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If $f_i$ and $g_i$ are local meromorphic equations for two divisors on a complex manifold, then $f_i{g}_i$ defines their sum. With local frames satisfying $e_j=(f_i/f_j)e_i$, the transition functions of a vector bundle for the tensor product are the products of the two ratios, exactly those for $f_i{g}_i$. Multiplication of local frames therefore gives the displayed holomorphic line bundle isomorphism.

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.

For the structure sheaf of a complex manifold, local holomorphic logarithms give the short exact sequence of sheavesThe connecting map $H^1(M,{\mathcal{O}}_M^{\ast })\to H^2(M,\mathbb{Z})$ sends a holomorphic line bundle to its First Chern class. In particular, it is onto when $H^2(M,{\mathcal{O}}_M)=0$.

The holomorphic Picard group consists of isomorphism classes of holomorphic line bundles, with tensor product as its group law. The trivial bundle is the identity and dualization gives inverses. Transition functions identify it with $H^1(X,{\mathcal{O}}_X^{\ast })$. On the complex projective line every class is $\mathcal{O}(n)$, so degree gives an isomorphism with $\mathbb{Z}$.

On $\mathbb{C}/(\mathbb{Z}+\mathbb{Z}i)$, a Hermitian form linear in its first argument has integral imaginary part on the period lattice exactly when its real scalar coefficient is integral. Positivity is equivalent to $t>0$. The associated holomorphic line bundle has degree $t$ because its First Chern class is represented by $(it/2)dz\wedge d\overline{z}$. In particular the oriented Chern pairing is minus the imaginary part under this convention.

The Kähler cone of a compact Kähler manifold is the open cone of real $(1,1)$ cohomology classes represented by positive closed $(1,1)$ forms. If every real degree-two cohomology class has type $(1,1)$, density of rational classes supplies a rational Kähler class. An integral multiple is the First Chern class of a positive holomorphic line bundle, and the Kodaira embedding theorem implies projectivity.

Let $X$ be a nonempty positive-dimensional compact complex submanifold of Complex projective space. On a positive-dimensional connected component choose a hyperplane meeting, but not containing, that component. Its linear equation gives a holomorphic section of ${\mathcal{O}}_X(1)$ with a zero and with a nonzero value. For $m>0$, its $m$th power cannot be a section of a trivial holomorphic line bundle, since such sections are constant on the compact connected component. Negative powers are also nontrivial by duality. Every positive-dimensional compact projective component meets every hyperplane: otherwise its affine coordinate functions would all be constant.

For a smooth complex vector bundle $E$, a connection is a complex-linear map $D:\Gamma (E)\to A^1(M;E)$ satisfying $D(fs)=df\otimes s+fDs$ for smooth complex functions $f$. Local trivializations supply flat local connections. A locally finite smooth partition of unity $({\rho }_i)$ subordinate to them gives $D={\sum }_i{\rho }_i{D}_i$; local finiteness makes the sum smooth and ${\sum }_i{\rho }_i=1$ gives the Leibniz rule. Thus connections on a vector bundle always exist on the usual paracompact smooth manifold.

Extend $D$ to bundle-valued forms by $D(\alpha \otimes s)=d\alpha \otimes s+(-1{)}^{\mathrm{deg}\alpha }\alpha \wedge Ds$. Squaring shows that $D^2(fs)=f{D}^{2}s$, so the curvature form is the endomorphism-valued two-form $\Theta =D^2$. With a row frame $e=(e_1,\dots ,e_r)$, write $De=eA$; for coefficient columns $s=ev$, $Ds=e(dv+Av)$. Applying $D$ again and using the graded Leibniz rule givesThe cross terms $-A\wedge dv$ and $A\wedge dv$ cancel. Hence the Cartan curvature matrix equation in this convention isFor a frame change $e^{\prime }=eg$, the connection matrix and curvature transform as $A^{\prime }=g^{-1}Ag+g^{-1}dg$ and ${\Theta }^{\prime }=g^{-1}\Theta g$. Trace is invariant under conjugation, so the local forms $\mathrm{Tr}\Theta$ patch into the global trace of vector-bundle curvature.

