In Deligne's paper "Hodge cycles on abelian varieties" (see page 11 of http://jmilne.org/math/Documents/Deligne82.pdf) he says that the following diagram fails to commute by a factor of $(2 \pi i)^m$, where $X$ is a smooth projective variety over $\mathbb{C}$:

$$ \require{AMScd} \begin{CD} H^n_B(X) \otimes \mathbb{C} @>{1 \mapsto (2\pi i)^m}>> H^n_B(X)(m) \otimes \mathbb{C} \\ @VVV @VVV \\ H^n_{dR}(X) @>{=}>> H^n_{dR}(X)(m) \end{CD} $$

Why? How is the vertical map on the right defined? To me the obvious definition is: the map which makes the above diagram commute.