论文标题
深度学习高斯 - 曼宁连接
Deep Learning Gauss-Manin Connections
论文作者
论文摘要
一系列Hypersurfaces家族的高斯 - 曼宁连接控制了家庭矩阵的变化。即使定义家庭的方程式看起来很简单，这种连接也可能很复杂。在这种情况下，计算在计算上很昂贵，可以通过同型延续来计算家庭中的各种矩阵。我们训练神经网络，可以快速可靠地猜测高斯铅笔的高斯 - 曼宁连接的复杂性。作为应用程序，我们计算了射影3空间中96％的平滑四分之一表面的周期，其定义方程为五个单元。从这些四分之一的表面的时期开始，我们提取了它们的PICARD数字和其先验晶格的内态字段。
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 equation is a sum of five monomials; from the periods of these quartic surfaces, we extract their Picard numbers and the endomorphism fields of their transcendental lattices.
