[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 $U_i \cap U_j$.
[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]
[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]
