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Mixed Hodge structure is introduced by Deligne and it's very useful for studying complex algebraic varieties. We know $\text{MHS}$, the category of mixed Hodge structures is an abelian category. Where can I find a good reference for the computation of the (higher) extension groups inside MHS (and it's variations)? Carlson's paper "Extensions of mixed Hodge structures" is helpful, but I hope a more general reference.

For instance, let $\text{MHS}^+_{\mathbb R}$ denote the category of mixed Hodge structures equipped with an involution $\phi_{\infty}$ preserving the weight filtration and such that $\phi_{\infty} \otimes c$ preserves the Hodge filtration (here $c$ means complex conjugation). Is there no $\operatorname{Ext}^{2}$ in this category?

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Beilinson, Notes on absolute Hodge cohomology is another reference. To answer your last question, the cohomological dimension of the category of mixed Hodge structures is $1$, i.e. there is no $Ext^i$ for $i>1$. See 1.10 of Beilinson's paper.

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