[proofplan] We prove the Chern-form identity locally in a holomorphic frame of $\mathcal{O}_{\mathbb{P}^N}(1)$. In such a frame, the first Chern form is obtained by applying the local differential operator $\frac{i}{2\pi}\bar{\partial}\partial \log$ to the metric weight, and the pullback metric has weight obtained by composing with $\Phi$. Since $\Phi$ is holomorphic, pullback commutes with $\partial$ and $\bar{\partial}$ on smooth functions, giving the identity. Positivity then follows by evaluating the pulled-back Fubini-Study form on tangent vectors and using injectivity of $d\Phi_x$ for a holomorphic embedding. [/proofplan]

[step:Compute the pulled-back Chern form in a local holomorphic frame]Let \begin{align*} L: \mathbb{P}^N &\to \mathbb{P}^N \end{align*} denote the holomorphic line bundle $\mathcal{O}_{\mathbb{P}^N}(1)$, equipped with the Hermitian metric $h_{\mathrm{FS}}$. Let $V \subset \mathbb{P}^N$ be an [open set](/page/Open%20Set) on which $L$ admits a nowhere-vanishing holomorphic frame \begin{align*} e: V &\to L|_V. \end{align*} Define the local metric weight \begin{align*} \rho: V &\to \mathbb{R}_{>0} \\ y &\mapsto h_{\mathrm{FS},y}(e(y), e(y)). \end{align*} By the local formula for the first Chern form of a Hermitian holomorphic line bundle, \begin{align*} c_1(L,h_{\mathrm{FS}})|_V = \frac{i}{2\pi}\bar{\partial}\partial \log \rho. \end{align*} Let $U := \Phi^{-1}(V) \subset X$. The pulled-back line bundle $\Phi^*L \to X$ has a nowhere-vanishing holomorphic frame over $U$ given by \begin{align*} \widetilde e: U &\to (\Phi^*L)|_U \\ x &\mapsto (x, e(\Phi(x))). \end{align*} The pulled-back metric $\Phi^*h_{\mathrm{FS}}$ has local metric weight \begin{align*} \widetilde \rho: U &\to \mathbb{R}_{>0} \\ x &\mapsto (\Phi^*h_{\mathrm{FS}})_x(\widetilde e(x), \widetilde e(x)) = \rho(\Phi(x)). \end{align*} Thus $\widetilde \rho = \rho \circ \Phi$. Applying the same local formula for the first Chern form gives \begin{align*} c_1(\Phi^*L,\Phi^*h_{\mathrm{FS}})|_U = \frac{i}{2\pi}\bar{\partial}\partial \log(\rho \circ \Phi). \end{align*} Because $\Phi$ is holomorphic, pullback by $\Phi$ commutes with $\partial$ and $\bar{\partial}$ on smooth functions. Therefore \begin{align*} \frac{i}{2\pi}\bar{\partial}\partial \log(\rho \circ \Phi) &= \Phi^*\left(\frac{i}{2\pi}\bar{\partial}\partial \log \rho\right) \\ &= \Phi^*\left(c_1(L,h_{\mathrm{FS}})|_V\right). \end{align*} Since these local identities hold on every such trivialising open set $V \subset \mathbb{P}^N$, they glue to the global identity \begin{align*} c_1(\Phi^*L,\Phi^*h_{\mathrm{FS}}) = \Phi^*c_1(L,h_{\mathrm{FS}}) = \Phi^*\omega_{\mathrm{FS}}. \end{align*}[/step]

[guided]The point is that the first Chern form of a Hermitian line bundle can be computed from any local holomorphic frame. We choose a trivialising open set $V \subset \mathbb{P}^N$ and a nowhere-vanishing holomorphic frame \begin{align*} e: V &\to L|_V, \end{align*} where $L = \mathcal{O}_{\mathbb{P}^N}(1)$. The metric is then encoded by the positive smooth function \begin{align*} \rho: V &\to \mathbb{R}_{>0} \\ y &\mapsto h_{\mathrm{FS},y}(e(y), e(y)). \end{align*} In this frame, the first Chern form is \begin{align*} c_1(L,h_{\mathrm{FS}})|_V = \frac{i}{2\pi}\bar{\partial}\partial \log \rho. \end{align*} Now pull everything back to $X$. On the open set $U := \Phi^{-1}(V)$, the pulled-back bundle $\Phi^*L$ has the induced holomorphic frame \begin{align*} \widetilde e: U &\to (\Phi^*L)|_U \\ x &\mapsto (x, e(\Phi(x))). \end{align*} The pulled-back metric is defined by evaluating the original metric at $\Phi(x)$, so its local weight is \begin{align*} \widetilde \rho: U &\to \mathbb{R}_{>0} \\ x &\mapsto (\Phi^*h_{\mathrm{FS}})_x(\widetilde e(x), \widetilde e(x)) = h_{\mathrm{FS},\Phi(x)}(e(\Phi(x)), e(\Phi(x))) = \rho(\Phi(x)). \end{align*} Hence $\widetilde \rho = \rho \circ \Phi$. The first Chern form of the pulled-back bundle is therefore \begin{align*} c_1(\Phi^*L,\Phi^*h_{\mathrm{FS}})|_U = \frac{i}{2\pi}\bar{\partial}\partial \log(\rho \circ \Phi). \end{align*} Because $\Phi$ is holomorphic, the operators $\partial$ and $\bar{\partial}$ commute with pullback on smooth functions. Applying this to the function $\log \rho: V \to \mathbb{R}$ gives \begin{align*} \frac{i}{2\pi}\bar{\partial}\partial \log(\rho \circ \Phi) &= \Phi^*\left(\frac{i}{2\pi}\bar{\partial}\partial \log \rho\right) \\ &= \Phi^*\left(c_1(L,h_{\mathrm{FS}})|_V\right). \end{align*} This proves the desired identity on $U = \Phi^{-1}(V)$. Since the trivialising open sets $V$ cover $\mathbb{P}^N$, the open sets $\Phi^{-1}(V)$ cover $X$, and the local identities agree on overlaps because both sides are globally defined $(1,1)$-forms. Thus \begin{align*} c_1(\Phi^*L,\Phi^*h_{\mathrm{FS}}) = \Phi^*c_1(L,h_{\mathrm{FS}}) = \Phi^*\omega_{\mathrm{FS}}. \end{align*}[/guided]

