Some remarks on decomposition theorem for proper K\"ahler morphisms

TL;DR

This paper provides a correct proof of the decomposition theorem for direct images of constant Hodge modules under proper Kähler morphisms, addressing previous issues and illustrating challenges with non-constant Hodge modules.

Contribution

It offers a verified proof for the decomposition theorem in the context of constant Hodge modules and discusses difficulties in extending it to non-constant cases.

Findings

Correct proof of the decomposition theorem for constant Hodge modules

Identification of challenges in non-constant Hodge module cases

Examples illustrating the complexities in non-constant Hodge modules

Abstract

We explain a correct proof of the decomposition theorem for direct images of constant Hodge modules by proper K\"ahler morphisms of complex manifolds. We also give some examples showing certain difficulty in the non-constant Hodge module case.