid threefolds correspond to a Hilbert modular form of weight [4,4] and to the twist of the restriction of a classical modular form of weight 4.
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Modularity of Maschke's octic and Calabi-Yau threefold
我们证明了模块Maschke的octic和两个Calabi-Yau三倍来自它作为双octic和商的海森堡群，如Bini和van Geemen所示。证明依赖于K3曲面的簇和同构的自同构。
Modularity of some non-rigid double octic Calabi-Yau threefolds
本文讨论了用$h^{12}=1$证明Calabi-Yau三重模性的四种方法：内部椭圆直纹曲面的存在性（Hulek-Verrill），与椭圆曲线和K3曲面的乘积的对应性（Livné-Yui），与（模）刚性Calabi-Yau三重模性的对应性，以及将四维表示分解为二维子表示的对合的存在性.我们应用这些方法来证明模块性的17个双octic卡-丘三倍，其中“模块性的数值证据”，发现在第二作者的论文。我们观察到，模块化持有铅笔中的那些元素有一些额外的几何属性。在证明中，我们使用所考虑的卡-丘三重作为库默尔纤维化的合理椭圆纤维化的纤维积的表示。
By the modularity theorem every rigid Calabi-Yau threefold $X$ has associated modular form $f$ such that the equality of $L$-functions $L(X,s)=L(f,s)$ holds. In this case period integrals of $X$ are expected to be expressible in terms of the special values $L(f,1)$ and $L(f,2)$. We propose a similar interpretation of period integrals of a nodal model of $X$. It is given in terms of certain variants of a Mellin transform of $f$. We provide numerical evidence towards this interpretation based on a case of double octics.
Modularity experiments on $S_4$-symmetric double octics
利用大量计算资源寻找与权4模形式相连的双八面体三维簇。
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).
Elliptic and K3 fibrations on double octic Calabi-Yau threefolds
研究由八次排列的奇异线和重数≥4的点确定的双八次Calabi-Yau三重折叠的椭圆与K3纤维化，得出每个双八次均存在多类纤维化，并据此构建了两个双八次家族元素间的双有理映射。
We present a method for numerical computation of period integrals of a rigid Calabi-Yau threefold using Picard-Fuchs operator of a one-parameter smoothing. Our method gives a possibility of computing the lattice of period integrals of a rigid double octic without any explicit knowledge of its geometric properties, employing only simple facts from the theory of Fuchsian equations and computations in MAPLE with a library for differential equations. As a surprising consequence we also get approximations of additional integrals related to a singular (nodal) model of considered Calabi-Yau threefold…
Doran and Morgan have introduced certain rational basis for the monodromy group of the Picard-Fuchs operator of a hypergeometric family of Calabi-Yau threefolds. In this paper we compute numerically the transition matrix between a generalization of the Doran-Morgan basis and the Frobenius basis at a half-conifold point of a one-parameter family of double octic Calabi-Yau threefolds. We identify the entries of this matrix as rational functions in the special values $L(f,1)$ and $L(f,2)$ of the corresponding modular form $f$ and one constant. We also present related results concerning the rank o…
We compute numerical approximations of the period integrals for eleven rigid double octic Calabi--Yau threefolds and compare them with the periods of corresponding weight our cusp forms and find, as to be expected, commensurabilities. These give information on character of the correspondences of these varieties with the associated Kuga-Sato modular threefolds.