The induced determinant connection on $\det E={\Lambda }^{r}E$ iswhere the one-form factor of each $D{s}_j$ is placed in front of the section factors. In the local determinant frame $e_1\wedge \cdots \wedge e_r$, only the diagonal terms survive, so its connection form is $\mathrm{Tr}A$. Its curvature is $d\mathrm{Tr}A$. Sinceby pairing off-diagonal terms and using $A_{ii}\wedge A_{ii}=0$, we obtainIt is closed locally by $d^2=0$, and thus closed globally.

For two connections, $B=D_1-D_0$ is a global endomorphism-valued one-form: the derivative terms cancel in their Leibniz rules. In a local frame $A_1=A_0+B$, soTaking traces cancels the mixed terms, because both factors are one-forms, and cancels $\mathrm{Tr}(B\wedge B)$. Therefore the trace curvature transgression formula isThe right-hand side is globally exact, proving independence of the de Rham cohomology class without needing a sheaf-cohomology identification.

Finally let $e$ be a holomorphic frame for the holomorphic line bundle, with $h=\Vert e{\Vert }^2>0$. Define $De=e\partial \log h$, extending by the Leibniz rule. If $e^{\prime }=eg$ for a nowhere-zero holomorphic function $g$, then $h^{\prime }=|g{|}^{2}h$ andThis is exactly the line-bundle frame-change law for a connection, so the local definitions glue globally. Its $(0,1)$ part is the Dolbeault operator. Metric compatibility follows from $d\log h=\partial \log h+\bar{\partial }\log h$, or $dh=h(A+\overline{A})$ for $A=\partial \log h$. These conditions uniquely force this $A$, identifying the Chern connection. Since a scalar one-form wedges with itself to zero,This agrees with the local formula for the Chern connection on a line bundle; keeping the order of the two Dolbeault differentials fixes the sign.

Let $X$ be a connected compact Riemann surface of genus $g$. Choose an oriented symplectic basis $a_1,\dots ,a_g,b_1,\dots ,b_g$ of $H_1(X,\mathbb{Z})$, with $a_i\cdot b_j={\delta }_{ij}$. For a closed complex differential form $\alpha$, write $A_i(\alpha )={\int }_{a_i}\alpha$ and $B_i(\alpha )={\int }_{b_i}\alpha$. The Riemann bilinear relations begin with the identityTo prove it, cut $X$ along the chosen cycles to obtain a fundamental polygon $P$. On this simply connected polygon, a closed differential form has a primitive $F$, with $dF=\alpha$. Stokes theorem gives ${\int }_P\alpha \wedge \beta ={\int }_{\partial P}F\beta$. The two copies of each edge have opposite orientations, and their values of $F$ differ by the period acquired in passing between them. Pairing the copies of the $a_i$ and $b_i$ edges gives, respectively, $-B_i(\alpha )A_i(\beta )$ and $A_i(\alpha )B_i(\beta )$. Summing proves the identity; the convention $a_i\cdot b_i=1$ fixes its sign.

