Smooth projective analytifications have rational Kähler forms

Let $x\in X(\mathbb{C})$ be a closed point. Since $X\to \mathrm{Spec}\mathbb{C}$ is smooth and locally of finite presentation, the Jacobian criterion [F4] at $x$ gives a standard smooth presentation $C_h\cong (\mathbb{C}[t_1,\dots ,t_m]\mathrm{/}(f_1,\dots ,f_r){)}_g$ in which some $r\times r$ minor of $\left(\mathrm{\partial }{f}_j\mathrm{/}\mathrm{\partial }{t}_i\right)$ is invertible, the components of the chart having dimension $m-r$; as $X$ has pure dimension $n$, this gives $m-r=n$ [F4]. The complex points of the chart model are $Z=\{t\in {\mathbb{C}}^m:f_1(t)=\cdots =f_r(t)=0,g(t)\mathrm{\ne }0\}$, and the point $x$ corresponds to a point of $Z$. If $r=0$, then $Z=\{g\mathrm{\ne }0\}\subseteq {\mathbb{C}}^m={\mathbb{C}}^n$ is an open coordinate set and the ambient coordinates are the chart. If $r\ge 1$ and $n\ge 1$, reorder the coordinates as $t=(t^{\mathrm{\prime }},t^{\mathrm{\prime }\mathrm{\prime }})\in {\mathbb{C}}^n\times {\mathbb{C}}^r$ so that $\det \left(\mathrm{\partial }{f}_j\mathrm{/}\mathrm{\partial }{t}_k^{\mathrm{\prime }\mathrm{\prime }}\right)$ is invertible at the point; the holomorphic implicit function theorem [F5] then gives a holomorphic $\phi :A\to {\mathbb{C}}^r$ on a neighbourhood $A$ of $x^{\mathrm{\prime }}$ with $f(t^{\mathrm{\prime }},\phi (t^{\mathrm{\prime }}))=0$ and, after shrinking and imposing $g\mathrm{\ne }0$, with $f(t)=0⟺t^{\mathrm{\prime }\mathrm{\prime }}=\phi (t^{\mathrm{\prime }})$ on $A\times B$, so that $t^{\mathrm{\prime }}\mapsto (t^{\mathrm{\prime }},\phi (t^{\mathrm{\prime }}))$ is a homeomorphism of a neighbourhood of $x^{\mathrm{\prime }}$ onto a neighbourhood of $x$ in $Z$, the inverse being the projection $(t^{\mathrm{\prime }},t^{\mathrm{\prime }\mathrm{\prime }})\mapsto t^{\mathrm{\prime }}$; this is a chart of a complex manifold of dimension $n$ [F2]. If $n=0$ and $m=r\ge 1$, the inverse function theorem in several variables [F5] applied to $(f_1,\dots ,f_r):{\mathbb{C}}^m\to {\mathbb{C}}^m$, whose Jacobian is invertible at $x$, shows that $Z$ is locally the single point $x$, a chart into ${\mathbb{C}}^0=\{0\}$; these singleton charts give the discrete structure on $X(\mathbb{C})$ in the $n=0$ convention of [F1], and if $X=\mathrm{\varnothing }$ there are no closed points and we set $X^{\mathrm{a}\mathrm{n}}=\mathrm{\varnothing }$. If $X\mathrm{\ne }\mathrm{\varnothing }$, then $X(\mathbb{C})\mathrm{\ne }\mathrm{\varnothing }$: a nonempty affine open has a nonzero finite-type complex coordinate ring, hence a maximal ideal by [F11]; its residue field is finite over the algebraically closed $\mathbb{C}$, hence equals $\mathbb{C}$. Such a rational point is closed in the whole finite-type scheme: any specialization lies in an affine neighbourhood that also contains the original point, which is maximal there, so the specialization equals that point. The same residue-field argument identifies all closed points with complex-valued coordinate points.