Classification of double octic Calabi-Yau threefolds defined by an arrangement of eight planes II
本文通过研究八平面排列，完整分类了双八次卡拉比-约当三重流形，并揭示了其几何结构和模性新实例。
The modularity conjecture for rigid Calabi-Yau threefolds over Q
本文给出了定义在有理数域Q上的刚性Calabi-Yau三重模的模猜想。我们建立了由根格A_3产生的刚性卡-丘三重模。我们的证明是基于几何分析。
On the modularity of Calabi-Yau threefolds containing elliptic ruled surfaces
我们证明了（不一定是刚性的）Calabi-Yau三重定义在有理数，其中包含足够多的椭圆规则面是模块化的（温和的限制下的素数的坏约化）。我们的证明使用Dieulefait和Manoharmayum的结果，他们证明了刚性Calabi-Yau三重模性。
Geometry and Arithmetic of certain Double Octic Calabi-Yau Manifolds
研究了由${\mathbb P}^3$沿八度曲面分支的双覆盖所构造的Calabi-Yau流形。我们给出了一个85个例子的列表，对应于在${\mathbb Q}$上定义的八个平面的排列。计算所有示例的霍奇数。有7个刚性Calabi-Yau流形和14个具有$h^{1,2}=1$的流形族。模块性猜想得到验证的所有刚性的例子。
Semistable degenerations of double octics
本文提出一种基于双八次图组合表示的算法，用于计算双八次 Calabi-Yau 三重流形的半稳定退化，并适用于经典与算术两种情形，通过三个显式例子验证其有效性。
On Hilbert modular threefolds of discriminant 49
设K是判别式49的全实三次域，O是它的整数环，p是7上的素数。设Gamma（p）\subset Gamma = SL_2（O）是水平p的主同余子群，本文研究了Gamma（p）上的Hilbert模三重子群及其相关簇的几何.特别地，我们发现了P^3中的一个八度图，它具有84个A_2型孤立奇点。
Orphan Calabi-Yau threefold with arithmetic monodromy group
本文研究了孤立Calabi-Yau三流形单参数族Picard-Fuchs算子的单值群，为4阶双重八次孤立算子构造有理辛基，证明其单值群在Sp(4,Z)中稠密，并证明其中一个算子的单值群是算术的。
The modularity of certain non-rigid Calabi-Yau threefolds
设$X$是一个在${\mathbb P}^1$上的Calabi-Yau三重非常数半稳定K3曲面，并达到Arakelov-Yau界.在[STZ]中，X. Sun，Sh. L. Tan和K. Zuo证明了$X$在一定意义下是模的.特别地，基弧是模曲线。在他们的结果中，他们区分了刚性和非刚性的情况。在[SY]和[V]中，构建了刚性实例。在本文中，我们构造明确的例子在非刚性的情况下。此外，我们证明了我们的三倍，“有趣”的一部分，他们的$L$系列是附加到一个自守形式，因此，他们是模块在另一种意义上。
On Calabi--Yau threefolds associated to a web of quadrics
研究了包含平面的七维复射影空间中二次曲面网的交与沿该网的判别式分支的双八度线之间的双有理映射的几何性质。
On the motive of Kummer varieties associated to $Γ_1(7)$ - Supplement to the paper: The modularity of certain non-rigid Calabi-Yau threefolds (by R. Livné and N. Yui)
在他们的论文Livné和Yui（数学AG/0304497）讨论了几个例子的非刚性卡-丘品种承认半稳定的K3纤维与6奇异纤维超过一个基地，这是一个合理的模曲线。他们还建立了这些例子的L-函数的模块性。本文的目的是指出，在他们的论文中列出的例子，但不导致半稳定纤维化，仍然是模块化的意义上，他们的L-功能是相关的模形式。我们详细讨论了与组Gamma_1（7）相关的情况，但我们的技术也适用于许多其他情况。我们还就一般椭圆曲面的纤维积的Kummer构造作了一些评论。
Flux compactification of IIB string theory associates special points in Calabi-Yau moduli space to choices of (pairs of) integral three-form fluxes. In this paper, we propose that supersymmetric flux vacua are modular. That is, to a supersymmetric flux vacuum arising in a variety defined over $\mathbb{Q}$, we associate a two-dimensional Galois representation that we conjecture to be modular. We provide numerical evidence for our conjecture by examining flux vacua arising on the octic hypersurface in $\mathbb{P}^{4}(1,1,2,2,2)$.