For holomorphic one-forms $\omega ,\eta$, their wedge product is zero because locally both are multiples of $dz$. Thus the first holomorphic Riemann bilinear relations isFor a nonzero holomorphic one-form $\omega =f(z)dz$, the second relation follows by applying the bilinear identity to $\omega ,\overline{\omega }$:In particular, a holomorphic one-form with all $a$-periods zero must vanish. Since $\dim H^0(X,K_X)=g$, the map taking a holomorphic one-form to its $a$-periods is an isomorphism. We can therefore choose normalized holomorphic one-forms ${\omega }_1,\dots ,{\omega }_g$ with ${\int }_{a_i}{\omega }_j={\delta }_{ij}$. Put $B_{ij}={\int }_{b_i}{\omega }_j$. The first relation applied to ${\omega }_j,{\omega }_k$ gives $B_{jk}=B_{kj}$, and the second applied to ${\sum }_j{c}_j{\omega }_j$ gives $2{\overline{c}}^{,t}(\mathrm{Im}B)c>0$ for $c\ne 0$. ConsequentlyThese conclusions make the construction of the Jacobian variety work. Integration identifies the image of $H_1(X,\mathbb{Z})$ in $H^0(X,K_X{)}^{\ast }$ with the period lattice $\Lambda ={\mathbb{Z}}^g+B{\mathbb{Z}}^g$. The real-linear map $(m,n)\mapsto m+Bn$ is invertible: its imaginary part is $Yn$, and $Y$ is invertible. Hence $\Lambda$ is a discrete lattice of full real rank $2g$, andis a compact complex torus, rather than a non-Hausdorff quotient. Changing the symplectic basis changes the displayed coordinates but not this intrinsic quotient.

The Riemann bilinear relations also supply its polarization of a complex torus. Use the Hermitian form $H(z,w)=z^t{Y}^{-1}\overline{w}$, linear in the first argument, and set $E=-\mathrm{Im}H$. Symmetry of $B$ givesThus $E$ is an integral positive Riemann form on a complex torus. Its matrix on the displayed lattice basis is the unimodular symplectic matrix. The corresponding positive holomorphic line bundle, equivalently the one constructed from theta automorphy in Solution 4, gives a principal polarization of a complex torus. By the Kodaira embedding theorem this makes the complex torus projective, hence an abelian variety. The normalization $E=-\mathrm{Im}H$ here is important because $H$ was chosen linear in its first argument. If $g=0$, the spaces of holomorphic one-forms and periods are zero and the Jacobian variety is a point.

For a symmetric matrix $B$ with $Y=\mathrm{Im}B$ positive definite, the Riemann theta function isOn any compact set of $z$, the absolute value of the summand is bounded by $\exp (-c|n{|}^2+C|n|)$ for some $c>0$, so the series and all its derivatives converge uniformly there. It is therefore entire. Its constant Fourier coefficient in $\mathrm{Re}z$ is $1$, so it is not identically zero. Reindexing $n$ gives its parity and automorphy laws:Although the theta function is not itself a function on $J={\mathbb{C}}^g/\Lambda$, these factors define a holomorphic line bundle on $J$, and its zeros define a divisor $\Theta$ there.

For the period matrix of $X$, the Riemann vanishing theorem asserts that a fixed class $\kappa \in J$, the vector of Riemann constants in the convention used here, satisfiesWe prove both its support statement and its reduced divisor structure. This will make the multiplicity calculation in Solution 5 independent of any assumed multiplicity theorem.

The essential contour calculation is the theta pullback zero-divisor identity. Put $f_z(p)=\theta (u(p)-z,B)$, regarding this as a local expression for a section on $X$. If it is not identically zero, let $D_z$ be its zero divisor. We claimUse the cut polygon from Solution 1 and choose cuts avoiding the zeros. There ${\alpha }_z=d_p\log f_z$ is a meromorphic differential whose residues are the zero multiplicities. Under continuation by an $a_j$ period, $u$ changes by the integer vector $e_j$, so ${\alpha }_z$ is unchanged. Under a $b_j$ period, $u$ changes by $B{e}_j$, so automorphy gives ${\alpha }_z\mapsto {\alpha }_z-2\pi i{\omega }_j$. Pairing opposite boundary sides therefore givesFor example, the two $a_j$ sides, traversed with opposite orientations and separated by a $b_j$ translation, contribute ${\int }_{a_j}({\alpha }_z-({\alpha }_z-2\pi i{\omega }_j))$. This fixes the sign of the count.

