[proofplan]
The proof is local, because positivity of a Hermitian line bundle is detected by the curvature form in local holomorphic coordinates. In a holomorphic frame $e$, the Chern curvature is represented by the real $(1,1)$-form $i\partial\bar\partial \varphi_e$, where $\varphi_e$ is the metric weight. We verify that changing the frame only adds a pluriharmonic function to the weight, so the Levi form is independent of the chosen frame. The positive, semipositive, and negative cases then become exactly the corresponding positivity conditions for the Levi form of $\varphi_e$.
[/proofplan]
[step:Compute the curvature form in a local holomorphic frame]
Let $U \subset X$ be an [open set](/page/Open%20Set) and let $e: U \to L$ be a nowhere-vanishing holomorphic frame. The weight $\varphi_e: U \to \mathbb{R}$ is the smooth function determined by
\begin{align*}
h(e,e)=e^{-\varphi_e}.
\end{align*}
For a smooth local section $s: U \to L$ written as $s=f e$, where $f: U \to \mathbb{C}$ is smooth, write the Chern connection in the frame $e$ as
\begin{align*}
\nabla e = A \otimes e,
\end{align*}
where $A$ is a smooth complex-valued one-form on $U$. Since $e$ is holomorphic and the Chern connection has $(0,1)$-part equal to the holomorphic structure operator $\bar\partial$, the one-form $A$ has type $(1,0)$. Metric compatibility gives
\begin{align*}
d(h(e,e))
= h(\nabla e,e)+h(e,\nabla e)
= (A+\overline{A})h(e,e).
\end{align*}
Taking the $(1,0)$-part and using $h(e,e)=e^{-\varphi_e}$ gives
\begin{align*}
A
= \partial \log h(e,e)
= \partial(-\varphi_e)
= -\partial \varphi_e.
\end{align*}
Therefore the curvature of the Chern connection on $L$ over $U$ is
\begin{align*}
\Theta_h(L)\big|_U
= F_\nabla
= \bar\partial(-\partial \varphi_e)
= \partial\bar\partial \varphi_e,
\end{align*}
where the last equality uses $\bar\partial\partial = -\partial\bar\partial$ on functions. Hence
\begin{align*}
i\Theta_h(L)\big|_U = i\partial\bar\partial \varphi_e.
\end{align*}
[guided]
We first translate the metric into a scalar function. Since $e: U \to L$ is a nowhere-vanishing holomorphic frame, every local section of $L$ over $U$ can be written uniquely as $s=f e$ for a smooth function $f: U \to \mathbb{C}$. The Hermitian metric is therefore encoded on this frame by the positive function $h(e,e)$, and the weight $\varphi_e: U \to \mathbb{R}$ is defined by
\begin{align*}
h(e,e)=e^{-\varphi_e}.
\end{align*}
The Chern connection is the unique connection compatible with both the holomorphic structure and the Hermitian metric. To compute its connection form, write
\begin{align*}
\nabla e = A \otimes e,
\end{align*}
where $A$ is a smooth complex-valued one-form on $U$. Because $e$ is holomorphic and the $(0,1)$-part of the Chern connection is the holomorphic structure operator $\bar\partial$, the $(0,1)$-part of $A$ vanishes. Hence $A$ has type $(1,0)$.
Metric compatibility determines $A$. Applying compatibility to the pair $(e,e)$ gives
\begin{align*}
d(h(e,e))
= h(\nabla e,e)+h(e,\nabla e)
= (A+\overline{A})h(e,e).
\end{align*}
Taking the $(1,0)$-part of this identity gives
\begin{align*}
\partial h(e,e) = A h(e,e),
\end{align*}
so, since $h(e,e)>0$,
\begin{align*}
A
= \partial \log h(e,e)
= \partial(-\varphi_e)
= -\partial \varphi_e.
\end{align*}
Thus
\begin{align*}
\nabla e = -\partial \varphi_e \otimes e.
