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.
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$\begingroup$ Didn't you ask this a few weeks ago? Your question answered here: mathoverflow.net/questions/42006/… . In summary, the category of all pure Hodge structures is not semisimple, but polarizable Hodge structures are. $\endgroup$– Sam GunninghamCommented Feb 1, 2021 at 11:16
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2$\begingroup$ @SamGunningham: This is not what I asked now. Now my question more general: it is about pure Hodge modules rather than pure Hodge structures. $\endgroup$– asvCommented Feb 1, 2021 at 12:03
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1$\begingroup$ Apologies, I read the question too hastily! In any case, the answer remains the same: if you restrict to polarizable objects you get a semisimple category, otherwise not. $\endgroup$– Sam GunninghamCommented Feb 1, 2021 at 12:47
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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 .
answered Feb 1, 2021 at 12:45
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2$\begingroup$ Did you mean "pure Hodge modules" rather than structures? $\endgroup$– asvCommented Feb 1, 2021 at 12:50
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