[proofplan] The proof is local, because curvature forms and the Ricci form are computed in holomorphic coordinate charts and agree on overlaps. In a holomorphic chart, we write the Hermitian metric by its coefficient matrix $G=(g_{j\bar{k}})$ and compute the induced metric on the canonical frame $dz_1\wedge\cdots\wedge dz_n$. The Chern curvature formula for a Hermitian line bundle then gives the curvature of $K_X$, while the dual metric reverses the sign. Finally, the first Chern class is represented by $\frac{i}{2\pi}\Theta_h(L)$ for any Hermitian metric $h$ on a holomorphic line bundle $L$. [/proofplan] [step:Compute the induced metric on the local canonical frame] Let $(U,z)$ be a holomorphic coordinate chart on $X$, where \begin{align*} z:U&\to z(U)\subseteq\mathbb{C}^n,\\ p&\mapsto (z_1(p),\dots,z_n(p)). \end{align*} Let $\operatorname{Herm}_n^{+}$ denote the set of positive definite Hermitian $n\times n$ complex matrices. Let \begin{align*} G:U&\to \operatorname{Herm}_n^{+},\\ p&\mapsto (g_{j\bar{k}}(p))_{1\le j,k\le n} \end{align*} be the positive Hermitian matrix representing $\omega$ in this chart, so that \begin{align*} \omega=i\sum_{j,k=1}^n g_{j\bar{k}}\,dz_j\wedge d\bar{z}_k \end{align*} on $U$. Define the local holomorphic frame \begin{align*} e_K:U&\to K_X|_U,\\ p&\mapsto dz_1|_p\wedge\cdots\wedge dz_n|_p. \end{align*} The metric induced by $\omega$ on $T^{*(1,0)}X$ has coefficient matrix $G^{-1}=(g^{j\bar{k}})$ in the coframe $(dz_1,\dots,dz_n)$. Therefore the induced metric on the determinant line $K_X=\Lambda^nT^{*(1,0)}X$ satisfies \begin{align*} |e_K|_{h_K}^2=\det(G^{-1})=(\det G)^{-1}. \end{align*} [guided] We work in one holomorphic coordinate chart because all objects in the theorem are tensorial: once the local formula is obtained in every chart, the resulting equality of forms holds globally. Let $(U,z)$ be a holomorphic coordinate chart, with \begin{align*} z:U&\to z(U)\subseteq\mathbb{C}^n,\\ p&\mapsto (z_1(p),\dots,z_n(p)). \end{align*} Let $\operatorname{Herm}_n^{+}$ denote the set of positive definite Hermitian $n\times n$ complex matrices. In this chart the Hermitian metric $\omega$ is represented by a smooth positive Hermitian matrix-valued map \begin{align*} G:U&\to \operatorname{Herm}_n^{+},\\ p&\mapsto (g_{j\bar{k}}(p))_{1\le j,k\le n}, \end{align*} and the local expression for the metric is \begin{align*} \omega=i\sum_{j,k=1}^n g_{j\bar{k}}\,dz_j\wedge d\bar{z}_k. \end{align*} The canonical line bundle is $K_X=\Lambda^nT^{*(1,0)}X$. The coordinate chart gives the local holomorphic frame \begin{align*} e_K:U&\to K_X|_U,\\ p&\mapsto dz_1|_p\wedge\cdots\wedge dz_n|_p. \end{align*} Since $G$ is the metric matrix on $T^{(1,0)}X$, the induced metric matrix on the dual bundle $T^{*(1,0)}X$ is $G^{-1}=(g^{j\bar{k}})$. Passing from a Hermitian [vector space](/page/Vector%20Space) to its top exterior power takes determinants of Gram matrices. Hence the squared norm of the determinant coframe is \begin{align*} |e_K|_{h_K}^2=\det(G^{-1})=(\det G)^{-1}. \end{align*} [/guided] [/step] [step:Apply the Chern curvature formula to the canonical bundle] For a Hermitian holomorphic line bundle with local holomorphic frame $e$ and local weight $|e|_h^2$, the Chern curvature is \begin{align*} i\Theta_h=-i\,\partial\bar{\partial}\log |e|_h^2. \end{align*} Applying this to $e_K$ gives \begin{align*} i\Theta_{h_K}(K_X) &=-i\,\partial\bar{\partial}\log |e_K|_{h_K}^2\\ &=-i\,\partial\bar{\partial}\log \bigl((\det G)^{-1}\bigr)\\ &=i\,\partial\bar{\partial}\log \det G. \end{align*} By the definition of the Ricci form of the Hermitian metric $\omega$, \begin{align*} \operatorname{Ric}(\omega)=-i\,\partial\bar{\partial}\log \det G. \end{align*} Therefore \begin{align*} i\Theta_{h_K}(K_X)=-\operatorname{Ric}(\omega). \end{align*} [guided] We now convert the metric computation into a curvature computation. For a Hermitian holomorphic line bundle with local holomorphic frame $e$, the local weight is the positive function $|e|_h^2$. The Chern curvature of that Hermitian line bundle is computed by \begin{align*} i\Theta_h=-i\,\partial\bar{\partial}\log |e|_h^2. \end{align*} Here the relevant frame is \begin{align*} e_K:U&\to K_X|_U,\\ p&\mapsto dz_1|_p\wedge\cdots\wedge dz_n|_p, \end{align*} and the previous step gives \begin{align*} |e_K|_{h_K}^2=(\det G)^{-1}. \end{align*} Substituting this positive local weight into the curvature