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The complex tautological line bundle is a holomorphic subbundle of the trivial bundle with fibre ${\mathbb{C}}^{n+1}$. A global holomorphic section therefore gives $n+1$ global holomorphic functions on compact connected Complex projective space. The maximum modulus principle makes them constant. The constant vector must lie in every projective line through the origin, whose intersection is zero for $n\ge 1$. For $n=0$, the base is one point and the section space is instead $\mathbb{C}$.

The dual of the complex tautological line bundle. On the complex projective line the induced frames satisfy $e_1=w{e}_0$ on the overlap of its standard affine charts.

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 $\boxed{h^0({\mathbf{P}}^n,{\Omega }^1(1))=0}$.

A rank-$k$ complex vector bundle is a smooth map $\pi :E\to B$ whose fibers are $k$-dimensional complex vector spaces, with local diffeomorphisms ${\Phi }_i:{\pi }^{-1}(U_i)\to U_i\times {\mathbb{C}}^k$ commuting with projection and linear on every fiber. Define transition functions of a vector bundle byThey are smooth and obeyon the appropriate double/triple overlaps. The structure group of a vector bundle is a Lie subgroup $G\subset GL(k,\mathbb{C})$ in which a chosen set of transitions takes values. It records allowed changes of fiber frame; different trivializations may exhibit different reductions of that group.

Using the same cocycle, glue the disjoint union of $U_i\times G$ byThe cocycle makes this an equivalence relation and makes its local charts compatible. Right multiplication $(b,h{)}_i\cdot a=(b,ha{)}_i$ is well-defined because it commutes with the left gluing matrices. It is free and transitive on each fiber. These charts therefore construct the associated right principal bundle $P\to B$ with group $G$.

For the natural line realization, take the complex tautological line bundle $L=\{(\ell ,v):\ell \in {\mathbb{CP}}^1,v\in \ell \subset {\mathbb{C}}^2\}$. Its nonzero vectors identify with ${\mathbb{C}}^2\setminus \{0\}$ by $(\ell ,v)\mapsto v$, with projection $v\mapsto [v]$. On the two standard charts, use $\zeta =z_2/z_1$ and $\eta =z_1/z_2=1/\zeta$. The local frames are $e_0=(1,\zeta )$ and $e_1=(\eta ,1)$, so $e_1=e_0/\zeta$. Normalize them using the ambient Hermitian norm:If $v=t_0{u}_0=t_1{u}_1$, then $t_0=(|\zeta |/\zeta )t_1$. Thus the unitary transitions of the tautological line over the projective line areThese are smooth on the overlap $\zeta \ne 0$. A Hermitian metric on a smooth complex vector bundle generally permits precisely this unitary reduction; here the ambient norm supplies it explicitly.

The corresponding principal circle bundle consists of unit vectors in the tautological line. Sending $(\ell ,v)$ to $v$ identifies its total space with $S^3\subset {\mathbb{C}}^2$, and the right circle action is $v\cdot e^{i\theta }=e^{i\theta }v$. Identify ${\mathbb{CP}}^1$ with $S^2$ usingIts affine-coordinate inverse on the chart $Z\ne -1$ is $\zeta =(X-iY)/(1+Z)$; the other chart handles the missing pole. This gives a diffeomorphism, and the projection becomes the Hopf fibration

There is a distinction if the stated punctured-bundle identification is interpreted only as a projection-preserving diffeomorphism, rather than a fiber-linear identification. It does not determine whether the complex line is tautological or dual: for a nonzero $\varphi \in L_{\ell }^{\ast }$, the unique $v\in \ell$ with $\varphi (v)=1$ gives a smooth map $L^{\ast }\setminus 0\to L\setminus 0$. Locally it is inversion of a nonzero complex number. Thus punctured line bundles do not determine their duality sign.

