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Over complex projective space, its fibre at a line $\ell \subseteq {\mathbb{C}}^{n+1}$ is $\ell$ itself. It is a holomorphic line bundle contained in the trivial bundle. Its dual is the hyperplane line bundle.

The First Chern class of the hyperplane line bundle. On Complex projective space it generates degree-two integral cohomology, and on ${\mathbb{CP}}^n$ its top power evaluates as one on the complex-oriented fundamental class.

At a point $[X]\in {\mathbf{P}}^n$, the complex tautological line bundle has fiber $\mathbb{C}X\subseteq {\mathbb{C}}^{n+1}$. The hyperplane line bundle is its dual, $[H]=\mathcal{O}(1)$, and $\mathcal{O}(m)=\mathcal{O}(1{)}^{\otimes m}$. On $U_i=\{X_i\ne 0\}$ let $t_i=X/X_i$ be the tautological frame and $e_i$ its dual. On overlaps $e_j=(X_j/X_i)e_i$. The sheaf of holomorphic sections of a vector bundle of this dual bundle is the sheaf ${\mathcal{O}}_{{\mathbf{P}}^n}(1)$.

If a nonzero linear form $\ell$ cuts out the projective hyperplane $H$, it gives a holomorphic section $s_H$ of $\mathcal{O}(1)$ with a simple zero along $H$. Dividing a local section $s$ by $s_H$ gives a meromorphic function with no poles away from $H$ and at most a simple pole on $H$. Conversely multiplication by $s_H$ turns such a function into a holomorphic section. These mutually inverse local constructions agree on overlaps and prove the holomorphic line bundle associated to a divisor identification

The nonnegative hyperplane twist cohomology by restriction calculation givesTo derive it from the permitted vanishing, use the hyperplane exact sequence for twisting sheaveswhere $i:H\hookrightarrow {\mathbf{P}}^n$ and $H\cong {\mathbf{P}}^{n-1}$. Local multiplication by a defining equation is injective, and restriction gives exactly its cokernel. Induct on $n$, starting from a point, and then on $m$. For $m=0$, higher sheaf cohomology vanishes by assumption and global holomorphic functions are constant by compactness and the maximum principle. For $m>0$, the long exact sequence in sheaf cohomology, the inductive vanishing for $\mathcal{O}(m-1)$ and the dimension induction on $H$ give vanishing in every positive degree andSumming this recurrence gives $\left(\genfrac{}{}{0ex}{}{n+m}{n}\right)$. Homogeneous degree-$m$ monomials give that many independent sections, so they form a basis. The printed extra equality to zero cannot hold in degree zero, already because $h^0({\mathbf{P}}^n,\mathcal{O})=1$; the boxed calculation gives the intended dimensions.

For the differential of the projective quotient map, the differential of a smooth map is computed by applying a tangent vector to coordinate functions. Since $x_j=X_j/X_0$,ThusUnder replacing $X$ by $\lambda X$, $d\pi$ on a constant ambient tangent vector scales by ${\lambda }^{-1}$, while a homogeneous linear form $L$ scales by $\lambda$. Therefore $d{\pi }_X(L(X){\partial }_{X_j})$ is independent of the representative. Its chart coefficients are holomorphic, so it is a global holomorphic vector field. The radial Euler vector field lies in the kernel of $d\pi$, either from the same formulas or because it differentiates a scaling orbit. Hence

Describe the Euler sequence on complex projective space fiberwise as well as in coordinates. At a line $\ell =\mathbb{C}X$, the middle bundle has fiber $\mathrm{Hom}(\ell ,{\mathbb{C}}^{n+1})$ and the holomorphic tangent bundle has fiber $\mathrm{Hom}(\ell ,{\mathbb{C}}^{n+1}/\ell )$. Send a homomorphism to its composition with the quotient. The map from $\mathcal{O}$ sends a scalar to that scalar times the inclusion $\ell \hookrightarrow {\mathbb{C}}^{n+1}$. In homogeneous notation these maps areA local section of $\mathcal{O}(1)$ is a local degree-one homogeneous function on the punctured cone, so the second expression is representative-independent. Quotienting is surjective, and its kernel consists exactly of scalar inclusion maps. This proves the short exact sequence of locally free sheavesSince it is locally split, dualizing and tensoring by $\mathcal{O}(1)$ preserves exactness and gives the cotangent Euler sequence in homogeneous coordinatesOn global sections the last map identifies ${\mathbb{C}}^{n+1}$ with the basis of linear forms in $H^0(\mathcal{O}(1))$. Its kernel is zero. This proves the vanishing of global sections of the once-twisted projective cotangent bundle; left exactness gives $\boxed{h^0({\mathbf{P}}^n,{\Omega }^1(1))=0}$.