For the first moment of the zeros, the residue theorem on the polygon gives, component by component,This is interpreted modulo the period lattice, with the lifts to the polygon fixed. Vary $z$ without letting zeros cross the cuts, and put $h=\delta \log f_z$. The coordinate $u_k$ does not vary, so integration by parts givesThere is no endpoint term: the closed polygon boundary returns both the Abel integral and $h$ to their starting values. Across an $a_j$ translation $h$ is unchanged, while across a $b_j$ translation it changes by $2\pi i\delta z_j$. Pairing the boundary edges consequently gives ${\int }_{\partial P}{\omega }_{k}h=-2\pi i{\sum }_j\delta z_j{\int }_{a_j}{\omega }_k=-2\pi i\delta z_k$. Thus $\delta u_g(D_z)=\delta z$, and $u_g(D_z)-z$ is constant. The calculation uses weighted zeros, so their collisions cause no problem. The parameters where $f_z$ is identically zero form a proper analytic subset: they are given by the vanishing of all its local Taylor coefficients, and parameters outside $\Theta$ are not in it. Its complement in the connected complex manifold $J$ is connected, so the constant is one class $\kappa$. This proves the pullback identity globally, with continuation of the zero divisors where necessary.

We must account for the parameters with identically zero pullback, rather than discard them. No irreducible component $C$ of $\Theta$ can consist entirely of such parameters. Otherwise parity would imply $C-u(p)\subset \Theta$ for every $p\in X$. The image of $C\times X$ under subtraction is irreducible, is contained in $\Theta$, and contains $C$ when $p=p_0$. Since $C$ is an irreducible component of the hypersurface, that image must be $C$. Hence $C-u(p)=C$ for all $p$. The surjectivity of the large-degree Abelian sum map, proved in Solution 3, says that these translations generate all of $J$. This would make the nonempty proper subset $C$ invariant under every translation, which is impossible.

A generic point $z$ of each component of $\Theta$ therefore has nonzero pullback. Since $\theta (-z)=\theta (z)=0$, its zero divisor contains $p_0$. Write $D_z=p_0+D^{\prime }$, with $D^{\prime }$ effective of degree $g-1$. The pullback identity gives $u_{g-1}(D^{\prime })=z+\kappa$. Each component of $\Theta$ is consequently contained in $W_{g-1}-\kappa$. By Solution 3 the latter is irreducible of dimension $g-1$, so any such component equals it. The zero locus is nonempty for $g>0$: any parameter with nonzero pullback has $g$ pullback zeros. We have proved equality of supports, in both directions.

Finally the automorphy factors make the First Chern class of the theta line bundle primitive. On the two-torus spanned by $a_i,b_j$, the logarithm of the $b_j$ factor changes by $-2\pi i{\delta }_{ij}$ after the $a_i$ translation; equivalently the transition for local frames has opposite winding. Thus $c_1(a_i,b_j)={\delta }_{ij}$, while $c_1(a_i,a_j)=c_1(b_i,b_j)=0$. In particular some integral pairing equals $1$. As its support is the single irreducible hypersurface $W_{g-1}-\kappa$, the zero divisor is $m(W_{g-1}-\kappa )$ for a positive integer $m$. Its integral First Chern class would then be divisible by $m$. Primitivity forces $m=1$, proving the displayed equality as reduced divisors.

The sign convention is $\kappa =u_g(D_z)-z$, so the translate is $W_{g-1}-\kappa$; a convention naming $-\kappa$ the Riemann constant reverses that displayed sign. For $g=1$, $\kappa =(1+B)/2$ modulo periods and the zero locus is the corresponding single point. For $g=0$, the empty-lattice theta series is $1$, its zero locus is empty, and the degree-$-1$ effective locus is empty; no negative symmetric product is required.

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.

