Hodge properties of Airy moments

TL;DR

This paper studies the Hodge structures of symmetric power moments of Airy differential equations, revealing that their Hodge numbers are either zero or one, and connects them to arithmetic and irregular Hodge filtrations.

Contribution

It demonstrates that the de Rham cohomologies of Airy moments form an arithmetic Hodge structure and computes their Hodge numbers using irregular Hodge filtration.

Findings

All Hodge numbers are either zero or one.

The Hodge structure is linked to an arithmetic Hodge structure.

The irregular Hodge filtration is indexed by rational numbers.

Abstract

We consider the complex analogues of symmetric power moments of cubic exponential sums. These are symmetric powers of the classical Airy differential equation. We show that their de Rham cohomologies underlie an arithmetic Hodge structure in the sense of Anderson and we compute their Hodge numbers by means of the irregular Hodge filtration, which is indexed by rational numbers, on their realizations as exponential mixed Hodge structures. The main result is that all Hodge numbers are either zero or one.