Let $[Z_0:Z_1]$ be homogeneous coordinates on the complex projective line. The complex tautological line bundle $\mathcal{O}(-1)$ has fibre the line $\mathbb{C}(Z_0,Z_1)\subset {\mathbb{C}}^2$ over that point. Its dual is the hyperplane line bundle $\mathcal{O}(1)$. Define the tensor powers of the hyperplane line bundle by $\mathcal{O}(n)=\mathcal{O}(1{)}^{\otimes n}$ for $n>0$, the trivial line bundle for $n=0$, and the appropriate power of $\mathcal{O}(-1)$ for $n<0$.

An explicit holomorphic line bundle construction uses the two charts $U_0=\{Z_0\ne 0\}$ and $U_{\infty }=\{Z_1\ne 0\}$ with coordinates $z=Z_1/Z_0$ and $w=Z_0/Z_1=1/z$. Choose holomorphic local frames related on the overlap byThis is consistent with the tautological frames $(1,z)$ and $(w,1)=z^{-1}(1,z)$ for $\mathcal{O}(-1)$. A holomorphic section of $\mathcal{O}(n)$ is represented by entire chart functions satisfyingExpand $f_0(z)={\sum }_{k\ge 0}{a}_k{z}^k$. Holomorphicity of $f_{\infty }$ at $w=0$ requires every power $w^{n-k}$ with $n-k<0$ to vanish. Thus for $n\ge 0$, $f_0$ is exactly a polynomial of degree at most $n$. Conversely every such polynomial gives an entire $f_{\infty }$. Its coefficients supply the explicit isomorphismEquivalently, the global holomorphic sections are the homogeneous polynomials ${\sum }_{k=0}^n{a}_k{Z}_0^{n-k}{Z}_1^k$. For $n<0$, the same chart argument gives no nonzero global section.

For the first cohomology, use the sheaf of holomorphic sections of a vector bundle and its Čech cohomology on this two-chart cover. A zero-cochain is a pair of entire chart functions $(h_0,h_{\infty })$. A one-cochain is a function $g(z)$ holomorphic on ${\mathbb{C}}^{\ast }$, multiplied by $e_0$. There are no alternating two-cochains on a two-set cover, so every one-cochain is a cocycle. The coboundaries areConsequently the holomorphic Laurent-series cohomology of twists on the projective line isHere the last term means the set of functions $z^{n}h(1/z)$ with $h$ entire. The charts and their intersection are Stein manifolds; Cartan theorem B makes this cover acyclic for the line-bundle section sheaf, so this Čech group computes the sheaf cohomology.

Every $g$ on ${\mathbb{C}}^{\ast }$ has a Laurent series ${\sum }_{k\in \mathbb{Z}}{a}_k{z}^k$ converging normally on compact annuli. Its nonnegative-power tail is an entire function of $z$ and can be removed by a finite-chart coboundary. Its tail with $k\le n$ can be removed by $z^n{h}_{\infty }(1/z)$: the series ${\sum }_{k\le n}{a}_k{w}^{n-k}$ is entire in $w$, because the inner Laurent radius is zero. Assign any overlapping removable powers to just one tail. The only possible remaining powers satisfy $n<k<0$.

If $n\ge -1$, there is no such integer, proving the requested vanishing:For completeness, if $n\le -2$, the remaining classes have basis $z^{n+1},\dots ,z^{-1}$ and dimension $-n-1$. None can be a coboundary, by uniqueness of Laurent coefficients. This also explains exactly why $-2$ is the threshold.

Finally, a global holomorphic section of $\mathcal{O}(1)\oplus \mathcal{O}(1)$ is a pair of linear homogeneous polynomials:It vanishes at a projective point precisely when the nonzero vector $(Z_0,Z_1{)}^T$ belongs to the kernel of the coefficient matrix $M=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$. If $M$ has rank two there is no zero; if $M=0$ the section vanishes everywhere. Exactly one projective zero occurs precisely when the kernel is a one-dimensional line. Thus the rank-one sections of two hyperplane line bundles are characterized byEquivalently, $s=v\ell$, where $v\in {\mathbb{C}}^2$ is a fixed nonzero vector and $\ell$ is a nonzero section of $\mathcal{O}(1)$. Its single simple zero is the zero of $\ell$. This includes the case when one component is identically zero and the other is a nonzero linear form.

A holomorphic line bundle $L\to X$ is a rank-one holomorphic vector bundle: it has local trivializations $L{|}_{U_i}\cong U_i\times \mathbb{C}$ whose fiberwise linear transition functions of a vector bundle are nowhere-zero holomorphic functions. Equivalently, choose local holomorphic frames $e_i$ and write $e_j=g_{ij}{e}_i$ on overlaps, with $g_{ij}\in {\mathcal{O}}^{\ast }(U_i\cap U_j)$ and the cocycle relation on triple overlaps. A holomorphic section over $U$ has coefficients $s_i$ holomorphic on $U\cap U_i$ in $s=s_i{e}_i$; consistency means $s_i=g_{ij}{s}_j$.

