[proofplan] We compute the transition functions of the canonical bundle on the standard affine cover of $\mathbb{P}^n$. On each chart $U_i = \{Z_i \neq 0\}$, the coordinate volume form gives a local frame of $K_{\mathbb{P}^n}$. A direct Jacobian determinant calculation shows that, after a harmless sign normalization of these frames, the transition functions are exactly those of $\mathcal{O}_{\mathbb{P}^n}(-n-1)$. Matching transition functions then constructs the desired holomorphic line bundle isomorphism. [/proofplan]

[step:Handle the zero-dimensional projective space] If $n = 0$, then $\mathbb{P}^0$ is a single point. The holomorphic cotangent bundle $\Omega_{\mathbb{P}^0}^{1}$ is the zero vector bundle, hence its determinant $K_{\mathbb{P}^0}$ is canonically isomorphic to the structure sheaf $\mathcal{O}_{\mathbb{P}^0}$. The tautological line bundle $\mathcal{O}_{\mathbb{P}^0}(-1)$ is also canonically isomorphic to the structure sheaf on a point, so \begin{align*} K_{\mathbb{P}^0} \simeq \mathcal{O}_{\mathbb{P}^0} \simeq \mathcal{O}_{\mathbb{P}^0}(-1). \end{align*} This proves the theorem for $n = 0$. For the remainder of the proof, assume $n \geq 1$. [/step]

[step:Choose the standard affine cover and its coordinate volume frames] Let homogeneous coordinates on $\mathbb{P}^n$ be written as $[Z_0:\cdots:Z_n]$. For each $i \in \{0,\dots,n\}$, define the standard affine [open set](/page/Open%20Set) \begin{align*} U_i := \{[Z_0:\cdots:Z_n] \in \mathbb{P}^n : Z_i \neq 0\}. \end{align*} Define the coordinate map \begin{align*} \varphi_i: U_i &\to \mathbb{C}^n \\ [Z_0:\cdots:Z_n] &\mapsto \bigl(z_{i,0},\dots,z_{i,i-1},z_{i,i+1},\dots,z_{i,n}\bigr), \end{align*} where, for each $k \neq i$, \begin{align*} z_{i,k} := \frac{Z_k}{Z_i}. \end{align*} The ordered coordinate volume form on $U_i$ is the local holomorphic frame \begin{align*} \omega_i := dz_{i,0} \wedge \cdots \wedge dz_{i,i-1} \wedge dz_{i,i+1} \wedge \cdots \wedge dz_{i,n} \end{align*} of $K_{\mathbb{P}^n}|_{U_i}$. [/step]

[step:Compute the Jacobian determinant on chart overlaps]Fix distinct indices $i,j \in \{0,\dots,n\}$. On the overlap $U_i \cap U_j$, the coordinate $z_{i,j}$ is nowhere zero, and the transition from the $i$-chart to the $j$-chart is given by \begin{align*} z_{j,i} &= \frac{1}{z_{i,j}}, \\ z_{j,k} &= \frac{z_{i,k}}{z_{i,j}} \qquad \text{for } k \neq i,j. \end{align*} With both coordinate lists ordered by increasing index, the Jacobian determinant of the map \begin{align*} \varphi_j \circ \varphi_i^{-1}: \varphi_i(U_i \cap U_j) \to \varphi_j(U_i \cap U_j) \end{align*} is \begin{align*} \det \left(\frac{\partial z_{j,a}}{\partial z_{i,b}}\right)_{a \neq j,\ b \neq i} = (-1)^{i+j}\bigl(z_{i,j}\bigr)^{-n-1}. \end{align*} Therefore the canonical frames satisfy \begin{align*} \omega_j = (-1)^{i+j}\bigl(z_{i,j}\bigr)^{-n-1}\omega_i \end{align*} on $U_i \cap U_j$.[/step]

[guided]We now compute exactly how the coordinate volume form changes from one affine chart to another. Fix $i \neq j$. On $U_i \cap U_j$, the coordinate $z_{i,j} = Z_j/Z_i$ is nonzero, because both $Z_i$ and $Z_j$ are nonzero there. The coordinate change is obtained by dividing homogeneous coordinates by $Z_j$ instead of by $Z_i$: \begin{align*} z_{j,i} &= \frac{Z_i}{Z_j} = \frac{1}{z_{i,j}}, \\ z_{j,k} &= \frac{Z_k}{Z_j} = \frac{Z_k/Z_i}{Z_j/Z_i} = \frac{z_{i,k}}{z_{i,j}} \qquad \text{for } k \neq i,j. \end{align*} Let the source coordinates be the ordered list $(z_{i,b})_{b \neq i}$ and the target coordinates be the ordered list $(z_{j,a})_{a \neq j}$. The Jacobian matrix has rows indexed by $a \neq j$ and columns indexed by $b \neq i$. For the row $a=i$, we have \begin{align*} \frac{\partial z_{j,i}}{\partial z_{i,j}} = -\bigl(z_{i,j}\bigr)^{-2}, \qquad \frac{\partial z_{j,i}}{\partial z_{i,b}} = 0 \quad \text{for } b \neq j. \end{align*} For a row $a \neq i,j$, the relation $z_{j,a} = z_{i,a}/z_{i,j}$ gives \begin{align*} \frac{\partial z_{j,a}}{\partial z_{i,a}} &= \bigl(z_{i,j}\bigr)^{-1}, \\ \frac{\partial z_{j,a}}{\partial z_{i,j}} &= -z_{i,a}\bigl(z_{i,j}\bigr)^{-2}, \\ \frac{\partial z_{j,a}}{\partial z_{i,b}} &= 0 \quad \text{for } b \neq a,j. \end{align*} Expanding the determinant along the row corresponding to $a=i$ leaves a diagonal matrix of size $n-1$ with diagonal entries $\bigl(z_{i,j}\bigr)^{-1}$. The product of the nonzero entries is therefore \begin{align*} -\bigl(z_{i,j}\bigr)^{-2}\bigl(z_{i,j}\bigr)^{-(n-1)} = -\bigl(z_{i,j}\bigr)^{-n-1}. \end{align*} The cofactor sign records the position of the row $a=i$ in the ordered target list and the column $b=j$ in the ordered source list. With the increasing-index convention, this total sign converts the preceding negative sign into $(-1)^{i+j}$. Hence \begin{align*} \det \left(\frac{\partial z_{j,a}}{\partial z_{i,b}}\right)_{a \neq j,\ b \neq i} = (-1)^{i+j}\bigl(z_{i,j}\bigr)^{-n-1}. \end{align*} Since the canonical bundle is the determinant of the holomorphic cotangent bundle, its local frame transforms by the Jacobian determinant of the coordinate change. Thus \begin{align*} \omega_j = (-1)^{i+j}\bigl(z_{i,j}\bigr)^{-n-1}\omega_i \end{align*} on $U_i \cap U_j$.[/guided]