Update on Modular Non-Rigid Calabi-Yau Threefolds
本文综述了非刚性Calabi-Yau三折叠模性的一些最新结果。
Modularity of a certain "rank-2 attractor" Calabi-Yau threefold
该研究证明了某种卡拉比-叶三种折面的4维伽罗华表示可约，并揭示其构成因子与特定模形式相关，为‘秩2吸引子’提供了理论支持。
On modularity of rigid and nonrigid Calabi-Yau varieties associated to the root lattice A_4
我们证明了根格A_4上的四个刚性和三个非刚性Calabi-Yau三重的模性。
Modularity of Calabi-Yau varieties
本文讨论了Calabi-Yau簇的模性研究的最新进展。我们主要关注曲面和三重折叠的情况。我们还将讨论与镜像对称有关的L-函数结构的一些进展。最后，我们解决一些问题和开放的问题。
On the modularity of three Calabi-Yau threefolds with bad reduction at 11
我们研究了三个非刚性的Calabi-Yau三重的模性，这些三重在11处具有坏约化，它们是有理椭圆曲面的纤维积。为此，我们应用Serre的方法比较了具有不均匀迹的二维2-adic伽罗瓦表示。因此，我们将三重中的两重（或者更准确地说，它们的中间上同调群的二维部分）分别与权重为4、水平为22和55的新形式相关联。这使我们能够计算这些变种的zeta函数，直到坏素数的欧拉因子。
Geometry and Arithmetic of Maschke's Calabi-Yau Threefold
Maschke的Calabi-Yau三重复盖是沿Maschke八次曲面分支的射影三空间的复盖。该曲面由作用于四维向量空间的某个有限群的最低度不变量定义。利用这个群，我们证明了三重的中间Betti上同调群分解为150个二维Hodge子结构的直和.我们展示了三维上的一维有理曲线族，并证明了相应的Abel-Jacobi映射是非平凡的。通过计算有限域上的点的个数，我们确定了Maschke曲面的Néron-Severi群的秩和它的某些子曲面在超越格上的伽罗瓦表示.我们还制定了精确的模块化的伽罗瓦表示Maschke的三倍和属33曲线参数化的有理曲线的三倍。
Modular Calabi-Yau threefolds of level eight
在研究刚性Calabi-Yau三重模型的模性猜想时，构造了几个具有唯一的8阶尖点形式的例子.根据Tate猜想，这些簇之间应该存在诱导中上同调同构的对应。在本文中，我们构建了几个这样的对应关系的例子。在这些结构中，椭圆纤维起着至关重要的作用。事实上，我们表明，所有，但三个例子在某种意义上是建立在两个模曲线从Beauville列表。
Modularity of the Consani-Scholten quintic
我们证明了QQ上的三重卡-丘型的Consani-Scholten五次曲线是Hilbert模的。为此，我们从模形式的上下文中改进了几种已知的技术。最值得注意的是，我们扩展的Faltings-Serre-Livne方法诱导四维伽罗瓦表示QQ。我们还需要希尔伯特模形式的Sturm界;这是由Jose Burgos Gil和第二作者在附录中开发的。
Degeneration of Calabi-Yau threefolds with Galois action induced by modular form of weight 3
研究双八次 Calabi-Yau 三折叠族的 K 型退化，构造半稳定模型并计算极限混合 Hodge 结构，证明极限 Galois 表示与奇异 K3 曲面相关的权 3 模形式及其 Tate 扭转同构，实现预期关系。
Mock modularity of Calabi-Yau threefolds
本研究针对IIA型弦论卡拉比-丘紧化中D4-D2-D0 BPS指标的生成函数所满足的模反常方程，提出用不定θ级数求解任意荷情形的方法，可通过有限个傅里叶系数确定生成函数对应的模形式。
In recent work, we conjectured that Calabi-Yau threefolds defined over $\mathbb{Q}$ and admitting a supersymmetric flux compactification are modular, and associated to (the Tate twists of) weight-two cuspidal Hecke eigenforms. In this work, we will address two natural follow-up questions, of both a physical and mathematical nature, that are surprisingly closely related. First, in passing from a complex manifold to a rational variety, as we must do to study modularity, we are implicitly choosing a "rational model" for the threefold; how do different choices of rational model affect our results?…
We establish the supercongruences for the fourteen rigid hypergeometric Calabi--Yau threefolds over $\mathbb Q$ conjectured by Rodriguez-Villegas in 2003. Our first method is based on Dwork's theory of $p$-adic unit roots and it allows us to establish the supercongruences between the truncated hypergeometric series and the corresponding unit roots for ordinary primes. The other method makes use of the theory of hypergeometric motives, in particular, adapts the techniques from the recent work of Beukers, Cohen and Mellit on finite hypergeometric sums over $\mathbb Q$. Essential ingredients in e…