\end{align*}
The curvature form is the curvature two-form $F_\nabla$ of this connection. Since $L$ has rank one, the connection form is scalar-valued, so no matrix commutator term appears. Thus
\begin{align*}
\Theta_h(L)\big|_U
= F_\nabla
= \bar\partial(-\partial \varphi_e)
= \partial\bar\partial \varphi_e,
\end{align*}
where $F_\nabla$ denotes the curvature two-form of the Chern connection $\nabla$. The last equality holds because $\bar\partial\partial = -\partial\bar\partial$ on smooth functions. Multiplying by $i$ gives the real curvature form used in positivity:
\begin{align*}
i\Theta_h(L)\big|_U = i\partial\bar\partial \varphi_e.
\end{align*}
This identity is the bridge between geometry and local potential theory: the curvature of the Hermitian line bundle is exactly the Levi form of the local metric weight.
[/guided]
[/step]
[step:Show that changing holomorphic frames only adds a pluriharmonic function]
Let $U_\alpha,U_\beta \subset X$ be two frame domains with nowhere-vanishing holomorphic frames $e_\alpha:U_\alpha\to L$ and $e_\beta:U_\beta\to L$. On the overlap $U_{\alpha\beta}:=U_\alpha\cap U_\beta$, define the holomorphic transition function
\begin{align*}
g_{\alpha\beta}: U_{\alpha\beta} &\to \mathbb{C}^{\times}
\end{align*}
by
\begin{align*}
e_\alpha = g_{\alpha\beta} e_\beta.
\end{align*}
Then
\begin{align*}
e^{-\varphi_\alpha}
= h(e_\alpha,e_\alpha)
= h(g_{\alpha\beta}e_\beta,g_{\alpha\beta}e_\beta)
= |g_{\alpha\beta}|^2 e^{-\varphi_\beta}.
\end{align*}
Taking logarithms gives
\begin{align*}
\varphi_\alpha=\varphi_\beta-\log |g_{\alpha\beta}|^2.
\end{align*}
The function $g_{\alpha\beta}$ is holomorphic and nowhere zero. For each point $p\in U_{\alpha\beta}$ there is an open neighbourhood $V\subset U_{\alpha\beta}$ and a [holomorphic function](/page/Holomorphic%20Function) $F:V\to\mathbb{C}$ such that $g_{\alpha\beta}=e^F$ on $V$. On $V$,
\begin{align*}
\log |g_{\alpha\beta}|^2 = F+\overline{F}.
\end{align*}
Since $F$ is holomorphic and $\overline{F}$ is antiholomorphic,
\begin{align*}
\partial\bar\partial \log |g_{\alpha\beta}|^2
=
\partial\bar\partial(F+\overline{F})
=0.
\end{align*}
Therefore
\begin{align*}
\partial\bar\partial\varphi_\alpha
=
\partial\bar\partial\varphi_\beta
\end{align*}
on $U_{\alpha\beta}$.
[guided]
The point of this step is to check that the Levi form of the weight is not an artifact of the chosen frame. Suppose $e_\alpha:U_\alpha\to L$ and $e_\beta:U_\beta\to L$ are two nowhere-vanishing holomorphic frames. On the overlap $U_{\alpha\beta}:=U_\alpha\cap U_\beta$, there is a unique holomorphic map
\begin{align*}
g_{\alpha\beta}: U_{\alpha\beta} &\to \mathbb{C}^{\times}
\end{align*}
such that
\begin{align*}
e_\alpha = g_{\alpha\beta} e_\beta.
\end{align*}
The codomain is $\mathbb{C}^{\times}$ because both frames are nowhere vanishing.
Now compare the two metric weights. By Hermitian homogeneity of $h$,
\begin{align*}
e^{-\varphi_\alpha}
= h(e_\alpha,e_\alpha)
= h(g_{\alpha\beta}e_\beta,g_{\alpha\beta}e_\beta)
= |g_{\alpha\beta}|^2 h(e_\beta,e_\beta)
= |g_{\alpha\beta}|^2 e^{-\varphi_\beta}.
\end{align*}
Taking logarithms yields
\begin{align*}
\varphi_\alpha=\varphi_\beta-\log |g_{\alpha\beta}|^2.