On the standard chart $U_j\cong {\mathbb{C}}^N$ of ${\mathbb{P}}^N(\mathbb{C})$ put ${\omega }_{\mathrm{F}\mathrm{S}}:=\frac{i}{2\pi }\mathrm{\partial }\bar{\mathrm{\partial }}\log (1+\mathrm{\mid }{z}^{(j)}{\mathrm{\mid }}^2)$, $\mathrm{\mid }{z}^{(j)}{\mathrm{\mid }}^2={\sum }_l\mathrm{\mid }{z}_l^{(j)}{\mathrm{\mid }}^2$. On an overlap one has $1+\mathrm{\mid }{z}^{(j)}{\mathrm{\mid }}^2=\mathrm{\mid }h{\mathrm{\mid }}^2(1+\mathrm{\mid }{z}^{(k)}{\mathrm{\mid }}^2)$ for the nonvanishing holomorphic function $h=Z_k\mathrm{/}{Z}_j$, and $\mathrm{\partial }\bar{\mathrm{\partial }}\log \mathrm{\mid }h{\mathrm{\mid }}^2=0$ because locally one can choose a holomorphic logarithm of $h$: near a point write $h=h_0(1+v)$ with $\mathrm{\mid }v\mathrm{\mid }<1$ and use the convergent power series for $\log (1+v)$; then $\log \mathrm{\mid }h{\mathrm{\mid }}^2$ is a constant plus that holomorphic logarithm and its conjugate; hence the local forms agree on overlaps and define a global smooth real form of type $(1,1)$ on ${\mathbb{P}}^N(\mathbb{C})$ [F1]. It is closed, since $d(\mathrm{\partial }\bar{\mathrm{\partial }}f)=\bar{\mathrm{\partial }}\mathrm{\partial }\bar{\mathrm{\partial }}f=0$ for smooth $f$. It is positive: in a chart its Hermitian matrix is $((1+\mathrm{\mid }z{\mathrm{\mid }}^2){\delta }_{jk}-{\bar{z}}_j{z}_k)\mathrm{/}(2\pi (1+\mathrm{\mid }z{\mathrm{\mid }}^2{)}^2)$, and for $v\mathrm{\ne }0$ the numerator quadratic form equals $(1+\mathrm{\mid }z{\mathrm{\mid }}^2)\mathrm{\mid }v{\mathrm{\mid }}^2-\mid {\sum }_j{z}_j{v}_j{\mid }^2>0$ by Cauchy-Schwarz. For $N=0$, ${\mathbb{P}}^0(\mathbb{C})$ is a point and ${\omega }_{\mathrm{F}\mathrm{S}}=0$.

On overlaps, regular coordinate functions are locally fractions with nonvanishing denominators, hence holomorphic on the graph charts of step 1.1; the reverse transition is obtained in the same way. Zariski-open subsets are open for this topology, because on an affine presentation their complements are polynomial zero loci. This gives compatible local holomorphic atlases. For a smooth separated finite-type complex scheme $Y$, choose a finite affine cover, available by finite type; each affine point locus is a subspace of a finite-dimensional complex coordinate space and is second-countable, so the resulting finite open cover makes $Y^{\mathrm{a}\mathrm{n}}$ second-countable. Its diagonal is closed: on each product of affine opens, [F12] realizes the scheme diagonal as a closed subscheme, whose complex point locus is cut out by polynomial equations and hence is Euclidean closed. A closed diagonal implies Hausdorffness, since its open complement supplies product neighbourhoods separating any two distinct points. Thus these charts define a complex manifold for $Y$ (with the dimension-zero and empty conventions of [F1]). Applying the same construction to a smooth separated finite-type source $Z$, a morphism $f:Z\to Y$ sends complex-valued points to complex-valued points by composition with $\mathrm{Spec}\mathbb{C}\to Z$, and its local expressions are regular functions, hence holomorphic. To see continuity before using those expressions, inverse images of target affine opens are Zariski-open and therefore open, and within them every target coordinate is continuous. Thus $f^{\mathrm{a}\mathrm{n}}$ is a holomorphic map.

