[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*}
[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]
[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.
[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]
[/step]
