[proofplan] We use the smooth Hermitian-metric definition of positivity for a holomorphic line bundle: positivity means that some Chern form $c_1(L,h)$ is a positive real closed $(1,1)$-form. The forward implication is then immediate because every Chern form represents the Bott-Chern first Chern class. Conversely, if a positive representative $\omega$ is Bott-Chern cohomologous to the Chern form of an arbitrary metric $h_0$, a global $\partial\bar\partial$-potential converts the reference metric into one whose curvature differs by exactly that term; multiplying $h_0$ by the corresponding exponential realises the positive representative as a Chern form. In the Kähler case, equality of de Rham classes for real closed $(1,1)$-forms is upgraded to equality in Bott-Chern cohomology by the $\partial\bar\partial$-lemma. [/proofplan]

[step:Fix the curvature and weight conventions] Let $h$ be a smooth Hermitian metric on $L$. On an [open set](/page/Open%20Set) $U\subset X$ with a nowhere-vanishing holomorphic frame $e:U\to L|_U$, define the local weight $\varphi_e:U\to\mathbb R$ by \begin{align*}|e|_h^2=\exp(-\varphi_e).\end{align*} With this convention, the Chern form of $(L,h)$ is locally \begin{align*}c_1(L,h)|_U=\frac{i}{2\pi}\partial\bar\partial\varphi_e.\end{align*} This expression is independent of the chosen holomorphic frame. Indeed, if $e'=g e$ for a nowhere-vanishing [holomorphic function](/page/Holomorphic%20Function) $g:U\to\mathbb C^\times$, then the corresponding weight is \begin{align*}\varphi_{e'}=\varphi_e-\log |g|^2.\end{align*} Since $\log |g|^2$ is pluriharmonic on $U$, one has \begin{align*}\partial\bar\partial \log |g|^2=0.\end{align*} Therefore $\frac{i}{2\pi}\partial\bar\partial\varphi_{e'}=\frac{i}{2\pi}\partial\bar\partial\varphi_e$, so the local formula glues to a globally defined real closed $(1,1)$-form on $X$. The Bott-Chern first Chern class $c_1^{BC}(L)$ is represented by $c_1(L,h)$ for any smooth Hermitian metric $h$ on $L$. If $h_0$ and $h_1$ are two such metrics, then locally $h_1=e^{-\psi}h_0$ for a globally defined smooth real function $\psi:X\to\mathbb R$, and the local weight formula gives \begin{align*}c_1(L,h_1)=c_1(L,h_0)+\frac{i}{2\pi}\partial\bar\partial\psi.\end{align*} Thus changing the metric changes the Chern form by a Bott-Chern exact real $(1,1)$-form. [/step]

[step:Show that positivity gives a positive Bott-Chern representative] Assume that $L$ is positive. By definition, there exists a smooth Hermitian metric $h$ on $L$ such that $c_1(L,h)$ is positive as a real $(1,1)$-form. From the previous step, $c_1(L,h)$ represents $c_1^{BC}(L)$. Hence the Bott-Chern class $c_1^{BC}(L)$ contains a positive real closed $(1,1)$-form. [/step]

[step:Construct a positive metric from a positive Bott-Chern representative]Assume that $c_1^{BC}(L)$ contains a positive real closed $(1,1)$-form. Let $\omega$ denote such a representative. Choose any smooth Hermitian metric $h_0$ on $L$, and let $c_1(L,h_0)$ be its Chern form. Since $\omega$ and $c_1(L,h_0)$ represent the same Bott-Chern class, there exists a smooth real function $\psi:X\to\mathbb R$ such that \begin{align*} \omega-c_1(L,h_0)=\frac{i}{2\pi}\partial\bar\partial\psi. \end{align*} Define a new smooth Hermitian metric $h$ on $L$ by \begin{align*} h=e^{-\psi}h_0. \end{align*} In a holomorphic frame $e$ on an open set $U\subset X$, if $\varphi_{0,e}:U\to\mathbb R$ is the local weight of $h_0$, then the local weight of $h$ is $\varphi_{0,e}+\psi|_U$. Therefore \begin{align*} c_1(L,h)|_U=\frac{i}{2\pi}\partial\bar\partial(\varphi_{0,e}+\psi|_U). \end{align*} Using the local curvature formula for $h_0$, this becomes \begin{align*} c_1(L,h)|_U=c_1(L,h_0)|_U+\frac{i}{2\pi}\partial\bar\partial\psi|_U. \end{align*} By the defining identity for $\psi$, the right-hand side is $\omega|_U$. Since this holds on every holomorphic trivialising open set $U$, we have \begin{align*}c_1(L,h)=\omega.\end{align*} The form $\omega$ is positive, so $h$ has positive Chern form. Hence $L$ is positive.[/step]

