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Hodge properties of Airy moments

Claude Sabbah and Jeng-Daw Yu

Vol. 5 (2023), No. 2, 215–271

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.

Airy differential equation, mixed Hodge structure, exponential mixed Hodge structure, irregular Hodge filtration
Mathematical Subject Classification
Primary: 14C30, 14D07, 32G20, 32S40, 34M35
Received: 3 January 2022
Revised: 2 October 2022
Accepted: 3 November 2022
Published: 4 June 2023
Claude Sabbah
Centre de Mathématiques Laurent Schwartz
CNRS, École polytechnique, Institut Polytechnique de Paris
Jeng-Daw Yu
Department of Mathematics
National Taiwan University