Enumerative geometry and modularity in two-modulus K3-fibered Calabi-Yau threefolds
本文研究了双模数K3纤维化Calabi-Yau三重展开的计数几何与模性，构造了39个Calabi-Yau镜像对，并展示了其模性下的自由能公式。
Fibering out Calabi-Yau motives
本研究通过“纤维化提取”方法，将特定卡拉比-丘三维曲面族奇异纤维的混合周期表示为K3曲面族周期的积分，证明了其模性，还得到了与留数非零亚纯模形式及CM点围道相关的新周期。
Numerical Calculation of Periods on Schoen's Class of Calabi-Yau Threefolds
研究Schoen类Calabi-Yau三维流形上全纯3-形式的周期积分，提出有效数值方法计算其周期，给出多示例结果并与模形式积分对比验证准确性。
New non-commutative resolutions of determinantal Calabi-Yau threefolds from hybrid GLSM
本文利用混合相的 GLSM 研究行列式 Calabi-Yau 三流形非交换解消上的拓扑弦，证明小解消必为非 Kähler 且引入 $\mathbb{Z}_2$ 规范对称性，通过镜像对称计算了 $\mathbb{Z}_2$-细化 GV 不变量。
Mock modularity at work, or black holes in a forest
本综述介绍拟模形式在弦论BPS黑洞微态数生成函数、拓扑不变量计算及卡拉比-丘三维簇模空间非交换结构构建中的应用。
Double solids, categories and non-rationality
本文以范畴论为基础，对三重合理性问题提出了一种新的研究方法。继M。Ballard，D.法韦罗湖Katzarkov（ArXiv：1012.0864）和D.法韦罗湖Katzarkov（Noether-Lefschetz Spectrum and Algebraic cycles，in preparation）我们改进了A. Kuznetsov（arXiv：0904.4330）通过引入Noether-Lefschetz谱--Orlov谱（C. Oliva，代数圈和Hodge理论上的广义Reye同余，Compos。92，No.1（1994）1- 22）和霍赫希尔德同调。本文的主要目标是提出一系列有趣的例子，其中上述技术可能适用。我们首先构造一个有35个结点的六次双立体$X$，它在$H^3(X,\mathbb Z)$中有挠。这是一个新的例子-在Artin和Mumford（1972）的经典例子之后，这是第二个在3阶整数同调群中具有挠率的Fano三重群的例子。特别是$X$是非理性的。我们考虑其他的例子，以及- $V_{10}$与10个奇点和双重覆盖的二次分歧在八次线与20个节点奇点。在分析了他们的朗道-金兹伯格模型的几何结构之后，我们提出了一个基于同调镜像对称和范畴论的一般非理性图景。
Let $X\to \mathbb P^2$ be the elliptic Calabi-Yau threefold given by a general Weierstrass equation. We answer the enumerative question of how many discrete rational curves lie over lines in the base, proving part of a conjecture by Huang, Katz, and Klemm. The key inputs are a modularity theorem of Kudla and Millson for locally symmetric spaces of orthogonal type and the deformation theory of $A_n$ singularities.
The Gromov-Witten theory of threefolds admitting a smooth K3 fibration can be solved in terms of the Noether-Lefschetz intersection numbers of the fibration and the reduced invariants of a K3 surface. Toward a generalization of this result to families with singular fibers, we introduce completed Noether-Lefschetz numbers using toroidal compactifications of the period space of elliptic K3 surfaces. As an application, we prove quasi-modularity for some genus 0 partition functions of Weierstrass fibrations over ruled surfaces, and show that they satisfy a holomorphic anomaly equation.
We study the limit of Calabi-Yau modular forms, and in particular, those resulting in classical modular forms. We then study two parameter families of elliptically fibred Calabi-Yau fourfolds and describe the modular forms arising from the degeneracy loci. In the case of elliptically fibred Calabi-Yau threefolds our approach gives a mathematical proof of many observations about modularity properties of topological string amplitudes starting with the work of Candelas, Font, Katz and Morrison. In the case of Calabi-Yau fourfolds we derive new identities not computed before.
