Equivalently, the complex-[linear map](/page/Linear%20Map)
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\begin{align*}
f^*:H^k(Y;\mathbb C)\to H^k(X;\mathbb C)
\end{align*}
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preserves the [Hodge decomposition](/theorems/2745) in every degree. With the underlying rational cohomology groups $H^k(Y;\mathbb Q)$ and $H^k(X;\mathbb Q)$ equipped with their compact Kähler pure Hodge structures of weight $k$, the rational pullback