The relative hermitian duality functor

TL;DR

This paper extends the Hermitian duality functor to relative regular holonomic modules on a manifold parametrized by a curve, establishing an equivalence with the conjugate manifold and introducing regular holonomic relative distributions.

Contribution

It generalizes the Hermitian duality functor to a relative setting and proves it induces an equivalence between categories on conjugate manifolds.

Findings

The Hermitian duality functor is an equivalence in the relative setting.

Introduction of regular holonomic relative distributions.

Extension of duality concepts to parametrized manifolds.

Abstract

We extend to the category of relative regular holonomic modules on a manifold $X$, parametrized by a curve $S$, the Hermitian duality functor (or conjugation functor) of Kashiwara. We prove that this functor is an equivalence with the similar category on the conjugate manifold $\overline X$, parametrized by the same curve. As a byproduct we introduce the notion of regular holonomic relative distribution.