On a complex manifold, a divisor on a complex manifold is a locally finite formal sum $D={\sum }_Y{m}_{Y}Y$, where $Y$ ranges over irreducible complex analytic hypersurfaces and $m_Y\in \mathbb{Z}$. The local algebra needed here is that the local ring ${\mathcal{O}}_{X,x}$, identified in coordinates with convergent power-series germs, is a regular local ring and a unique factorization domain. In particular, each height-one prime is generated by one irreducible germ, and a nonzero meromorphic germ is a quotient of holomorphic germs. Reduced local equations of the same hypersurface differ by a holomorphic unit. These facts include local branches when a globally irreducible hypersurface has several branches at a singular point.

The order of a meromorphic function along a hypersurface is the exponent of its reduced local equation in the quotient's factorization. Equivalently, at a generic smooth point of $Y$, write $f=t^{m}u$, with $t$ a transverse local equation of $Y$ and $u$ a holomorphic unit; then ${\mathrm{ord}}_Y(f)=m$. Changing a reduced equation multiplies it by a unit and leaves $m$ unchanged. These local orders agree on overlaps and are constant along the generic smooth locus of an irreducible hypersurface. A nonzero global meromorphic function therefore defines its principal divisor on a complex manifoldPositive coefficients record zeros and negative coefficients record poles.

A local defining function $f_i$ for $D$ on $U_i$ is a nonzero meromorphic function whose orders there equal the coefficients of $D$. It can be constructed as the product of local reduced hypersurface equations to the prescribed integer powers. On an overlap, $f_i/f_j$ has no zero or pole along any hypersurface. The unique factorization domain property then makes it a nowhere-zero holomorphic function, after choosing representatives of the germs. The holomorphic line bundle associated to a divisor is obtained by gluing holomorphic frames usingThese ratios satisfy the cocycle condition, and the expressions $f_i{e}_i$ agree on overlaps. They form the canonical meromorphic section of a holomorphic line bundle with divisor $D$.

If $D=\mathrm{div}(f)$ globally, $f_i/f$ is a holomorphic unit on $U_i$, and $(f_i/f)e_i$ is a global nowhere-zero holomorphic section. Hence $[D]$ is holomorphically trivial. Conversely, if $[D]$ has a nowhere-zero global holomorphic frame $e$, write $e_i=a_{i}e$ with $a_i$ holomorphic units. The coefficients $f_i{a}_i$ of the canonical meromorphic section of a holomorphic line bundle agree on overlaps and give a nonzero global meromorphic function $f$. Since multiplication by a unit does not change orders, $\mathrm{div}(f)=D$. Therefore

For a smooth hypersurface $Y\subset X$, the adjunction formula isIndeed, the holomorphic conormal sequence has first term $N_{Y/X}^{\ast }\cong [-Y]{|}_Y$; taking its determinant yields $K_X{|}_Y\cong N_{Y/X}^{\ast }\otimes K_Y$, which gives the formula. Here the canonical bundle is the top exterior power of the holomorphic cotangent bundle.

Apply this first to $W\subset {\mathbb{P}}^4$. Its degree-$d_1$ equation is a section of $\mathcal{O}(d_1)$, so $[W]\cong \mathcal{O}(d_1)$. Since $K_{{\mathbb{P}}^4}\cong \mathcal{O}(-5)$, we obtain $K_W\cong {\mathcal{O}}_W(d_1-5)$. The restricted second equation cuts out $S$ in $W$ and has associated bundle ${\mathcal{O}}_W(d_2)$. A second application of adjunction formula gives the canonical bundle of a regular projective complete intersection

It remains to prove that a nonzero exponent cannot yield a trivial holomorphic line bundle on $S$. Choose a connected component $S_0$ and a hyperplane meeting but not containing it. Such a hyperplane exists: a positive-dimensional compact projective submanifold cannot lie entirely in an affine chart, since all its affine coordinate functions would be global holomorphic functions and hence constant. A hyperplane through one point and not another consequently meets $S_0$ properly. Its equation gives a holomorphic section of ${\mathcal{O}}_{S_0}(1)$ that has both a zero and a nonzero value. For $m>0$, the $m$th power has the same two properties. Were ${\mathcal{O}}_{S_0}(m)$ trivial, this section would be represented by a global holomorphic function, contradicting its constancy on compact connected $S_0$. For $m<0$, triviality would imply triviality of its positive dual power. This proves nontrivial hyperplane powers on a compact projective submanifold and henceThe degrees are positive because the two equations define a hypersurface and a surface of the specified dimensions.

