When $X$ is a $k$-variety the category $MHM(X_{\mathbb{C}})$ of mixed Hodge modules on $X\otimes_k \mathbb{C}$ doesn't remember the $k$-structure. For example you have $Ext^1_{MHM(X)}(Q_X, Ext^1_{MHM(X_{\mathbb{C}})}(Q_X, Q_X(1)) = \mathbb{C}(X)^\times \otimes_{\mathbb{Z}} \mathbb{Q}$. In a category of mixed Hodge modules with de Rham $k$-structure this group would be $k(X)^\times \otimes_{\mathbb{Z}} \mathbb{Q}$ instead.
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When $X$ is a $k$-variety the category $MHM(X_{\mathbb{C}})$ of mixed Hodge modules on $X\otimes_k \mathbb{C}$ doesn't remember the $k$-structure. For example you have $Ext^1_{MHM(X)}(Q_X, Q_X(1)) = \mathbb{C}(X)^\times \otimes_{\mathbb{Z}} \mathbb{Q}$. In a category of mixed Hodge modules with de Rham $k$-structure this group would be $k(X)^\times \otimes_{\mathbb{Z}} \mathbb{Q}$ instead. |
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