\end{align*}
It remains to show that the correction term has zero Levi form. Because $g_{\alpha\beta}$ is holomorphic and nowhere zero, each point $p\in U_{\alpha\beta}$ has an open neighbourhood $V\subset U_{\alpha\beta}$ on which a holomorphic logarithm exists. Thus there is a holomorphic function $F:V\to\mathbb{C}$ such that $g_{\alpha\beta}=e^F$ on $V$. On this neighbourhood,
\begin{align*}
\log |g_{\alpha\beta}|^2 = F+\overline{F}.
\end{align*}
The function $F$ is holomorphic, so $\bar\partial F=0$. The function $\overline{F}$ is antiholomorphic, so $\partial \overline{F}=0$. Hence
\begin{align*}
\partial\bar\partial \log |g_{\alpha\beta}|^2
=
\partial\bar\partial(F+\overline{F})
=0.
\end{align*}
Applying $\partial\bar\partial$ to the transformation law for the weights gives
\begin{align*}
\partial\bar\partial\varphi_\alpha
=
\partial\bar\partial\varphi_\beta.
\end{align*}
Thus the local Levi form of the weight is independent of the holomorphic frame, even though the weight itself changes by the real part of a holomorphic function.
[/guided]
[/step]
[step:Identify positivity with strict plurisubharmonicity of the weights]
Let $p\in X$, choose a holomorphic coordinate chart $(U,z)$ around $p$ with
\begin{align*}
z=(z_1,\dots,z_n):U\to z(U)\subset\mathbb{C}^n,
\end{align*}
and choose a nowhere-vanishing holomorphic frame $e:U\to L$. Define $T_p^{1,0}X$ to be the holomorphic tangent space at $p$, namely the complex [vector space](/page/Vector%20Space) spanned in the chosen chart by $\frac{\partial}{\partial z_1}\big|_p,\dots,\frac{\partial}{\partial z_n}\big|_p$. In these coordinates,
\begin{align*}
i\partial\bar\partial\varphi_e
=
i\sum_{j,k=1}^n
\frac{\partial^2\varphi_e}{\partial z_j\partial\bar z_k}
\, dz_j\wedge d\bar z_k.
\end{align*}
For a tangent vector $\xi=\sum_{j=1}^n \xi_j \frac{\partial}{\partial z_j}\in T_p^{1,0}X$, the associated Hermitian form is
\begin{align*}
(i\partial\bar\partial\varphi_e)_p(\xi,\overline{\xi})
=
\sum_{j,k=1}^n
\frac{\partial^2\varphi_e}{\partial z_j\partial\bar z_k}(p)\,
\xi_j\overline{\xi_k}.
\end{align*}
By the previous step this Hermitian form is independent of the chosen holomorphic frame.
By definition, $(L,h)$ is positive exactly when $i\Theta_h(L)$ is positive definite on $T_p^{1,0}X$ for every $p\in X$. Since $i\Theta_h(L)=i\partial\bar\partial\varphi_e$ in every local holomorphic frame, this is equivalent to the Levi form of every local weight $\varphi_e$ being positive definite at every point of its domain. For smooth functions, positive definiteness of the Levi form is exactly strict plurisubharmonicity. Therefore $(L,h)$ is positive if and only if every $\varphi_e$ is strictly plurisubharmonic.
[guided]
We now translate the curvature condition into the standard local condition for plurisubharmonicity. Fix $p\in X$. Choose a holomorphic coordinate chart $(U,z)$ around $p$, where
\begin{align*}
z=(z_1,\dots,z_n):U\to z(U)\subset\mathbb{C}^n,
\end{align*}
and choose a nowhere-vanishing holomorphic frame $e:U\to L$. The holomorphic tangent space $T_p^{1,0}X$ is the complex vector space spanned in this chart by
\begin{align*}
\frac{\partial}{\partial z_1}\bigg|_p,\dots,\frac{\partial}{\partial z_n}\bigg|_p.
\end{align*}
In these coordinates, the local curvature identity from the first step gives
\begin{align*}
i\Theta_h(L)\big|_U
=
i\partial\bar\partial\varphi_e
=
i\sum_{j,k=1}^n
\frac{\partial^2\varphi_e}{\partial z_j\partial\bar z_k}
\, dz_j\wedge d\bar z_k.
