Hypergeometric periods for a tame polynomial

The monodromy representation and twisted period relations for Appell's hypergeometric function F_4

We consider the system $\mathcal{F}_4(a,b,c)$ of differential equations annihilating Appell's hypergeometric series $F_4(a,b,c;x)$. We find the integral representations for four linearly independent solutions expressed by the hypergeometric series $F_4$. By using the intersection forms of twisted (co)homology groups associated with them, we provide the monodromy representation of $\mathcal{F}_4(a,b,c)$ and the twisted period relations for the fundamental systems of solutions of $\mathcal{F}_4$.

Twisted period relations for Lauricella's hypergeometric function F_A

We study Lauricella's hypergeometric function $F_A$ of $m$-variables and the system $E_A$ of differential equations annihilating $F_A$, by using twisted (co)homology groups. We construct twisted cycles with respect to an integral representation of Euler type of $F_A$. These cycles correspond to $2^m$ linearly independent solutions to $E_A$, which are expressed by hypergeometric series $F_A$. Using intersection forms of twisted (co)homology groups, we obtain twisted period relations which give quadratic relations for Lauricella's $F_A$.

Twisted cycles and twisted period relations for Lauricella's hypergeometric function F_C

We study Lauricella's hypergeometric function F_C by using twisted (co)homology groups. We construct twisted cycles with respect to an Euler-type integral representation of F_C. These cycles correspond to 2^m linearly independent solutions to the system of differential equations annihilating F_C. Using intersection forms of twisted (co)homology groups, we obtain twisted period relations which give quadratic relations for Lauricella's F_C.

Intersection numbers and twisted period relations for the generalized hypergeometric function ${}_{m+1} F_m$

We study the generalized hypergeometric function ${}_{m+1} F_m$ and the differential equation ${}_{m+1}E_m$ satisfied by it. We use the twisted (co)homology groups associated with an integral representation of Euler type. We evaluate the intersection numbers of some twisted cocycles which are defined as $m$-th exterior products of logarithmic $1$-forms. We also give twisted cycles corresponding to the series solutions to ${}_{m+1}E_m$, and evaluate the intersection numbers of them. These intersection numbers of the twisted (co)cycles lead twisted period relations which give relations for two f…

Determinants of the hypergeometric period matrices of an arrangement and its dual

We fix three natural numbers $k, n, N$, such that $n+k+1=N$, and introduce the notion of two dual arrangements of hyperplanes. One of the arrangements is an arrangement of $N$ hyperplanes in a $k$-dimensional affine space, the other is an arrangement of $N$ hyperplanes in an $n$-dimensional affine space. We assign weights $\al_1, ..., \al_N$ to the hyperplanes of the arrangements and for each of the arrangements consider the associated period matrices. The first is a matrix of $k$-dimensional hypergeometric integrals and the second is a matrix of $n$-dimensional hypergeometric integrals. The s…

The Determinant of a Hypergeometric Period Matrix

We consider a function $U=e^{-f_0}\prod_j^N f_j^{α_j}$ on a real affine space, here $f_0,..,f_N$ are linear functions, $α_1, ...,α_N$ complex numbers. The zeros of the functions $f_1, ..., f_N$ form an arrangement of hyperplanes in the affine space. We study the period matrix of the hypergeometric integrals associated with the arrangement and the function $U$ and compute its determinant as an alternating product of gamma functions and critical points of the functions $f_0,..., f_N$ with respect to the arrangement. In the simplest example, $N=1, f_0=f_1=t$, the determinant formula takes the for…

Hodge 圈的周期与 Gauss 超几何函数的特殊值

Periods of Hodge cycles and special values of the Gauss' hypergeometric function

本文通过计算 Fermat 畴扰动的周期，研究了由算术条件定义的 Hodge 圈子空间，给出了其维数上界，并推导出在单变量有理函数域上为代数的 Gauss 超几何函数的显式表达式。

A-hypergeometric functions in transcendental questions of algebraic geometry

We generalize the known constructions of A-hypergeometric functions. In particular, we show that periods of middle dimension on affine or projective complex algebraic varieties are A-hypergeometric functions of coefficients of polynomial equations of these varieties.

