[proofplan]
We prove the analytic direction as a direct corollary of the high tensor-power Kodaira theorem proved separately in these notes: sufficiently high powers of a positive line bundle are very ample. We then unpack very ampleness into point separation, tangent-vector separation, and the Kodaira embedding. Conversely, if $L$ is ample, the definition gives a high tensor power that is very ample, so it is the pullback of the hyperplane bundle on projective space; pulling back the Fubini-Study metric and taking a tensor root gives a Hermitian metric on $L$ with positive curvature.
[/proofplan]
[step:Apply the high tensor-power Kodaira theorem]
Let $h$ be a smooth Hermitian metric on $L$ whose Chern curvature form $\Theta_h(L)$ is positive, and define the positive real $(1,1)$-form
\begin{align*}
\omega_h := \frac{i}{2\pi}\Theta_h(L).
\end{align*}
The high tensor-power Kodaira theorem for positive line bundles, proved independently by Kodaira vanishing on blow-ups, states that if a holomorphic line bundle on a compact complex manifold admits a Hermitian metric with positive curvature, then some integer $m_0$ has the property that $L^{\otimes m}$ is very ample for every $m\ge m_0$. The hypotheses are exactly satisfied by $X$, $L$, and $h$. Hence there exists $m_0\in\mathbb N$ such that, for every $m\ge m_0$, the complete linear system of $L^{\otimes m}$ defines a holomorphic embedding into projective space.
[/step]
[step:Record the separation consequences of very ampleness]
Fix $m \geq m_0$. By very ampleness, the complete linear system $H^0(X,L^{\otimes m})$ is base-point free and separates distinct points. It also separates tangent vectors: for every $x\in X$ and every nonzero tangent vector $v\in T_xX$, some global section vanishing at $x$ has nonzero first derivative in the direction $v$. Equivalently, the differential of the Kodaira map associated to $H^0(X,L^{\otimes m})$ is injective at every point. These are precisely the separation properties asserted in the statement.
[/step]
[step:Build the projective embedding from the separating sections]
Choose a complex basis $s_0,\dots,s_N$ of the finite-dimensional [vector space](/page/Vector%20Space) $H^0(X,L^{\otimes m})$. This basis identifies $\mathbb{P}(H^0(X,L^{\otimes m})^*)$ with $\mathbb{P}^N$. Under a local holomorphic frame of $L^{\otimes m}$, the section values $s_j(x)$ become complex numbers, and changing the frame multiplies all coordinates by the same nonzero scalar; hence the projective point is independent of the frame. Since $L^{\otimes m}$ is base-point free, the map
\begin{align*}
\Phi_m: X &\longrightarrow \mathbb{P}^N \\
x &\longmapsto [s_0(x):\cdots:s_N(x)]
\end{align*}
is the coordinate expression of the intrinsic evaluation map $x \mapsto \{s \in H^0(X,L^{\otimes m}) : s(x)=0\}$ and is well-defined and holomorphic. Since the sections separate points, $\Phi_m$ is injective. Since the sections separate tangent vectors, the differential
\begin{align*}
d(\Phi_m)_x: T_xX &\longrightarrow T_{\Phi_m(x)}\mathbb{P}^N
\end{align*}
is injective for every $x \in X$. Thus $\Phi_m$ is a holomorphic injective immersion.
Because $X$ is compact and $\mathbb{P}^N$ is Hausdorff, a continuous injective map $\Phi_m: X \to \mathbb{P}^N$ is a homeomorphism onto its image. Therefore $\Phi_m$ is a holomorphic embedding. This proves that positivity of $L$ implies ampleness and, more precisely, that all sufficiently large powers $L^{\otimes m}$ define projective embeddings.
[/step]
[step:Pull back the Fubini Study metric to prove the converse]
Assume conversely that $L$ is ample. We use the standard convention for ampleness of a holomorphic line bundle on a compact complex manifold: there exists an integer $r \in \mathbb{N}$ such that $L^{\otimes r}$ is very ample. Hence there is a holomorphic embedding
\begin{align*}
\iota: X &\longrightarrow \mathbb{P}^N
\end{align*}
for some $N \in \mathbb{N}$ such that
\begin{align*}
L^{\otimes r} &\cong \iota^*\mathcal{O}_{\mathbb{P}^N}(1).
\end{align*}
Let $h_{FS}$ denote the Fubini-Study Hermitian metric on $\mathcal{O}_{\mathbb{P}^N}(1)$. The standard curvature computation for the Fubini-Study metric gives that its curvature form is the positive Fubini-Study form $\omega_{FS}$. Pulling back gives a smooth Hermitian metric $h_r := \iota^*h_{FS}$ on $L^{\otimes r}$ with curvature form
\begin{align*}
\frac{i}{2\pi}\Theta_{h_r}(L^{\otimes r}) = \iota^*\omega_{FS}.
\end{align*}
Since $\iota$ is an embedding and $\omega_{FS}$ is positive on $\mathbb{P}^N$, the pullback $\iota^*\omega_{FS}$ is positive on $X$.
Choose local holomorphic frames $e$ for $L$. Define a local metric $h_L$ on $L$ by
\begin{align*}
h_L(e,e) := h_r(e^{\otimes r}, e^{\otimes r})^{1/r}.
\end{align*}
This definition is independent of the frame because both sides transform by $|g|^2$ under the change of frame $e \mapsto ge$ for a nowhere-vanishing [holomorphic function](/page/Holomorphic%20Function) $g$. Therefore the local definitions glue to a smooth Hermitian metric $h_L$ on $L$. Its curvature satisfies
\begin{align*}
\frac{i}{2\pi}\Theta_{h_L}(L) = \frac{1}{r}\frac{i}{2\pi}\Theta_{h_r}(L^{\otimes r}) = \frac{1}{r}\iota^*\omega_{FS},
\end{align*}
which is positive because $r>0$ and $\iota^*\omega_{FS}$ is positive. Thus $L$ is positive in the sense of Kodaira.
[/step]
[step:Conclude the equivalence]
The first three steps show that a holomorphic line bundle admitting a smooth Hermitian metric with positive curvature has sufficiently high tensor powers whose global sections define holomorphic embeddings into projective space. This is precisely ampleness. The fourth step shows that ampleness produces a smooth Hermitian metric with positive curvature. Hence $L$ is positive in the sense of Kodaira if and only if $L$ is ample.
[/step]