[guided]The point of the argument is to turn a positive form in the correct cohomology class into the curvature form of an actual Hermitian metric. We begin with an arbitrary smooth Hermitian metric $h_0$ on $L$. This gives a reference Chern form $c_1(L,h_0)$ representing $c_1^{BC}(L)$. The hypothesis gives another representative $\omega$ of the same Bott-Chern class, and the meaning of equality in Bott-Chern cohomology is exactly that their difference is globally of the form \begin{align*}\omega-c_1(L,h_0)=\frac{i}{2\pi}\partial\bar\partial\psi\end{align*} for some smooth real function $\psi:X\to\mathbb R$. Now define $h=e^{-\psi}h_0$. This is again a smooth Hermitian metric because $e^{-\psi}$ is a smooth positive real-valued function on $X$. To compute its curvature, work on a holomorphic trivialising open set $U\subset X$ with frame $e:U\to L|_U$. Let $\varphi_{0,e}:U\to\mathbb R$ be the local weight of $h_0$, so \begin{align*}|e|_{h_0}^2=\exp(-\varphi_{0,e}).\end{align*} For the new metric $h$, we have \begin{align*}|e|_h^2=e^{-\psi}|e|_{h_0}^2.\end{align*} Substituting the expression for $|e|_{h_0}^2$ gives \begin{align*} |e|_h^2=\exp(-\psi)\exp(-\varphi_{0,e})=\exp(-(\varphi_{0,e}+\psi|_U)). \end{align*} Thus the local weight of $h$ is $\varphi_{0,e}+\psi|_U$. Using the Chern-form formula for the local weight, we obtain \begin{align*} c_1(L,h)|_U=\frac{i}{2\pi}\partial\bar\partial(\varphi_{0,e}+\psi|_U). \end{align*} Linearity of $\partial\bar\partial$ gives \begin{align*} c_1(L,h)|_U=\frac{i}{2\pi}\partial\bar\partial\varphi_{0,e}+\frac{i}{2\pi}\partial\bar\partial\psi|_U. \end{align*} The first term is $c_1(L,h_0)|_U$, so \begin{align*} c_1(L,h)|_U=c_1(L,h_0)|_U+\frac{i}{2\pi}\partial\bar\partial\psi|_U. \end{align*} By the Bott-Chern potential identity, this is exactly $\omega|_U$. Since the computation is local and the forms agree on every holomorphic trivialisation, the global equality is \begin{align*}c_1(L,h)=\omega.\end{align*} Because $\omega$ was assumed positive, the metric $h$ has positive Chern form. Therefore $L$ is positive.[/guided]

[step:Deduce the de Rham version on compact Kähler manifolds] Now assume that $X$ is compact Kähler and that the de Rham class $c_1(L)\in H^2(X,\mathbb R)$ contains a positive real closed $(1,1)$-form $\omega$. Choose a smooth Hermitian metric $h_0$ on $L$. The Chern form $c_1(L,h_0)$ represents the same de Rham class $c_1(L)$ as $\omega$, so the real closed $(1,1)$-form \begin{align*} \eta=\omega-c_1(L,h_0) \end{align*} is de Rham exact. By the $\partial\bar\partial$-lemma for compact Kähler manifolds, applied to the real $d$-exact closed $(1,1)$-form $\eta$, there exists a smooth real function $\psi:X\to\mathbb R$ such that \begin{align*} \eta=\frac{i}{2\pi}\partial\bar\partial\psi. \end{align*} Thus $\omega$ and $c_1(L,h_0)$ represent the same Bott-Chern class. The previous step constructs a smooth Hermitian metric $h=e^{-\psi}h_0$ with \begin{align*}c_1(L,h)=\omega.\end{align*} Since $\omega$ is positive, $L$ is positive. This proves the Kähler de Rham-class assertion and completes the proof. [/step]