For completeness, these are the only possible signs here. Complex line bundles over $S^2$ are described by the clutching construction with a loop in ${\mathbb{C}}^{\ast }$, reducible to $S^1$, and its integer winding number determines the bundle: a zero-degree ratio has a logarithm and can be removed by changes of trivializations extending over the two disks. For winding number $d$, the unit circle bundle has fundamental group $\mathbb{Z}/d\mathbb{Z}$, as the two solid-torus trivializations identify the common fiber generator and impose its $d$th power as the equatorial gluing relation. There is a fiberwise radial deformation retraction from the punctured line bundle onto its unit circle bundle. Since ${\mathbb{C}}^2\setminus 0$ retracts to simply connected $S^3$, the hypothesis forces $|d|=1$.

Consequently the allowed unitary transitions are the displayed $g_{01}$ or its reciprocal, according to the bundle's sign. For the dual, the same underlying Hopf projection has the inverse principal action $v\cdot e^{i\theta }=e^{-i\theta }v$. In either case the principal total space and projection are $S^3\to S^2$; the purely diffeomorphic hypothesis alone leaves the circle-action sign unspecified.

On a complex manifold, holomorphic coordinates give the decomposition $d=\partial +\bar{\partial }$, withFor $\alpha ={\sum }_{I,J}{a}_{I\bar{J}}d{z}^I\wedge d{\bar{z}}^J$, the conjugate Dolbeault operator $\partial$ and the Dolbeault operator $\bar{\partial }$ are explicitlyThese are the type components of the intrinsic exterior derivative, so are independent of coordinates. The identity $d^2=0$, separated by type, gives ${\partial }^2={\bar{\partial }}^2=0$ and $\partial \bar{\partial }+\bar{\partial }\partial =0$. A holomorphic p-form on a complex manifold is a $(p,0)$-form whose coefficients are holomorphic functions; equivalently it is a $(p,0)$-form killed by the Dolbeault operator.

A rank-$r$ holomorphic vector bundle has local trivializations $E{|}_{U_i}\cong U_i\times {\mathbb{C}}^r$ whose changes are $(x,v)\mapsto (x,g_{ij}(x)v)$, with $g_{ij}:U_i\cap U_j\to {\mathrm{GL}}_r(\mathbb{C})$ holomorphic. These transition functions of a vector bundle satisfy $g_{ii}=I$ and $g_{ij}{g}_{jk}=g_{ik}$. Equivalently, for local frames we use the convention $e_j=e_i{g}_{ij}$. A holomorphic section is a section having holomorphic coordinate functions in every such holomorphic local frame; the condition is unchanged by the holomorphic invertible transitions.

The holomorphic cotangent bundle has local frames $d{z}^1,\dots ,d{z}^n$. Under another holomorphic coordinate system $w$, the chain rule givesThe Jacobian matrix is holomorphic and invertible, with holomorphic inverse supplied by the inverse coordinate change. These are precisely holomorphic bundle transitions. A section written as ${\sum }_b{f}_b(z)d{z}^b$ is holomorphic exactly when every $f_b$ is holomorphic, sothe space of holomorphic one-forms. In particular, being a holomorphic section is not the same as being a $d$-closed one-form on an arbitrary complex manifold.

The complex tautological line bundle isOn $U_i=\{Z_i\ne 0\}$ use the nowhere-zero frame $s_i([Z])=Z/Z_i$. Its components are holomorphic affine coordinates and the constant one in position $i$. On overlapsso the frame changes are nowhere-zero holomorphic functions. This proves that $\mathcal{O}(-1)$ is a holomorphic line bundle, and its inclusion into the trivial bundle is a holomorphic bundle map.

A global holomorphic section composed with that inclusion has $n+1$ component holomorphic functions on compact connected Complex projective space. Each is constant: its absolute value attains a maximum; in a coordinate ball the one-variable maximum modulus principle along complex lines makes it locally constant, and connectedness propagates that constant globally. Thus the section is a fixed vector $v\in {\mathbb{C}}^{n+1}$ lying in every fibre, namely every line $\mathbb{C}Z$. For $n\ge 1$ the intersection of all these lines is zero. HenceThe positive-dimension convention is necessary: ${\mathbb{CP}}^0$ is one point and this section space is $\mathbb{C}$.

