We give an algorithm to compute the periods of smooth projective hypersurfaces of any dimension. This is an improvement over existing algorithms which could only compute the periods of plane curves. Our algorithm reduces the evaluation of period integrals to an initial value problem for ordinary differential equations of Picard-Fuchs type. In this way, the periods can be computed to extreme-precision in order to study their arithmetic properties. The initial conditions are obtained by an exact determination of the cohomology pairing on Fermat hypersurfaces with respect to a natural basis.
We calculate the period integrals for a special class of affine hypersurfaces (deformed Delsarte hypersurfaces) in an algebraic torus by the aid of their Mellin transforms. A description of the relation between poles of Mellin transforms of period integrals and the mixed Hodge structure of the cohomology of the hypersurface is given. By interpreting the period integrals as solutions to Pochhammer hypergeometric differential equation, we calculate concretely the irreducible monodromy group of period integrals that correspond to the compactification of the affine hypersurface in a complete simpl…
Computing Milnor fiber monodromy for some projective hypersurfaces
我们在第二页描述了一种计算极序谱序列退化的$\mathbb{P}^n$超曲面顶部Milnor纤维上同调的单值性和极序过滤的算法。在超平面排列和自由的局部拟齐次超曲面的情况下，并假设一个关键猜想，该算法比上述超曲面快得多。我们的猜想得到了L. Narv\' ez Macarro和M.齐藤根的伯恩斯坦佐藤多项式的超曲面，所有的例子计算到目前为止，和一个部分结果。对于来自反射群的超平面排列，在我们的例子中显示了它们的极序谱在上同调上的惊人对称性。在平面曲线的情况下，我们也改进了我们以前的结果。
Given a hypersurface, $X$, prime $p$, the zeta function is a generating function for the number of $\mathbb{F}_{p}$ rational points of $X$. Until now, there is no algorithm for computing hypersurfaces with ADE singularities. Scott Stetson and Vladimir Baranovsky provided an algorithm with Mathematica for the ordinary double point case. In this paper, I go over a Sage algorithm for computing the zeta function of a hypersurface with ADE singularities over 3-dimensional projective space. To make the algorithm more efficient, I established an equivalence between a polynomial belonging to the Jacob…
We consider the residual B-model variation of Hodge structure of Iritani defined by a family of toric Calabi--Yau hypersurfaces over a punctured disk $D \setminus \{ 0\}$. It is naturally extended to a logarithmic variation of polarized Hodge structure of Kato--Usui on the whole disk $D$. By restricting it to the origin, we obtain a polarized logarithmic Hodge structure (PLH) on the standard log point. In this paper, we describe the PLH in terms of the integral affine structure of the dual intersection complex of the toric degeneration in the Gross--Siebert program.
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 $CP^{N-1}$
For every even number $n$, and every $n$-dimensional smooth hypersurface of $\mathbb{P}^{n+1}$ of degree $d$, we compute the periods of all its $\frac{n}{2}$-dimensional complete intersection algebraic cycles. Furthermore, we determine the image of the given algebraic cycle under the cycle class map inside the De Rham cohomology group of the corresponding hypersurface in terms of its Griffiths basis and the polarization. As an application, we use this information to address variational Hodge conjecture for a non complete intersection algebraic cycle. We prove that the locus of general hypersur…
Hasse-Witt matrices, unit roots and period integrals
受Candelas，de la Ossa和Rodriguez-Villegas [6]工作的启发，本文研究了复曲面簇和部分旗簇中的Calabi-Yau超曲面的周期积分与Hasse-Witt矩阵之间的关系.我们证明了Vlasenko [23]关于复曲面的高阶Hasse-Witt矩阵的一个猜想，遵循Katz的局部展开方法[14，15]。高阶Hasse-Witt矩阵与周期积分也有密切的关系。该证明给出了一种从Katz关于展开系数的同余关系[15]到Dwork关于周期的同余关系[8]的途径。
Let $k$ be a field and let $\text{GW}(k)$ be the Grothendieck-Witt ring of virtual non-degenerate symmetric bilinear forms over $k$. We develop methods for computing the quadratic Euler characteristic $χ(X/k)\in \text{GW}(k)$ for $X$ a smooth hypersurface in a projective space or a weighted projective space. We raise the question of a quadratic refinement of classical conductor formulas and find such a formula for the degeneration of a smooth hypersurface $X$ in $\mathbb{P}^{n+1}$ to the cone over a smooth hyperplane section of $X$; we also find a similar formula in the weighted homogeneous ca…
We study the equivalence between infinitesimal Torelli theorem for smooth hypersurfaces in rational homogeneous varieties with Picard number one and the theory of generalized Massey products. This equivalence shows that the differential of the period map vanishes on an infinitesimal deformation if and only if certain twisted differential forms are elements of the Jacobian ideal of the hypersurface. We also prove an infinitesimal Torelli theorem result for smooth hypersurfaces in log parallelizable varieties.
We analyze the bit complexity of an algorithm for the computation of at least one point in each connected component of a smooth real algebraic set. This work is a continuation of our analysis of the hypersurface case (On the bit complexity of finding points in connected components of a smooth real hypersurface, ISSAC'20). In this paper, we extend the analysis to more general cases. Let $F=(f_1,..., f_p)$ in $\mathbb{Z}[X_1, ... , X_n]^p$ be a sequence of polynomials with $V = V(F) \subset \mathbb{C}^n$ a smooth and equidimensional variety and $\langle F \rangle \subset \mathbb{C}[X_1, ..., X_n…
Harvard University（哈佛大学）；Dartmouth College（达特茅斯学院）；Leibniz Universität Hannover（汉诺威莱布尼茨大学）
The Gauss-Manin connection of a family of hypersurfaces governs the change of the period matrix along the family. This connection can be complicated even when the equations defining the family look simple. When this is the case, it is computationally expensive to compute the period matrices of varieties in the family via homotopy continuation. We train neural networks that can quickly and reliably guess the complexity of the Gauss-Manin connection of a pencil of hypersurfaces. As an application, we compute the periods of 96% of smooth quartic surfaces in projective 3-space whose defining equat…
We give a new practical method for computing subvarieties of projective hypersurfaces. By computing the periods of a given hypersurface X, we find algebraic cohomology cycles on X. On well picked algebraic cycles, we can then recover the equations of subvarieties of X that realize these cycles. In practice, a bulk of the computations involve transcendental numbers and have to be carried out with floating point numbers. However, if X is defined over algebraic numbers then the coefficients of the equations of subvarieties can be reconstructed as algebraic numbers. A symbolic computation then ver…
We propose a new method to compute asymptotics of periods using tropical geometry, in which the Riemann zeta values appear naturally as error terms in tropicalization. Our method suggests how the Gamma class should arise from the Strominger-Yau-Zaslow conjecture. We use it to give a new proof of (a version of) the Gamma Conjecture for Batyrev pairs of mirror Calabi-Yau hypersurfaces.