An algorithm of computing cohomology intersection number of hypergeometric integrals

We show that the cohomology intersection number of a twisted Gauss-Manin connection with regularization condition is a rational function. As an application, we obtain a new quadratic relation associated to period integrals of a certain family of K3 surfaces.

D-brane masses at special fibres of hypergeometric families of Calabi-Yau threefolds, modular forms, and periods

We consider the fourteen families $W$ of Calabi-Yau threefolds with one complex structure parameter and Picard-Fuchs equation of hypergeometric type, like the mirror of the quintic in $\mathbb{P}^4$. Mirror symmetry identifies the masses of even--dimensional D--branes of the mirror Calabi-Yau $M$ with four periods of the holomorphic $(3,0)$-form over a symplectic basis of $H_3(W,\mathbb{Z})$. It was discovered by Chad Schoen that the singular fiber at the conifold of the quintic gives rise to a Hecke eigenform of weight four under $Γ_0(25)$, whose Hecke eigenvalues are determined by the Hasse-…

超几何周期 $p^s$-近似的幽灵与同余

Ghosts and congruences for $p^s$-approximations of hypergeometric periods

本文证明洛朗多项式常数项的 Dwork 型同余，并借此研究超几何与 KZ 方程模 $p^s$ 多项式解的算术和 $p$-进解析性质，进而展示 $p$-进 KZ 联络存在不变线子丛而复数情形没有。

Jacobian elliptic Kummer surfaces and special function identities

We derive formulas for the construction of all inequivalent Jacobian elliptic fibrations on the Kummer surface of two non-isogeneous elliptic curves from extremal rational elliptic surfaces by rational base transformations and quadratic twists. We then show that each such decomposition yields a description of the Picard-Fuchs system satisfied by the periods of the holomorphic two-form as either a tensor product of two Gauss' hypergeometric differential equations, an Appell hypergeometric system, or a GKZ differential system. As the answer must be independent of the fibration used, identities r…

Generalized Hypergeometric Functions and Rational Curves on Calabi-Yau Complete Intersections in Toric Varieties

We formulate general conjectures about the relationship between the A-model connection on the cohomology of a $d$-dimensional Calabi-Yau complete intersection $V$ of $r$ hypersurfaces $V_1, \ldots, V_r$ in a toric variety ${\bf P}_Σ$ and the system of differential operators annihilating the special hypergeometric function $Φ_0$ depending on the fan $Σ$. In this context, the Mirror Symmetry phenomenon can be interpreted as the following twofold characterization of the series $Φ_0$. First, $Φ_0$ is defined by intersection numbers of rational curves in ${\bf P}_Σ$ with the hypersurfaces $V_i$ and…

Thomae type formula for K3 surfaces given by double covers of the projective plane branching along six lines

In this paper, we give Thomae type formula for \KK surfaces $\cS$ given by double covers of the projective plane branching along six lines. This formula gives relations between theta constants on the bounded symmetric domain of type $I_{22}$ and period integrals of $X$. Moreover, we express the period integrals by using the hypergeometric function $F_S$ of four variables. As an application of our main theorem, we define $\R^4$-valued sequences by mean iterations of four terms, and express their common limits by the hypergeometric function $F_S$.

Moduli spaces of K3 surfaces and complex ball quotients

These are lecture notes based on a series of talks given by the authors at the CIMPA Summer School on Algebraic Geometry and Hypergeometric Functions held in Istanbul in Summer of 2005. They provide an introduction to a recent work on the complex ball uniformization of the moduli spaces of Del Pezzo surfaces, K3 surfaces and algebraic curves of lower genus. We discuss the relationship of these constructions with the Deligne-Mostow theory of periods of hypergeometric differentianl forms. For convenience to a non-expert reader we include an introduction to the theory of periods of integrals on a…

Picard-Fuchs Equations for Relative Periods and Abel-Jacobi Map for Calabi-Yau Hypersurfaces

We study the variation of relative cohomology for a pair consisting of a smooth projective hypersurface and an algebraic subvariety in it. We construct an inhomogeneous Picard-Fuchs equation by applying a Picard-Fuchs operator to the holomorphic top form on a toric Calabi-Yau hypersurface, and deriving a general formula for the $d$-exact form on one side of the equation. We also derive a double residue formula, giving a purely algebraic way to compute the inhomogeneous Picard-Fuchs equations for Abel-Jacobi map, which has played an important role in recent study of D-branes [25]. Using the var…