The calculation uses “non-singular” in the usual smooth complete-intersection sense: the equations define reduced smooth divisors, with independent differentials along $S$. If only their reduced zero sets are required to be smooth, polynomial multiplicities cannot determine the canonical bundle. For example, $Q_1=Z_0^m$ and $Q_2=Z_1^4+Z_2^4+Z_3^4+Z_4^4$ have the same smooth quartic surface in the hyperplane $Z_0=0$ for every $m\ge 1$. That surface has trivial canonical bundle, while the displayed polynomial degree sum is $m+4$. Thus unreduced equations must not be substituted into the adjunction formula for the reduced surface.

An irreducible complex analytic hypersurface is a closed analytic subset of pure complex codimension one which cannot be expressed as the union of two proper closed analytic subsets. Here a local defining function of a complex analytic hypersurface means a reduced local equation: a holomorphic germ generating its vanishing ideal, with each local branch occurring once. The word reduced matters. Merely prescribing the zero set would allow both $f$ and $f^2$, and the requested uniqueness would then be false.

The local facts used are these: in holomorphic coordinates the local ring ${\mathcal{O}}_{X,p}$ is $\mathbb{C}\{z_1,\dots ,z_n\}$, a regular local ring and a unique factorization domain; its units are exactly the germs nonzero at $p$; and the analytic Nullstellensatz identifies the zero-set ideal of a germ $f$ with $\sqrt{(f)}$. A reduced germ is a product of distinct irreducible factors. Unique factorization makes its principal ideal radical: if $f\mid h^r$, every irreducible factor of $f$ divides $h$, so $f\mid h$. Thus two reduced germs with the same hypersurface germ have $(f)=(g)$. Write $g=uf$, $f=vg$. The local ring is a domain, so $uv=1$. Hence $u$ is a holomorphic unit and $u(p)\ne 0$. This proves reduced local defining functions differ by a holomorphic unit. Global irreducibility does not imply local irreducibility at a singular point; the same argument uses the product of all local branches and therefore covers that case too.

A divisor on a complex manifold is a locally finite integer combination of irreducible analytic hypersurfaces. On a compact manifold only finitely many have nonzero coefficients. Choose local meromorphic equations $f_i$ for a divisor $D$, by products of powers of reduced local equations. Their ratios are holomorphic units on overlaps. For the holomorphic line bundle associated to a divisor, choose frames satisfyingThe ratios obey the cocycle identity and so define a holomorphic line bundle $[D]$. This convention makes the local meromorphic sections $f_i{e}_i$ agree, with divisor $D$. Changing the equations by units changes frames and gives an isomorphic bundle.

For hyperplanes $H_1,H_2$ in Complex projective space, choose nonzero linear equations ${\ell }_1,{\ell }_2$. The degree-zero quotient $r={\ell }_1/{\ell }_2$ is a global meromorphic function and has divisor $H_1-H_2$. A principal divisor on a complex manifold gives a trivial associated bundle: using the same global meromorphic equation on every chart makes all frame transitions equal to one. The allowed tensor-product rule now givesIf the hyperplanes coincide the same conclusion is immediate. Equivalently, their common bundle is the hyperplane bundle $\mathcal{O}(1)$.

Locally at the centre of a blowup of a complex manifold at a point, replace a coordinate neighbourhood in ${\mathbb{C}}^n$ byAway from zero the direction is uniquely $[z]$, so this map is an isomorphism there; glue it to $X\setminus \{p\}$. In the chart ${\ell }_j\ne 0$, write $z_j=u$ and $z_i=u{t}_i$ for $i\ne j$. These are smooth holomorphic coordinates on the blowup. The exceptional divisor is $u=0$, globally $E={\sigma }^{-1}(p)\cong \mathbb{P}(T_{p}X)$.

