Fourier transform and Radon transform for mixed Hodge modules

TL;DR

This paper generalizes the relationship between microlocalization and vanishing cycles for mixed Hodge modules, establishing an isomorphism between localization triangles and comparing Radon and Fourier-Laplace transforms, with applications to GKZ systems.

Contribution

It introduces a new generalization linking microlocalization and vanishing cycles for mixed Hodge modules, and compares Radon and Fourier-Laplace transforms in this context.

Findings

Established an isomorphism between localization triangles.

Compared Radon and Fourier-Laplace transforms for mixed Hodge modules.

Applied results to the Hodge structure of GKZ systems.

Abstract

We give a generalization to bi-filtered $\mathcal D$-modules underlying mixed Hodge modules of the relation between microlocalization along $f_1,...,f_r \in \mathcal O_X(X)$ and vanishing cycles along $g = \sum_{i=1}^r y_i f_i$. This leads to an interesting isomorphism between localization triangles. As an application, we use these results to compare the $k$-plane Radon transform and the Fourier-Laplace transform for mixed Hodge modules. This is then applied to the Hodge module structure of certain GKZ systems.