关于 Calabi–Yau 动机的 $L$-导数与双扩张

On $L$-derivatives and biextensions of Calabi-Yau motives

本文证明特定 $DLD$ 微分算子为 Picard–Fuchs 型，并给出解析秩推测为 1 的秩 4、权 3 Calabi–Yau 动机双扩张周期矩阵子式的闭式超几何表达式，且将其数值与相应 $L$-函数在 $s=2$ 处的导数比较。

Chowla-Selberg 时期线性独立性度量

Linear independence measures for Chowla--Selberg periods

通过同时Padé近似估计超几何函数，研究特定d和D下1、π√d、Ω_D/π和π/Ω_D的线性形式，揭示其线性独立性。

On the Family of Pentagonal curves of genus 6 and associated modular forms on the Ball

In this article we study the inverse of the period map for the family $\mathcal{F}$ of complex algebraic curves of genus 6 equipped with an automorphism of order 5. This is a family with 2 parameters, and is fibred over a certain type of Del Pezzo surace. The period satisfies the hypergeometric differential equation for Appell's $F_1({3/5},{3/5},{2/5},{6/5})$ of two variables after a certain normalization of the variable parameter. This differential equation and the family $\mathcal{F}$ are studied by G. Shimura (1964), T. Terada (1983, 1985), P. Deligne - G.D. Mostow (1986) and T. Yamazaki- M…

$CP^{N-1}$ 中 Calabi-Yau 超曲面的周期积分（Givental 的 I-函数）与从带两个标记点的 $CP^1$ 到 $CP^{N-1}$ 的拟映射模空间的相交数

Period Integrals (Givental's I-function) of Calabi-Yau Hypersurface in $CP^{N-1}$ and Intersection Numbers of Moduli Space of Quasimaps from $CP^1$ with Two Marked Points to $CP^{N-1}$

本文将 $CP^{N-1}$ 中 Calabi-Yau 超曲面镜像计算的周期积分导出为从带两标记点的 $CP^1$ 到 $CP^{N-1}$ 的拟映射模空间相交数的生成函数。

On Semi-Periods

The periods of the three-form on a Calabi-Yau manifold are found as solutions of the Picard-Fuchs equations; however, the toric varietal method leads to a generalized hypergeometric system of equations which has more solutions than just the periods. This same extended set of equations can be derived from symmetry considerations. Semi-periods are solutions of this extended system. They are obtained by integration of the three-form over chains; these chains can be used to construct cycles which, when integrated over, give periods. In simple examples we are able to obtain the complete set of solu…

The transcendental part of the regulator map for K_1 on a mirror family of K3 surfaces

We compute the transcendental part of the normal function corresponding to the Deligne class of a cycle in K_1 of a mirror family of quartic K3 surfaces. The resulting multivalued function does not satisfy the hypergeometric differential equation of the periods and we conclude that the cycle is indecomposable for most points in the mirror family. The occurring inhomogenous Picard-Fuchs equation are related to Painlevé VI type differential equations.

伽马函数、单值化与弗罗贝尼乌斯常数

Gamma functions, monodromy and Frobenius constants

本文建立弗罗贝尼乌斯常数与广义伽马函数泰勒系数的关系，证明 Picard–Fuchs 算子的弗罗贝尼乌斯常数是周期，并将其推广到一般正则奇点情形，给出超几何连接的生成函数。

费曼积分的模性与阿普尔F2系统的因子分解

Modularity of Feynman Integrals and Factorization of Appell F2 Systems

本文通过构造高斯超几何系统证明了二维共形轨积分的模性，揭示了费曼积分与Calabi-Yau manifold周期之间的关系。

与费曼积分相关的扭曲微分形式的 Picard-Fuchs 方程

Picard-Fuchs Equations of Twisted Differential forms associated to Feynman Integrals

本文提出了一种改进的 Griffiths-Dwork 极点减少算法，用于从费曼积分导出扭曲微分形式的 D-模块，展示了其在超几何、椭圆和卡拉比-扬微分动机中的应用。