[proofplan] We estimate the energy density of the holomorphic map, namely the trace of $f^*\omega_Y$ with respect to $\omega_X$, and define it explicitly in the first step. The Chern--Lu Bochner computation gives a logarithmic differential inequality on the set where this energy density is positive. We then use Yau's generalized maximum principle in the precise form designed for nonnegative functions satisfying such a logarithmic inequality, avoiding any direct application to a logarithm that may be non-global or unbounded. The resulting scalar trace bound gives the tensor inequality because $f^*\omega_Y$ is semipositive, and when $A=0$ the differential of $f$ vanishes everywhere. [/proofplan]

[step:Define the energy density and reduce the form estimate to a scalar estimate] Let \begin{align*} u:X&\to [0,\infty)\\ x&\mapsto \operatorname{tr}_{\omega_X}(f^*\omega_Y)(x) \end{align*} be the trace of the semipositive Hermitian form $f^*\omega_Y$ with respect to $\omega_X$. Equivalently, if $(z_1,\dots,z_m)$ are holomorphic coordinates on $X$ and $(w_1,\dots,w_n)$ are holomorphic coordinates on $Y$, and if \begin{align*} \omega_X &= i\sum_{\alpha,\beta=1}^m g_{\alpha\bar\beta}\,dz_\alpha\wedge d\bar z_\beta,\\ \omega_Y &= i\sum_{i,j=1}^n h_{i\bar j}\,dw_i\wedge d\bar w_j, \end{align*} then \begin{align*} u=\sum_{\alpha,\beta=1}^m\sum_{i,j=1}^n g^{\alpha\bar\beta}h_{i\bar j}(f)\,\frac{\partial f_i}{\partial z_\alpha}\,\overline{\frac{\partial f_j}{\partial z_\beta}}. \end{align*} At each point $x\in X$, the Hermitian endomorphism associated to $f^*\omega_Y$ relative to $\omega_X$ has nonnegative eigenvalues $\lambda_1(x),\dots,\lambda_m(x)$ and \begin{align*} u(x)=\sum_{\alpha=1}^m \lambda_\alpha(x). \end{align*} Hence $0\leq \lambda_\alpha(x)\leq u(x)$ for every $\alpha$, and therefore the scalar estimate \begin{align*} u\leq \frac{A}{B} \end{align*} implies \begin{align*} f^*\omega_Y\leq \frac{A}{B}\,\omega_X. \end{align*} Thus it remains to prove the scalar bound for $u$. [/step]

[step:Apply the Chern--Lu inequality to $\log u$ on the nonzero locus]Let \begin{align*} U:=\{x\in X:u(x)>0\}. \end{align*} On $U$, the function \begin{align*} \log u:U&\to \mathbb{R}\\ x&\mapsto \log(u(x)) \end{align*} is smooth. The Chern--Lu Bochner formula for holomorphic maps between Kähler manifolds gives \begin{align*} \Delta_{\omega_X}\log u \geq \frac{1}{u}\left( \operatorname{Ric}(\omega_X)(\partial f,\overline{\partial f}) - R^Y(\partial f,\overline{\partial f},\partial f,\overline{\partial f}) \right), \end{align*} where $\Delta_{\omega_X}$ is the nonnegative complex Laplacian. More explicitly, at a point $x\in U$, choose a $\omega_X$-unitary frame $(e_1,\dots,e_m)$ for $T_x^{1,0}X$ and a $\omega_Y$-unitary frame $(E_1,\dots,E_n)$ for $T_{f(x)}^{1,0}Y$. Write the complex differential as \begin{align*} \partial f_x(e_\alpha)=\sum_{i=1}^n f_{\alpha i}E_i, \end{align*} where $f_{\alpha i}\in\mathbb{C}$ are the frame components of $\partial f_x$. Then \begin{align*} u(x)=\sum_{\alpha=1}^m\sum_{i=1}^n |f_{\alpha i}|^2, \end{align*} and the contractions in the Chern--Lu formula mean \begin{align*} \operatorname{Ric}(\omega_X)(\partial f,\overline{\partial f}) &=\sum_{\alpha,\beta=1}^m\sum_{i=1}^n \operatorname{Ric}^X_{\alpha\bar\beta}\, f_{\alpha i}\,\overline{f_{\beta i}},\\ R^Y(\partial f,\overline{\partial f},\partial f,\overline{\partial f}) &=\sum_{\alpha,\beta=1}^m R^Y\bigl(\partial f_x(e_\alpha),\overline{\partial f_x(e_\alpha)},\partial f_x(e_\beta),\overline{\partial f_x(e_\beta)}\bigr). \end{align*} These definitions are independent of the chosen unitary frames because they are tensor contractions. The lower Ricci bound gives \begin{align*} \operatorname{Ric}(\omega_X)(\partial f,\overline{\partial f})\geq -A\,u. \end{align*} The holomorphic bisectional curvature bound of $\omega_Y$ gives \begin{align*} -R^Y(\partial f,\overline{\partial f},\partial f,\overline{\partial f})\geq B\,u^2. \end{align*} Combining these two curvature estimates yields \begin{align*} \Delta_{\omega_X}\log u\geq Bu-A \end{align*} on $U$.[/step]