Let $\mathcal{O}(1)$ be the holomorphic line bundle on ${\mathbb{P}}^N(\mathbb{C})$ with transition functions $g_{jk}=Z_j\mathrm{/}{Z}_k$ on the standard cover, and give it the Hermitian metric whose local weight in the $j$-th trivialization is ${\psi }_j=(1+\mathrm{\mid }{z}^{(j)}{\mathrm{\mid }}^2{)}^{-1}>0$; on overlaps ${\psi }_j=\mathrm{\mid }{g}_{jk}{\mathrm{\mid }}^2{\psi }_k$, so the weights define a Hermitian metric [F2]. The local forms ${\omega }_j:={\psi }_j^{-1}\mathrm{\partial }{\psi }_j$ satisfy ${\omega }_j={\omega }_k+g_{jk}^{-1}d{g}_{jk}$ because $\mathrm{\partial }\log \mathrm{\mid }{g}_{jk}{\mathrm{\mid }}^2=g_{jk}^{-1}\mathrm{\partial }{g}_{jk}$ for holomorphic nonvanishing $g_{jk}$, so they glue to a complex connection $\mathrm{\nabla }$ on $\mathcal{O}(1)$ [F8]; and $\mathrm{\nabla }$ is Hermitian, because ${\omega }_j+{\bar{\omega }}_j={\psi }_j^{-1}\mathrm{\partial }{\psi }_j+{\psi }_j^{-1}\bar{\mathrm{\partial }}{\psi }_j={\psi }_j^{-1}d{\psi }_j$ is exactly the metric-compatibility identity $d{\psi }_j={\omega }_j{\psi }_j+{\psi }_j{\bar{\omega }}_j$ in the rank-one holomorphic frame [F8]. Its curvature is $\mathrm{\Omega }=d{\omega }_j+{\omega }_j\wedge {\omega }_j=d{\omega }_j$, the wedge term vanishing in rank one, and $d{\omega }_j=\bar{\mathrm{\partial }}({\psi }_j^{-1}\mathrm{\partial }{\psi }_j)=\bar{\mathrm{\partial }}\mathrm{\partial }\log {\psi }_j=-\mathrm{\partial }\bar{\mathrm{\partial }}\log {\psi }_j$ [F8]; for ${\psi }_j=(1+\mathrm{\mid }{z}^{(j)}{\mathrm{\mid }}^2{)}^{-1}$ this gives $\mathrm{\Omega }=\mathrm{\partial }\bar{\mathrm{\partial }}\log (1+\mathrm{\mid }{z}^{(j)}{\mathrm{\mid }}^2)$, so $-\mathrm{\Omega }\mathrm{/}(2\pi i)=\frac{i}{2\pi }\mathrm{\partial }\bar{\mathrm{\partial }}\log (1+\mathrm{\mid }{z}^{(j)}{\mathrm{\mid }}^2)={\omega }_{\mathrm{F}\mathrm{S}}$. The first-Chern comparison [F7] applied to $\mathcal{O}(1)$ with this Hermitian metric and Hermitian connection therefore gives $J_{{\mathbb{P}}^N(\mathbb{C})}[{\omega }_{\mathrm{F}\mathrm{S}}]=\rho (c_1(\mathcal{O}(1)))$.

