Triangulated structure on complexes of mixed Hodge structures

Question

I'm trying to read parts of the Peters Tata Lectures on "Motivic Aspects of Mixed Hodge structures" One aspect I don't really understand is the construction of the ''mixed cone'' for morphisms of complexes of mixed Hodge structures, and the associated long-exact sequence.

The setup is roughly of follows: A complex of mixed Hodge structures is a tuple $((K,W),(K_{\mathbb{C}},W,F),\alpha)$, where:

• $(K,W) \in DF^+(\mathbb{Q})$ is a object in the derived category of filtered $\mathbb{Q}$-complexes
• $(K_{\mathbb{C}},W,F)) \in {DF_2}^+(\mathbb{C})$ is an object in the derived category of bi-filtered $\mathbb{C}$-modules
• $\alpha: (K,W)\otimes \mathbb{C} \xrightarrow{\sim} (K_{\mathbb{C}}),W)$ is an isomorphism in $DF^+(\mathbb{C})$. This

and this data is required to satisfy (roughly) that the graded pieces have Hodge structure. One can then define a morphim of mixed Hodge structures to consist of a pair $(f,f_\mathbb{C})$ where $f$ is a morphism in $DF^+(\mathbb{Q})$ and $f_{\mathbb{C}}$ a morphism ${DF_2}^+(\mathbb{C})$. The important part is that for $(f,f_\mathbb{C}): ((K,W),(K_{\mathbb{C}},W,F),\alpha) \to ((K',W'),(K_{\mathbb{C}}',W',F'),\alpha') $ is a morphism of mixed Hodge structures, one requires the compatibility of $f$ with the comparison maps $\alpha$ and $\alpha'$ in the sense that \begin{equation} \tag{$\ast$} \alpha' \circ (f\otimes \mathrm{id}) = f_{\mathbb{C}} \circ \alpha \end{equation} holds in $DF^+(\mathbb{C})$

Even though all the appearing derived categories are triangulated, this definition makes it difficult to define a triangulated structure on the category of complexes of mixed Hodge structures, because to get a cone, one would need to fix homotopies witnessing $(\ast)$ in $DF^+(\mathbb{C})$. I think the notes claim that by considering sheaves of complexes of mixed Hodge structures instead one can make this issue disappear. But I don't really see why this is the case.

I was wondering if there is a reference where this issue is discussed, and how to endow this category with a structure in which it makes sense to take about mapping cones and associated long-exact sequences in the (abelian) category of mixed Hodge modules.

On a second note, I think that one still gets these notions if one just doesn't pass to the derived category but sticks to complexes and 'actual' morphisms. If we do this, do we still get the result that the cohomology of a (complex) variety has a mixed Hodge structure, or is the passage really necessary (I don't have a full overview of the details of Deligne's proof).

Answer 1

I haven't looked at Peters' notes, so I don't have any comments about what he does. But I agree that the mapping cone is tricky as usual. You would need lifts of $(f,f_{\mathbb{C}})$ to the (bi)filtered complex level before it is really well defined. You can look at the way Deligne defines the total complex of a DG mixed Hodge complex in section 8 of his Hodge III (the mapping cone is a special case). Deligne does not discuss a triangulated structure on these objects, but Beilinson does in his Notes on absolute Hodge cohomology. So you might want to look at that too.

Finally, as ZM comments, the $\infty$-category framework is probably the right way to handle these issues nowadays. Hopefully, someone will work out the theory of mixed Hodge complexes from this perspective, if it hasn't been done already.