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

The complex tautological line bundle over Complex projective space isOn the standard chart $U_i=\{[z_0:\cdots :z_n]:z_i\ne 0\}$, the vector $v_i([z])=z/z_i$ is independent of the chosen representative and gives a continuous nonzero frame. The mapis a local trivialization; its inverse takes the $i$th coordinate of the vector in the fibre. Hence $\lambda$ is a locally trivial complex line bundle.

Its unit sphere bundle is $S^{2n+1}$: a unit vector $v\in {\mathbb{C}}^{n+1}$ corresponds to $([v],v)$. The projection is the Hopf fibration. Regard $\lambda$ as an oriented real rank-two vector bundle using its complex orientation, and put $t=e(\lambda )\in H^2({\mathbb{CP}}^n;\mathbb{Z})$. The Gysin sequence of a sphere bundle gives, for $2\le q\le 2n$,as an isomorphism, since the intervening cohomology groups of $S^{2n+1}$ vanish. It also gives $H^1({\mathbb{CP}}^n)=0$. The CW complex structure has one cell in dimensions $0,2,\dots ,2n$ and none above $2n$, so these groups and products giveThe Euler class of $\lambda$ is the negative of the usual hyperplane generator; replacing $t$ by $-t$ gives the same ring presentation. For $n=0$ the formula reads $\mathbb{Z}$.

Now let $\alpha$ generate $H^2(S^2;\mathbb{Z})$ and let $u_i$ be its pullback from the $i$th factor of $(S^2{)}^n$. The Künneth theorem gives $H^2((S^2{)}^n;\mathbb{Z})={⨁}_i\mathbb{Z}{u}_i$. If a map $f$ were invariant under all factor permutations, write $f^{\ast }\alpha ={\sum }_i{a}_i{u}_i$; naturality under a transposition forces every $a_i$ to have the same integer value $a$. Let $\Delta :S^2\to (S^2{)}^n$ be the diagonal. ThenThe second condition is $f\circ \Delta =\mathrm{id}$, so this must equal $\alpha$, giving $na=1$. For $n>1$ this is impossible in $\mathbb{Z}$, and $\boxed{\text{no such continuous map exists}.}$ This is the diagonal-degree obstruction to a symmetric sphere retraction.

Use integral cohomology and let $k\ge 1$. On the projective bundle $Y=\mathbb{P}(E)$ define the complex tautological line bundlePut $t=e(\mathcal{L})\in H^2(Y;\mathbb{Z})$, using the canonical complex orientation. On each fibre, $t$ is the Euler class of the tautological line over ${\mathbb{CP}}^{k-1}$, so $1,t,\dots ,t^{k-1}$ restrict to an integral basis of its cohomology.

Here is the finite-cover Leray-Hirsch theorem proof in this case. For each open set $U\subseteq X$, defineIf $E$ is trivial on $U$, its projective bundle is $U\times {\mathbb{CP}}^{k-1}$ and $\mathcal{L}$ is pulled back from the tautological line on the second factor. The Künneth theorem makes ${\Phi }_U$ an isomorphism, since the fibre has finite free integral cohomology. The same holds for every open subset of $U$.

Compactness of $X$ provides a finite trivializing cover $U_1,\dots ,U_N$. Induct on its size. If the result holds on $V=U_1\cup \cdots \cup U_{N-1}$, it holds on $U_N$ and on $V\cap U_N$, both lying in a trivializing chart. Form the diagram of Mayer–Vietoris sequences for the base, with the finitely many degree shifts on the left, and the total space on the right. Naturality of pullback and multiplication by the global even-degree classes $t^i$ makes the diagram commute. The Five lemma gives the isomorphism on $V\cup U_N$. ThusThis is the claimed free module statement, with its graded degree shifts made explicit.