\end{align*}
For a tangent vector $\xi=\sum_{j=1}^n \xi_j \frac{\partial}{\partial z_j}\in T_p^{1,0}X$, the Hermitian form associated to this $(1,1)$-form is
\begin{align*}
(i\partial\bar\partial\varphi_e)_p(\xi,\overline{\xi})
=
\sum_{j,k=1}^n
\frac{\partial^2\varphi_e}{\partial z_j\partial\bar z_k}(p)\,
\xi_j\overline{\xi_k}.
\end{align*}
The previous step shows that changing the holomorphic frame replaces $\varphi_e$ by $\varphi_e$ plus a pluriharmonic correction, so this Hermitian form is independent of the frame.
By definition, $(L,h)$ is positive exactly when $i\Theta_h(L)$ is positive definite on $T_p^{1,0}X$ for every point $p\in X$. Using the displayed identity, this means exactly that the Hermitian matrix
\begin{align*}
\left(\frac{\partial^2\varphi_e}{\partial z_j\partial\bar z_k}(p)\right)_{j,k=1}^n
\end{align*}
is positive definite at every point of every frame domain. For a smooth real-valued function, that positive definiteness of the Levi form is precisely strict plurisubharmonicity. Therefore $(L,h)$ is positive if and only if every local weight $\varphi_e$ is strictly plurisubharmonic.
[/guided]
[/step]
[step:Identify semipositivity and negativity with the corresponding weight conditions]
The same local formula gives the semipositive case. The bundle $(L,h)$ is semipositive exactly when $i\Theta_h(L)$ is positive semidefinite on $T_p^{1,0}X$ for every $p\in X$. Since
\begin{align*}
i\Theta_h(L)=i\partial\bar\partial\varphi_e
\end{align*}
on every frame domain, this is equivalent to the Levi form of every weight $\varphi_e$ being positive semidefinite. For smooth functions, this is exactly plurisubharmonicity. Hence $(L,h)$ is semipositive if and only if every $\varphi_e$ is plurisubharmonic.
Finally, $(L,h)$ is negative exactly when $-i\Theta_h(L)$ is positive definite. On a frame domain,
\begin{align*}
-i\Theta_h(L)
=
i\partial\bar\partial(-\varphi_e).
\end{align*}
Thus negativity of $(L,h)$ is equivalent to positive definiteness of the Levi form of $-\varphi_e$ for every local weight $\varphi_e$, which is exactly strict plurisubharmonicity of $-\varphi_e$. This proves all three equivalences.
[guided]
The semipositive case uses the same local curvature computation, with positive definiteness replaced by positive semidefiniteness. By definition, $(L,h)$ is semipositive exactly when $i\Theta_h(L)$ is positive semidefinite on $T_p^{1,0}X$ for every $p\in X$. On each frame domain, the first step gives
\begin{align*}
i\Theta_h(L)=i\partial\bar\partial\varphi_e.
\end{align*}
Therefore semipositivity is equivalent to the Hermitian form associated to $i\partial\bar\partial\varphi_e$ being positive semidefinite at every point. For smooth real-valued functions, positive semidefiniteness of the Levi form is exactly plurisubharmonicity. Hence $(L,h)$ is semipositive if and only if every local weight $\varphi_e$ is plurisubharmonic.
For negativity, the sign of the curvature is reversed. By definition, $(L,h)$ is negative exactly when $-i\Theta_h(L)$ is positive definite on $T_p^{1,0}X$ for every $p\in X$. Using the local curvature formula,
\begin{align*}
-i\Theta_h(L)
=
-i\partial\bar\partial\varphi_e
=
i\partial\bar\partial(-\varphi_e).
\end{align*}
Thus the negative curvature condition for $(L,h)$ is the positive definiteness of the Levi form of the function $-\varphi_e$. For smooth real-valued functions, this is precisely strict plurisubharmonicity of $-\varphi_e$. This proves the semipositive and negative equivalences, and together with the previous step proves the theorem.
[/guided]
[/step]
