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Let $V, W$ be real (pure) Hodge structures of weight $m, n$. Is there a natural Hodge structure on $\text{Hom}(V,W)$?

As I understand, there is one in the case $V = W$, although the definition I found (via the existence of a mapping from the Deligne torus) seems quite indirect. Is there a simple definition in this case that makes it clear what the $(p,q)$ components are?

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    $\begingroup$ Yes, there is. As you observe, this can be seen via Deligne's torus, but it can be done directly. The $(p,q)$ components are sums of $Hom(V^{i,j}, W^{i+p, j+q})$ over $(i,j)$. $\endgroup$
    – Donu Arapura
    Commented May 3, 2019 at 15:49

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