The canonical bundle formula for a point blowup isThe local chart also explains the exponent: pulling back a coordinate volume form gives $\pm u^{n-1}du\wedge {\bigwedge }_{i\ne j}d{t}_i$, so the Jacobian section has divisor $(n-1)E$.

Since $Y$ is smooth at the centre, choose coordinates with $Y=\{z_1=0\}$. Its strict transform is a genuine analytic hypersurface. In a blowup chart $j\ne 1$, it is the closed set $t_1=0$: off $E$, the original equation $u{t}_1=0$ reduces to $t_1=0$, and taking the closure restores the points with $u=0$. In chart $j=1$, there are no such off-exceptional points, and the strict transform is empty there. These descriptions agree on overlaps and show analyticity, with $\tilde{Y}\cap E\cong {\mathbb{CP}}^{n-2}$. The total transform has the extra factor $u$ exactly once, soUnder the given anticanonical assumption, this yieldsTo prove the only-if direction, one must rule out an accidental trivial power of $[E]$. The local blowup is the tautological line bundle over ${\mathbb{CP}}^{n-1}$, with $E$ its zero section. Its normal bundle is therefore ${\mathcal{O}}_E(-1)$. The normal bundle of a smooth analytic hypersurface identifies $[E]{|}_E$ with this bundle. If $n\ge 2$, restrict to a projective line inside $E$; the restriction of $[E{]}^{\otimes (2-n)}$ has degree $n-2$. A trivial line bundle has degree zero, so it is trivial only if $n=2$. Conversely the displayed tensor factor is literally trivial when $n=2$. This is the anticanonical strict transform under a point blowup criterion:For completeness, dimension one cannot satisfy the premise in this compact setting: an irreducible analytic hypersurface is one point, and $[-Y]$ has degree $-1$ on its curve component, whereas $K_X$ has degree $2g-2$. Thus no exceptional dimension-one case invalidates the equivalence.

A holomorphic line bundle is a complex rank-one bundle locally trivialized over holomorphic coordinate opens, with nowhere-zero holomorphic transition functions. A local holomorphic section has a holomorphic coefficient in a holomorphic local frame. Let $h$ be a smooth positive Hermitian metric on its fibres. A Chern connection is a connection compatible with $h$ whose $(0,1)$ part is the bundle's Dolbeault operator.

Fix the Hermitian-product convention linear in the first variable and a nonvanishing holomorphic local frame $e$. Put $h_e=h(e,e)>0$ and write $\nabla e=Ae$. Compatibility with the holomorphic structure forces $A^{0,1}=0$. Metric compatibility givesso taking its $(1,0)$ part determines $A$ uniquely:Define the connection by this formula. To check it exists globally, change frame to $e^{\prime }=ge$ for a nonzero holomorphic function $g$. Then $h_{e^{\prime }}=|g{|}^2{h}_e$ andwhich is exactly the connection transformation law obtained from $\nabla (ge)=dge+g\nabla e$. Hence the local connections glue. The formula and its conjugate give metric compatibility, and $A^{0,1}=0$ gives the required holomorphic structure. Since either condition already forced this unique formula in every frame, existence and uniqueness are proved.

For a line bundle, the curvature form of a connection is $F={\nabla }^2=dA$, as the scalar one-form satisfies $A\wedge A=0$. The local formula for the Chern connection on a line bundle therefore givesThis formula uses a nowhere-zero local holomorphic section; arbitrary sections with zeros are not frames there. It has type $(1,1)$, is $d$-closed and satisfies ${\bar{F}}_h=-F_h$, so $i{F}_h$ is a real $(1,1)$-form. The transformation calculation makes its frame independence explicit.