formula yields \begin{align*} i\Theta_{h_K}(K_X) &=-i\,\partial\bar{\partial}\log |e_K|_{h_K}^2\\ &=-i\,\partial\bar{\partial}\log \bigl((\det G)^{-1}\bigr)\\ &=-i\,\partial\bar{\partial}\bigl(-\log\det G\bigr)\\ &=i\,\partial\bar{\partial}\log\det G. \end{align*} The Ricci form of the Hermitian metric $\omega$ is defined in this same holomorphic chart by \begin{align*} \operatorname{Ric}(\omega)=-i\,\partial\bar{\partial}\log\det G. \end{align*} Comparing the two displayed formulas gives \begin{align*} i\Theta_{h_K}(K_X)=-\operatorname{Ric}(\omega). \end{align*} The equality is independent of the chosen chart because both sides are globally defined real $(1,1)$-forms. [/guided] [/step] [step:Use the dual metric to compute the anticanonical curvature] Let \begin{align*} e_K^{-1}:U&\to K_X^{-1}|_U \end{align*} be the dual frame determined by $e_K^{-1}(e_K)=1$. The dual metric satisfies \begin{align*} |e_K^{-1}|_{h_K^{-1}}^2=|e_K|_{h_K}^{-2}=\det G. \end{align*} Applying the Chern curvature formula to $K_X^{-1}$ gives \begin{align*} i\Theta_{h_K^{-1}}(K_X^{-1}) &=-i\,\partial\bar{\partial}\log |e_K^{-1}|_{h_K^{-1}}^2\\ &=-i\,\partial\bar{\partial}\log\det G\\ &=\operatorname{Ric}(\omega). \end{align*} [guided] The anticanonical bundle is the dual line bundle $K_X^{-1}=(K_X)^*$. Let \begin{align*} e_K^{-1}:U&\to K_X^{-1}|_U \end{align*} be the dual holomorphic frame, characterized by the identity $e_K^{-1}(e_K)=1$ on $U$. The dual Hermitian metric inverts squared norms on dual frames. Since \begin{align*} |e_K|_{h_K}^2=(\det G)^{-1}, \end{align*} we obtain \begin{align*} |e_K^{-1}|_{h_K^{-1}}^2=|e_K|_{h_K}^{-2}=\det G. \end{align*} Now we apply the same Chern curvature formula to the Hermitian line bundle $(K_X^{-1},h_K^{-1})$: \begin{align*} i\Theta_{h_K^{-1}}(K_X^{-1}) &=-i\,\partial\bar{\partial}\log |e_K^{-1}|_{h_K^{-1}}^2\\ &=-i\,\partial\bar{\partial}\log\det G. \end{align*} By the defining local formula for the Ricci form, \begin{align*} \operatorname{Ric}(\omega)=-i\,\partial\bar{\partial}\log\det G. \end{align*} Therefore \begin{align*} i\Theta_{h_K^{-1}}(K_X^{-1})=\operatorname{Ric}(\omega). \end{align*} [/guided] [/step] [step:Identify the resulting cohomology classes] With the curvature convention used above, for a Hermitian holomorphic line bundle $(L,h)$, the real de Rham cohomology class of the closed real form $\frac{i}{2\pi}\Theta_h(L)$ is $c_1(L)$. Applying this to $(K_X^{-1},h_K^{-1})$ and using the curvature identity just proved gives \begin{align*} c_1(K_X^{-1}) =\left[\frac{i\Theta_{h_K^{-1}}(K_X^{-1})}{2\pi}\right] =\left[\frac{\operatorname{Ric}(\omega)}{2\pi}\right]. \end{align*} Since first Chern classes are additive under tensor products and $K_X^{-1}$ is the dual line bundle of $K_X$, we have \begin{align*} c_1(K_X^{-1})=-c_1(K_X). \end{align*} Combining the two identities yields \begin{align*} \left[\frac{\operatorname{Ric}(\omega)}{2\pi}\right]=c_1(K_X^{-1})=-c_1(K_X). \end{align*} This proves the theorem. [guided] The curvature identities are identities of differential forms. To convert them into the stated topological conclusion, we use the Chern-Weil definition of the first Chern class with the same curvature convention used in the local formula above: if $(L,h)$ is a Hermitian holomorphic line bundle, then \begin{align*} c_1(L)=\left[\frac{i\Theta_h(L)}{2\pi}\right]\in H^2(X;\mathbb{R}). \end{align*} Applying this to the anticanonical bundle with its dual metric gives \begin{align*} c_1(K_X^{-1}) =\left[\frac{i\Theta_{h_K^{-1}}(K_X^{-1})}{2\pi}\right]. \end{align*} The previous step proved \begin{align*} i\Theta_{h_K^{-1}}(K_X^{-1})=\operatorname{Ric}(\omega), \end{align*} so substitution gives \begin{align*} c_1(K_X^{-1}) =\left[\frac{\operatorname{Ric}(\omega)}{2\pi}\right]. \end{align*} Finally, $K_X^{-1}$ is the dual line bundle of $K_X$. First Chern classes are additive under tensor products, and the [tensor product](/page/Tensor%20Product) $K_X\otimes K_X^{-1}$ is holomorphically isomorphic to the product line bundle $X\times\mathbb{C}$. Hence \begin{align*} c_1(K_X)+c_1(K_X^{-1})=c_1(K_X\otimes K_X^{-1})=0, \end{align*} which gives \begin{align*} c_1(K_X^{-1})=-c_1(K_X). \end{align*} Combining the two cohomological identities yields \begin{align*} \left[\frac{\operatorname{Ric}(\omega)}{2\pi}\right]=c_1(K_X^{-1})=-c_1(K_X). \end{align*} This is exactly the asserted cohomology statement. [/guided] [/step]