Let $(X,J_X)$ and $(Y,J_Y)$ be compact Kähler manifolds. Let $H^2(X,\mathbb C)=\bigoplus_{p+q=2}H^{p,q}(X)$ and $H^2(Y,\mathbb C)=\bigoplus_{p+q=2}H^{p,q}(Y)$ denote their compact Kähler Hodge decompositions, and regard $H^2(X,\mathbb R)$ and $H^2(Y,\mathbb R)$ as real subspaces of complex de Rham cohomology by extension of scalars. Let $f:(X,J_X)\to (Y,J_Y)$ be a smooth holomorphic map, so that for every $x\in X$ the differential $df_x:T_xX\to T_{f(x)}Y$ satisfies $df_x\circ J_X=J_Y\circ df_x$. Let $\omega\in A^{1,1}(Y)$ be a smooth real Kähler form on $Y$, meaning that $\omega$ is a closed real two-form of type $(1,1)$ and $\omega_y(w,J_Yw)>0$ for every $y\in Y$ and every nonzero $w\in T_yY$. Let $[\omega]\in H^2(Y,\mathbb R)$ be its de Rham cohomology class, viewed in $H^{1,1}(Y)\cap H^2(Y,\mathbb R)$ under the [Hodge decomposition](/theorems/2745). Then $f^*[\omega]\in H^{1,1}(X)\cap H^2(X,\mathbb R)$. Moreover, this cohomology class is represented by the closed semipositive real $(1,1)$-form $f^*\omega\in A^{1,1}(X)$; explicitly, for every $x\in X$ and every $v\in T_xX$, $(f^*\omega)_x(v,J_Xv)\geq 0$.