Deep Learning Gauss-Manin Connections
This work addresses a computational bottleneck in algebraic geometry for researchers studying hypersurfaces, though it is incremental as it applies existing deep learning methods to a new domain.
The paper tackled the problem of computationally expensive period matrix calculations for hypersurfaces by training neural networks to predict the complexity of Gauss-Manin connections, achieving 96% success in computing periods for smooth quartic surfaces.
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.