F-theory compactifications on appropriate local elliptic Calabi-Yau manifolds engineer six dimensional superconformal field theories and their mass deformations. The partition function $Z_{top}$ of the refined topological string on these geometries captures the particle BPS spectrum of this class of theories compactified on a circle. Organizing $Z_{top}$ in terms of contributions $Z_β$ at base degree $β$ of the elliptic fibration, we find that these, up to a multiplier system, are meromorphic Jacobi forms of weight zero with modular parameter the Kaehler class of the elliptic fiber and ellipti…
Revisiting the Quantum Geometry of Torus-fibered Calabi-Yau Threefolds
本文证明拓扑弦振幅的模性质等价于相对锥形单值下的波函数性质，并引入全纯模协变版本，揭示GV不变量与DT指数的拟模关系，分析大量新例子。
Modular bootstrap for D4-D2-D0 indices on compact Calabi-Yau threefolds
该研究针对b₂=1的完全交Calabi-Yau三维流形上的IIA型弦论，通过模自举方法分析D4-D2-D0束缚态BPS指标的生成级数，得到r=1时的无穷DT不变量预测，并将r=2情形的极项确定留作开放问题。
Invariants of pure 2-dimensional sheaves inside threefolds and modular forms
受弦论中的S-对偶模块性理论的启发，我们研究了非奇异三重X中纯二维层在三种不同情况下的Donaldson-Thomas型不变量：（1）. X是曲线上的K3纤维化。研究了X中二维Gieseker稳定层在纤维上的Donaldson-Thomas不变量。类似于Maulik-Pandharipande工作中建立的Gromov-Witten理论公式，我们用K3曲面上点的Hilbert格式的欧拉示性数和纤维化的Noether-莱夫谢茨数来表示这些不变量，并证明了这些不变量具有模性质.（二）、X是P^2的典范丛的全空间。本文研究了X上第一Chern类为X的零截面类的k倍的2维Gieseker半稳定层的模空间的Joyce-Song定义的广义Donaldson-Thomas不变量.当k= 1，2或3时，半稳定性蕴涵稳定性，我们用已知的模形式表示不变量。在严格半稳定层存在的情况下，我们证明了k=2时不变量的一个组合公式，并在某些情况下验证了Joyce-Song的BPS完整性猜想.（三）、（与Richard Thomas联合）X是Calabi-Yau三重的，L是充分正的线丛。我们定义了新的不变量计数的限制类的2维扭层，枚举对（Z，H）在X中，其中H是一个成员的线性系|L|和Z是一个1维的子计划H。伴随层是Z在H中的理想层，被向前推到X，并被认为是X的导出范畴中的某个Joyce-Song对。我们用X的MN…
Motivated by S-duality modularity conjectures in string theory, we define new invariants counting a restricted class of 2-dimensional torsion sheaves, enumerating pairs $Z\subset H$ in a Calabi-Yau threefold X. Here H is a member of a sufficiently positive linear system and Z is a 1-dimensional subscheme of it. The associated sheaf is the ideal sheaf of $Z\subset H$, pushed forward to X and considered as a certain Joyce-Song pair in the derived category of X. We express these invariants in terms of the MNOP invariants of X.
Quantum geometry and mock modularity
本研究将基于GV与DT不变量的分析拓展到带2单位D4膜荷的D4-D2-D0指标情形，验证了两类Calabi-Yau三维流形的对应生成级数符合S对偶预测的mock模形式，并得到拓扑弦振幅全纯歧义性的新边界条件，大幅提升了可确定的最高亏格。
We discuss Donaldson-Thomas (DT) invariants of torsion sheaves with 2 dimensional support on a smooth projective surface in an ambient non-compact Calabi Yau fourfold given by the total space of a rank 2 bundle on the surface. We prove that in certain cases, when the rank 2 bundle is chosen appropriately, the universal truncated Atiyah class of these codimension 2 sheaves reduces to one, defined over the moduli space of such sheaves realized as torsion codimension 1 sheaves in a noncompact divisor (threefold) embedded in the ambient fourfold. Such reduction property of universal Atiyah class e…
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