The dual bundle has fiber $L_x^{\ast }={\mathrm{Hom}}_{\mathbb{C}}(L_x,\mathbb{C})$. Its dual frames $e_i^{\ast }$ are characterized by $e_i^{\ast }(e_i)=1$, and the frame relation becomes $e_j^{\ast }=g_{ij}^{-1}{e}_i^{\ast }$. Reciprocals of nowhere-zero holomorphic functions are holomorphic, and their cocycle relations persist. This defines the holomorphic dual vector bundle structure on $L^{\ast }$, with holomorphic evaluation pairing $L^{\ast }\otimes L\to {\mathcal{O}}_X$.

Assume now that $X$ is compact and connected, and let $s$ and $t$ be nonzero global holomorphic sections of $L$ and $L^{\ast }$, respectively. The function $t(s)$ is a global holomorphic function. The maximum modulus principle, applied in local complex coordinate lines near a maximum and then propagated on the connected manifold, makes it constant. Moreover it is not identically zero: the zero locus of a nonzero holomorphic section has empty interior by the identity theorem, so the two nonvanishing loci are dense open sets and intersect. Thus $t(s)=c\ne 0$ everywhere.

In particular, $s$ never vanishes. The map $(x,a)\mapsto as(x)$ is a holomorphic bundle isomorphism $X\times \mathbb{C}\to L$; in a local frame its inverse divides by the nowhere-zero holomorphic coefficient of $s$. Conversely, a holomorphic trivialization supplies the constant unit sections of both $L$ and $L^{\ast }$. We have proved the two-section criterion for holomorphic triviality:

Write points of Complex projective space as lines $[Z]\subset {\mathbb{C}}^{n+1}$. The complex tautological line bundle iswith fiber that very line. Its dual is the hyperplane line bundle $\mathcal{O}(1)$. On the affine chart $U_i=\{Z_i\ne 0\}$, the vector $e_i([Z])=Z/Z_i$ is a nowhere-zero holomorphic frame of $\mathcal{O}(-1)$. On $U_i\cap U_j$,Both ratios are nowhere-zero holomorphic functions of the affine coordinates. They exhibit the holomorphic transition functions of a vector bundle for the two bundles explicitly.

Choose a nonzero linear functional $\ell :{\mathbb{C}}^{n+1}\to \mathbb{C}$ whose zero hyperplane is $H$. At $[Z]$, its restriction to $\mathbb{C}Z$ is an element of that line's dual and therefore defines a holomorphic section $s_{\ell }$ of $\mathcal{O}(1)$. In the dual frame above its coefficient isThis expression is holomorphic even along $H$, and its zero locus is exactly $H$. Consequently $s_{\ell }{|}_{{\mathbb{P}}^n\setminus H}$ is the requested nowhere-zero section; its global holomorphic extension vanishes precisely on $H$. The extension need not, and does not, stay nowhere zero on the hyperplane.

The divisor determines the bundle through its local equations. Choose an open cover on which $V$ has a reduced holomorphic defining equation $f_i$, using $f_i=1$ on sets disjoint from $V$. Because $V$ is a complex submanifold of codimension of a submanifold one, its ideal is locally generated by a coordinate; consequently $f_i/f_j$ is a nowhere-zero holomorphic function on each overlap. Inside the sheaf of meromorphic functions on a complex manifold, consider the locally free sheaf of rank oneThese local descriptions agree on overlaps. With $e_i=1/f_i$, its transition functions of a vector bundle are specified byThe ratios satisfy the Čech cocycle condition, and the associated holomorphic line bundle associated to a divisor is $[V]$. The section $1=f_i{e}_i$ is holomorphic and vanishes to order one along $V$. Replacing $f_i$ by $u_i{f}_i$, with $u_i$ a holomorphic unit, replaces the frame by $u_i^{-1}{e}_i$ without changing the subsheaf of meromorphic functions. Refining the cover changes no local sections either. This proves choice independence up to isomorphism rather than merely producing a bundle for one cover.

