Deep Learning Gauss-Manin Connections

TL;DR

This paper introduces neural networks to efficiently estimate the complexity of Gauss-Manin connections in hypersurface families, enabling rapid computation of period matrices and geometric invariants like Picard numbers.

Contribution

It presents a novel neural network approach to predict Gauss-Manin connection complexity, facilitating faster period computations for hypersurfaces.

Findings

Successfully computed periods for 96% of certain quartic surfaces.

Extracted Picard numbers and endomorphism fields from period data.

Demonstrated neural networks' effectiveness in complex algebraic geometry computations.

Abstract

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.