The module basis expresses $t^k$ uniquely using the lower powers, with homogeneous coefficients. Define the Chern classes $c_j(E)\in H^{2j}(X;\mathbb{Z})$ by the unique relationThe pullbacks are suppressed when $f$ is regarded as a polynomial over $H^{\ast }(X)$. Evaluation at $t$ gives a surjective map $H^{\ast }(X)[T]\to H^{\ast }(Y)$. Since $f(T)$ is monic, monic polynomial division over a ring writes any polynomial as $q(T)f(T)+r(T)$ with degree of $r$ below $k$. If its evaluation is zero, module independence forces every coefficient of $r$ to vanish. Hence the kernel is exactly the ideal generated by $f$, provingThe even-degree coefficients are central in the graded commutative algebra, so this division and ideal statement also apply when the base has odd-degree cohomology. Uniqueness of the coefficients proves their naturality under pullback, by pulling back the relation and using the same module basis. With the hyperplane convention $u=-t$, the relation has the usual all-positive Chern coefficients; the alternating signs here correspond to the tautological line itself.

Now suppose $E={⨁}_i{L}_i$. The sections of a projective bundle $s_i:X\to Y$ choose the line $(L_i{)}_x\subset E_x$. They satisfy $s_i^{\ast }\mathcal{L}\cong L_i$, so $s_i^{\ast }t=e_i$ and pulling back the relation gives $f(e_i)=0$.

To obtain the full factorization over the possibly torsion-containing base ring, also use the associated open chartsThey contain the images of the sections $s_i$ and cover $Y$. Projection identifies $\mathcal{L}{|}_{V_i}$ with ${\pi }^{\ast }{L}_i{|}_{V_i}$, so $t-{\pi }^{\ast }{e}_i$ restricts to zero on $V_i$. The long exact sequence of the pair lifts this class to $H^2(Y,V_i)$. The relative cup product of the $k$ lifted classes lies inIts absolute image is ${\prod }_i(t-{\pi }^{\ast }{e}_i)$, so that product vanishes. The polynomial ${\prod }_i(T-e_i)$ is monic of degree $k$ and lies in the kernel of evaluation. Subtracting the monic generator $f(T)$ leaves degree below $k$, and module independence again makes the difference zero. ThereforeThe open-cover argument proves the factorization without a non-zero-divisor assumption on the differences $e_i-e_j$.

The complex tautological line bundle on ${\mathbb{P}}^1$ has fibre at a line $\ell \subseteq {\mathbb{C}}^2$ equal to $\ell$ itself:Define $\mathcal{O}(1)=\mathcal{O}(-1{)}^{\ast }$, $\mathcal{O}(k)=\mathcal{O}(1{)}^{\otimes k}$ for $k>0$, $\mathcal{O}(0)$ to be trivial, and $\mathcal{O}(k)=\mathcal{O}(-1{)}^{\otimes (-k)}$ for $k<0$. These are the tensor powers of the hyperplane line bundle.

On the two given charts, use the holomorphic local frames $e_0(w)=(1,w)$ and $e_1(z)=(z,1)$ of $\mathcal{O}(-1)$. The maps $([1:w],a)\mapsto ([1:w],a(1,w))$ and $([z:1],b)\mapsto ([z:1],b(z,1))$ are its trivializations. On the overlap $z=1/w$,Thus the frame-transition factor from $e_0$ to $e_1$ is $w^{-1}$, while the fibre-coordinate transition from chart 0 to chart 1 is $w$. Specifying both avoids an inverse-convention ambiguity.

For $\mathcal{O}(k)$, let $e_i^{(k)}$ be the induced frames; they satisfy $e_1^{(k)}=w^k{e}_0^{(k)}$. Identifying overlap sections using $e_0^{(k)}$, the Čech cochain groups and Čech coboundary for $k\ge 0$ areHere $\mathcal{O}(\mathbb{C})$ means entire functions in the coordinate of the corresponding chart. Global sections are $\ker \delta$, so $a(w)=w^{k}b(1/w)$, or $b(z)=z^{k}a(1/z)$. Expanding the entire function $a(w)={\sum }_{j\ge 0}{a}_j{w}^j$ shows that $b$ is holomorphic at zero exactly when $a_j=0$ for $j>k$. Thus a global section is determined by a polynomial of degree at most $k$, with basis $1,w,\dots ,w^k$ in the chart-0 frame. Therefore $\boxed{\dim H^0({\mathbb{P}}^1,\mathcal{O}(k))=k+1(k\ge 0)}$.