For a complex manifold of dimension $n$, the canonical bundle isIt is the holomorphic line bundle of holomorphic top-degree differential forms. To compute it on Complex projective space, let $S=\mathcal{O}(-1)$ be the complex tautological line bundle, so that $S^{\ast }=[H]=\mathcal{O}(1)$ is the hyperplane line bundle. At a line $\ell \subset {\mathbb{C}}^{n+1}$, the tangent space is $\mathrm{Hom}(\ell ,{\mathbb{C}}^{n+1}/\ell )$. Tensoring the tautological quotient sequence by $S^{\ast }$ gives the Euler sequence on complex projective spaceTaking the top exterior power gives $\det T^{1,0}{\mathbb{P}}^n=\mathcal{O}(n+1)$; taking its dual bundle yieldsHere a negative tensor power means the corresponding positive tensor power of the dual bundle.

The proposed sections over $U$ form a sheaf: compatible maps into $L$ glue uniquely, and their projections remain $\varphi$. Addition and multiplication by holomorphic functions are defined in the fibres of $L$, making this a sheaf of ${\mathcal{O}}_X$-modules. If $e_i$ is a holomorphic local frame of $L$ on $W_i\subset Y$, every section over $U\subset {\varphi }^{-1}(W_i)$ has the unique expressionThus the sheaf is a locally free sheaf of rank one. Its transition functions of a vector bundle are those of $L$ composed with $\varphi$, so the corresponding pullback vector bundle is precisely ${\varphi }^{\ast }L$. This proves invertibility locally and compatibility globally.

For the local blowup of a complex manifold at a point, work on the projective chart $Y_i\ne 0$ and writeAll the incidence equations reduce to $x_j=u{t}_j$. Therefore $(u,(t_j{)}_{j\ne i})$ gives a holomorphic coordinate chart isomorphic to ${\mathbb{C}}^n$. On an overlap with $Y_{\ell }\ne 0$,which is a biholomorphism where $t_{\ell }\ne 0$. The incidence subset is closed in the product of two Hausdorff spaces and is covered by these charts; hence it is a complex manifold of dimension $n$. In each chart the exceptional fibre is $u=0$, so it is a smooth complex submanifold, with its induced projective coordinates givingAway from the origin the inverse of the projection is $x\mapsto (x,[x])$, which is holomorphic; the projection is therefore a biholomorphism outside $E$.

To find the canonical bundle, pull back the nowhere-zero top form on ${\mathbb{C}}^n$. In the chart above, its Jacobian determinant givesThe sign depends only on the ordering of the coordinates. This is a section of $K_X$ with divisor on a complex manifold exactly $(n-1)E$: it has that vanishing order in every exceptional chart and no zeros elsewhere. A nonzero meromorphic section of a holomorphic line bundle with divisor $D$ identifies its holomorphic line bundle with $[D]$, by sending the local generator $1/f_i$ of $\mathcal{O}(D)$ to the nowhere-zero frame obtained by dividing the section by $f_i$. ConsequentlyThis also includes $n=1$, when the local map is an isomorphism and the exponent is zero.

For a general complex manifold, choose a coordinate neighbourhood $U$ of $P$, with $P$ sent to $0$, and replace $U$ by the inverse image of this coordinate neighbourhood in the local blowup of a complex manifold at a point. Glue this space to $M\setminus \{P\}$ along $U\setminus \{P\}$ using the preceding biholomorphism. The resulting charts give a complex manifold $\tilde{M}$ and a holomorphic map $\varphi$; the exceptional fibre is ${\mathbb{P}}^{n-1}$ and the map is a biholomorphism off that fibre. For completeness, the local projection is proper because its inverse image over a compact subset is closed in that subset times compact Complex projective space. This ensures the gluing is Hausdorff: separate points with distinct images downstairs, and points over $P$ inside the local blowup. The atlas is second countable as well.

The top exterior power of the differential defines a global holomorphic bundle map ${\varphi }^{\ast }{K}_M\to K_{\tilde{M}}$. Equivalently, it is a section of $K_{\tilde{M}}\otimes ({\varphi }^{\ast }{K}_M{)}^{-1}$. The same local Jacobian determinant has vanishing order $n-1$ on $E$, and the differential is invertible elsewhere. Applying the holomorphic line bundle associated to a divisor construction gives the canonical bundle formula for a point blowup

For positive $k$ take the $k$th tensor power of the hyperplane line bundle; for negative $k$ take the corresponding power of its dual; for $k=0$ take the trivial line bundle. On the complex projective line, global sections for $k\ge 0$ correspond to polynomials of degree at most $k$ and form a space of dimension $k+1$.

Dualize the Euler sequence on complex projective space and tensor by the hyperplane line bundle to obtain $0\to {\Omega }^1(1)\to {\mathcal{O}}^{n+1}\to \mathcal{O}(1)\to 0$. On global sections the last map sends $(c_i)$ to ${\sum }_i{c}_i{X}_i$, an isomorphism onto the linear forms. Left exactness of sheaf cohomology therefore gives the asserted vanishing. No higher-cohomology assumption is needed for this final kernel calculation.