By [F6] choose $d\ge 1$, $N\ge 0$ and a closed immersion $j:X\to {\mathbb{P}}_{\mathbb{C}}^N$ with $j^{\ast }\mathcal{O}(1)\cong A^{\otimes d}$. Its complex point map is injective and its image is closed in ${\mathbb{P}}^N(\mathbb{C})$: in each standard affine chart [F12] presents the image as the polynomial zero locus of a quotient ideal. The atlas topology agrees with this subspace topology. Indeed, on such an affine chart, the source coordinate algebra is a quotient of the ambient polynomial algebra by [F12]. Localizing and choosing any finite presentation of that algebra does not change its Euclidean point topology: each generator in either presentation is a polynomial or a fraction in the other generators with nonvanishing denominator, so the induced point maps are continuous inverses; localized principal opens are Euclidean open. The graph neighbourhoods of step 1.1 are precisely open subsets of these affine point loci. Hence $j^{\mathrm{a}\mathrm{n}}$ is a homeomorphism onto its closed image, and [F3] gives compactness, Hausdorffness and second-countability. Its holomorphic differential is injective: the free local coordinates $t_1^{\mathrm{\prime }},\dots ,t_n^{\mathrm{\prime }}$ of step 1.1 lift through the localized quotient map to regular functions $q_1,\dots ,q_n$ on a neighbourhood of $j(x)$ in the ambient affine chart; for the local holomorphic expression $g$ of $j$, $(q_1,\dots ,q_n)\circ g=t^{\mathrm{\prime }}$ in those coordinates, so differentiation gives a left inverse to $d{g}_x$. If $n=0$, the topology is discrete by step 1.1 and compactness now makes $X(\mathbb{C})$ finite; the empty case is immediate.

Pull the Hermitian metric and the Hermitian connection of step 2.2 back along the holomorphic map $j^{\mathrm{a}\mathrm{n}}:X^{\mathrm{a}\mathrm{n}}\to {\mathbb{P}}^N(\mathbb{C})$ to the line bundle $A^{\otimes d}$ on $X^{\mathrm{a}\mathrm{n}}$; in a pulled-back local frame its connection form is $j^{\ast }{\omega }_j$, whose gauge law is the pullback of that in step 2.2; pulling back the metric-compatibility identity proves it is Hermitian, and exterior differentiation gives curvature $d(j^{\ast }{\omega }_j)=j^{\ast }d{\omega }_j=j^{\ast }\mathrm{\Omega }$ [F8], and ${\eta }_{A^{\otimes d}}:=-j^{\ast }\mathrm{\Omega }\mathrm{/}(2\pi i)=j^{\ast }{\omega }_{\mathrm{F}\mathrm{S}}$ is a smooth closed real $(1,1)$-form, positive because the differential of $j$ is injective at every point of $X(\mathbb{C})$ by step 3.1 while ${\omega }_{\mathrm{F}\mathrm{S}}$ is positive. The first-Chern comparison [F7] applied on the compact manifold $X^{\mathrm{a}\mathrm{n}}$ to $(A^{\otimes d},j^{\ast }\mathrm{\nabla })$ gives $J_{X^{\mathrm{a}\mathrm{n}}}[{\eta }_{A^{\otimes d}}]=\rho (c_1(A^{\otimes d}))$, and the tensor-power additivity [F9] gives $c_1(A^{\otimes d})=d{c}_1(A)$, by [F13] the smooth line bundles are numerable and each connected manifold component is path-connected, paracompact Hausdorff, compactly generated weak Hausdorff and of CW homotopy type. Components are open, so the identity applies componentwise and hence globally; compactness permits only finitely many components. With the natural comparisons of [F10], ${\eta }_A:=d^{-1}{\eta }_{A^{\otimes d}}$ is again smooth, closed, real, of type $(1,1)$ and positive, and $J_{X^{\mathrm{a}\mathrm{n}}}[{\eta }_A]=d^{-1}\rho (d{c}_1(A))=\rho (c_1(A))$. For $X=\mathrm{\varnothing }$ or $n=0$ all forms of positive degree and all degree-two classes are zero, and ${\eta }_A=0$ satisfies the identity trivially. This proves the lemma for every ample $A$, the Axiom of Choice being used through the declared algebraic and comparison suppliers [F4, F6, F7, F9, F10, F11, F12, F13].