[step:Evaluate the pulled-back form on tangent vectors]Assume now that $\Phi$ is a holomorphic embedding. Let $x \in X$, and let $v \in T_x^{1,0}X$ be nonzero. The complex differential of $\Phi$ at $x$ is the complex-[linear map](/page/Linear%20Map) \begin{align*} d\Phi_x^{1,0}: T_x^{1,0}X &\to T_{\Phi(x)}^{1,0}\mathbb{P}^N. \end{align*} Since $\Phi$ is a holomorphic embedding, $d\Phi_x^{1,0}$ is injective, so \begin{align*} d\Phi_x^{1,0}(v) \neq 0. \end{align*} Using the identity proved above and the definition of pullback of a $(1,1)$-form, \begin{align*} c_1(\Phi^*L,\Phi^*h_{\mathrm{FS}})_x(v,\overline v) &= (\Phi^*\omega_{\mathrm{FS}})_x(v,\overline v) \\ &= \omega_{\mathrm{FS},\Phi(x)} \left(d\Phi_x^{1,0}(v), \overline{d\Phi_x^{1,0}(v)}\right). \end{align*} The Fubini-Study form $\omega_{\mathrm{FS}}$ is positive on $\mathbb{P}^N$, and $d\Phi_x^{1,0}(v)$ is nonzero. Therefore \begin{align*} \omega_{\mathrm{FS},\Phi(x)} \left(d\Phi_x^{1,0}(v), \overline{d\Phi_x^{1,0}(v)}\right) > 0. \end{align*} Hence \begin{align*} c_1(\Phi^*L,\Phi^*h_{\mathrm{FS}})_x(v,\overline v) > 0. \end{align*} Since this holds for every $x \in X$ and every nonzero $v \in T_x^{1,0}X$, the Hermitian holomorphic line bundle $(\Phi^*L,\Phi^*h_{\mathrm{FS}})$ is positive.[/step]

[guided]We now prove positivity. Fix a point $x \in X$ and a nonzero tangent vector $v \in T_x^{1,0}X$. The holomorphic map $\Phi$ has complex differential \begin{align*} d\Phi_x^{1,0}: T_x^{1,0}X &\to T_{\Phi(x)}^{1,0}\mathbb{P}^N. \end{align*} The embedding hypothesis is used exactly here: a holomorphic embedding is an immersion, so this differential is injective at every point. Therefore, because $v \neq 0$, \begin{align*} d\Phi_x^{1,0}(v) \neq 0. \end{align*} From the first part of the proof, the Chern form of the pulled-back Hermitian line bundle is the pulled-back Fubini-Study form: \begin{align*} c_1(\Phi^*L,\Phi^*h_{\mathrm{FS}}) = \Phi^*\omega_{\mathrm{FS}}. \end{align*} Evaluating both sides on $(v,\overline v)$ gives \begin{align*} c_1(\Phi^*L,\Phi^*h_{\mathrm{FS}})_x(v,\overline v) &= (\Phi^*\omega_{\mathrm{FS}})_x(v,\overline v). \end{align*} By the definition of pullback of a $(1,1)$-form, the right-hand side is obtained by applying $\omega_{\mathrm{FS}}$ to the pushed-forward tangent vector: \begin{align*} (\Phi^*\omega_{\mathrm{FS}})_x(v,\overline v) = \omega_{\mathrm{FS},\Phi(x)} \left(d\Phi_x^{1,0}(v), \overline{d\Phi_x^{1,0}(v)}\right). \end{align*} The Fubini-Study form is a positive $(1,1)$-form on $\mathbb{P}^N$. Since $d\Phi_x^{1,0}(v)$ is nonzero, positivity gives \begin{align*} \omega_{\mathrm{FS},\Phi(x)} \left(d\Phi_x^{1,0}(v), \overline{d\Phi_x^{1,0}(v)}\right) > 0. \end{align*} Combining the displayed equalities, we obtain \begin{align*} c_1(\Phi^*L,\Phi^*h_{\mathrm{FS}})_x(v,\overline v) > 0. \end{align*} Because $x \in X$ and nonzero $v \in T_x^{1,0}X$ were arbitrary, the first Chern form of $(\Phi^*L,\Phi^*h_{\mathrm{FS}})$ is positive at every point. This is precisely the positivity of the Hermitian holomorphic line bundle.[/guided]