[guided]The point of introducing $u$ is that it measures exactly the squared norm of the holomorphic differential $\partial f$ with respect to $\omega_X$ and $\omega_Y$. On the [open set](/page/Open%20Set) \begin{align*} U:=\{x\in X:u(x)>0\}, \end{align*} we may take the logarithm, so we define \begin{align*} \log u:U&\to \mathbb{R}\\ x&\mapsto \log(u(x)). \end{align*} The Chern--Lu Bochner computation is the analytic core of the proof. It applies here because $f$ is holomorphic and both metrics are Kähler. It states that, on the locus where the energy density is positive, \begin{align*} \Delta_{\omega_X}\log u \geq \frac{1}{u}\left( \operatorname{Ric}(\omega_X)(\partial f,\overline{\partial f}) - R^Y(\partial f,\overline{\partial f},\partial f,\overline{\partial f}) \right). \end{align*} The two curvature assumptions now enter with opposite signs. Since \begin{align*} \operatorname{Ric}(\omega_X)\geq -A\,\omega_X, \end{align*} contracting both sides against the semipositive tensor determined by $\partial f$ gives \begin{align*} \operatorname{Ric}(\omega_X)(\partial f,\overline{\partial f})\geq -A\,u. \end{align*} Similarly, the assumption that the holomorphic bisectional curvature of $\omega_Y$ is at most $-B$ means that the target curvature term contributes with a favourable sign: \begin{align*} -R^Y(\partial f,\overline{\partial f},\partial f,\overline{\partial f})\geq B\,u^2. \end{align*} Substituting these two estimates into the Chern--Lu inequality gives \begin{align*} \Delta_{\omega_X}\log u \geq \frac{-A\,u+B\,u^2}{u} = Bu-A. \end{align*} Thus, wherever $u>0$, the logarithmic energy density is subharmonic up to the zeroth-order error $A$ and gains the positive term $Bu$.[/guided]

[step:Use Yau's generalized maximum principle for logarithmic inequalities]We use the following consequence of the [Omori--Yau maximum principle](/page/Omori-Yau%20Maximum%20Principle): if $(M,\omega)$ is a complete Kähler manifold whose Ricci curvature is bounded below, and if $v:M\to[0,\infty)$ is smooth and satisfies \begin{align*} \Delta_\omega\log v\geq c\,v-d \end{align*} on the open set $\{v>0\}$ for constants $c>0$ and $d\geq 0$, then \begin{align*} \sup_M v\leq \frac{d}{c}. \end{align*} This formulation is applied to the nonnegative smooth function $u:X\to[0,\infty)$ itself, not to $\log u$ as a globally defined bounded function. Its hypotheses are satisfied: the Kähler metric associated to $\omega_X$ is complete by assumption, the Ricci inequality \begin{align*} \operatorname{Ric}(\omega_X)\geq -A\,\omega_X \end{align*} is a lower Ricci curvature bound for the underlying Riemannian metric, and the previous step gives \begin{align*} \Delta_{\omega_X}\log u\geq B u-A \end{align*} on $\{u>0\}$. Taking $v=u$, $c=B$, and $d=A$ gives \begin{align*} \sup_X u\leq \frac{A}{B}, \end{align*} and hence \begin{align*} u\leq \frac{A}{B} \end{align*} on $X$.[/step]

[guided]The differential inequality from the Chern--Lu computation holds only on the positive locus $\{u>0\}$, because $\log u$ is not defined where $u=0$. We therefore do not apply the maximum principle directly to $\log u$ on $X$. Instead, we use the logarithmic form of Yau's generalized maximum principle, which is designed for this situation. The precise version needed here is the following consequence of the [Omori--Yau maximum principle](/page/Omori-Yau%20Maximum%20Principle). Let $(M,\omega)$ be a complete Kähler manifold with Ricci curvature bounded below. Let $v:M\to[0,\infty)$ be a smooth nonnegative function. If there are constants $c>0$ and $d\geq 0$ such that \begin{align*} \Delta_\omega\log v\geq c\,v-d \end{align*} on the open set $\{v>0\}$, then \begin{align*} \sup_M v\leq \frac{d}{c}. \end{align*} This statement avoids both possible failures of the naive argument: it does not require $\log v$ to be defined on all of $M$, and it does not assume in advance that $\log v$ is bounded above. We verify the hypotheses with $M=X$, $\omega=\omega_X$, and $v=u$. The function $u:X\to[0,\infty)$ is smooth because it is the trace, with respect to the smooth Kähler metric $\omega_X$, of the smooth semipositive form $f^*\omega_Y$. The metric is complete by the hypothesis that $(X,\omega_X)$ is complete. The curvature assumption \begin{align*} \operatorname{Ric}(\omega_X)\geq -A\,\omega_X \end{align*} is exactly a lower Ricci curvature bound for the Kähler metric associated to $\omega_X$. Finally, the Chern--Lu step proved that on $\{u>0\}$, \begin{align*} \Delta_{\omega_X}\log u\geq B u-A. \end{align*} Thus the generalized maximum principle applies with $c=B$ and $d=A$. Since $B>0$ and $A\geq 0$, it yields \begin{align*} \sup_X u\leq \frac{A}{B}. \end{align*} Therefore \begin{align*} u\leq \frac{A}{B} \end{align*} at every point of $X$.[/guided]

[step:Convert the scalar bound into the tensor inequality and handle the case $A=0$] From the first step, the pointwise scalar estimate \begin{align*} u\leq \frac{A}{B} \end{align*} implies \begin{align*} f^*\omega_Y\leq \frac{A}{B}\,\omega_X. \end{align*} If $A=0$, then $u\leq 0$. Since $u\geq 0$ by definition, we have $u=0$ on $X$. Therefore all eigenvalues of $f^*\omega_Y$ with respect to $\omega_X$ vanish, so $f^*\omega_Y=0$. Because $\omega_Y$ is positive definite, this implies $df_x=0$ for every $x\in X$. Since $X$ is connected and $f$ has zero differential everywhere, $f$ is constant. [/step]