[step:Normalize the canonical frames to remove the sign] For each $i \in \{0,\dots,n\}$, define a new local frame \begin{align*} \eta_i := (-1)^i \omega_i \end{align*} of $K_{\mathbb{P}^n}|_{U_i}$. Using the transition formula for the $\omega_i$, on $U_i \cap U_j$ we obtain \begin{align*} \eta_j &= (-1)^j \omega_j \\ &= (-1)^j(-1)^{i+j}\bigl(z_{i,j}\bigr)^{-n-1}\omega_i \\ &= (-1)^i\bigl(z_{i,j}\bigr)^{-n-1}\omega_i \\ &= \bigl(z_{i,j}\bigr)^{-n-1}\eta_i. \end{align*} Thus the normalized frames of $K_{\mathbb{P}^n}$ have transition functions \begin{align*} g_{ij} = \bigl(z_{i,j}\bigr)^{-n-1}. \end{align*} [/step]

[step:Identify these transitions with those of $\mathcal{O}_{\mathbb{P}^n}(-n-1)$] Let $\mathcal{O}_{\mathbb{P}^n}(-1)$ be the tautological line bundle on $\mathbb{P}^n$. For each $i$, define the local holomorphic frame \begin{align*} e_i: U_i &\to \mathcal{O}_{\mathbb{P}^n}(-1)|_{U_i} \\ [Z_0:\cdots:Z_n] &\mapsto \frac{1}{Z_i}(Z_0,\dots,Z_n). \end{align*} This vector lies in the tautological line over $[Z_0:\cdots:Z_n]$ and has $i$-th coordinate equal to $1$. On $U_i \cap U_j$, \begin{align*} e_j = \frac{1}{Z_j}(Z_0,\dots,Z_n) = \frac{Z_i}{Z_j}\frac{1}{Z_i}(Z_0,\dots,Z_n) = \bigl(z_{i,j}\bigr)^{-1}e_i. \end{align*} Therefore the local frame \begin{align*} e_i^{\otimes(n+1)} \end{align*} of \begin{align*} \mathcal{O}_{\mathbb{P}^n}(-n-1) = \mathcal{O}_{\mathbb{P}^n}(-1)^{\otimes(n+1)} \end{align*} satisfies \begin{align*} e_j^{\otimes(n+1)} = \bigl(z_{i,j}\bigr)^{-n-1}e_i^{\otimes(n+1)}. \end{align*} These are exactly the transition functions computed for the normalized canonical frames $\eta_i$. [/step]

[step:Glue the local frame identifications into a global isomorphism] For each $i \in \{0,\dots,n\}$, define a local holomorphic line bundle isomorphism \begin{align*} \Phi_i: K_{\mathbb{P}^n}|_{U_i} &\to \mathcal{O}_{\mathbb{P}^n}(-n-1)|_{U_i} \end{align*} by declaring \begin{align*} \Phi_i(\eta_i) := e_i^{\otimes(n+1)} \end{align*} and extending fiberwise by complex linearity. On $U_i \cap U_j$, the transition identities give \begin{align*} \Phi_j(\eta_j) &= e_j^{\otimes(n+1)} \\ &= \bigl(z_{i,j}\bigr)^{-n-1}e_i^{\otimes(n+1)} \\ &= \Phi_i\left(\bigl(z_{i,j}\bigr)^{-n-1}\eta_i\right) \\ &= \Phi_i(\eta_j). \end{align*} Thus the maps $\Phi_i$ agree on overlaps and glue to a global holomorphic line bundle isomorphism \begin{align*} \Phi: K_{\mathbb{P}^n} \to \mathcal{O}_{\mathbb{P}^n}(-n-1). \end{align*} This proves \begin{align*} K_{\mathbb{P}^n} \simeq \mathcal{O}_{\mathbb{P}^n}(-n-1), \end{align*} as required. [/step]