I am a beginner in the subject, and at the moment I am trying to understand basic properties of the main objects of the M. Saito's theory of the mixed Hodge modules in general.The question in the title might be trivially wrong or correct.



Is the abelian category of pure Hodge modules semi-simple?
The category of polarizable pure Hodge modules (of a given weight) on an algebraic variety $X$ is semi-simple. This is Theorem 14.37 in the book "Mixed Hodge Structures" by Peters and Steenbrink.


If you don't insist on polarizability the category is not semisimple even for $X=pt$ (where we just get pure Hodge structures) as explained in this question: [On polarized (pure) Hodge structures](https://mathoverflow.net/questions/42006/on-polarized-pure-hodge-structures) .


