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Let $Y$ be a smooth complex analytic hypersurface through the blowup centre with $[-Y]\cong K_X$. Its multiplicity at that point is one, so ${\sigma }^{\ast }Y=\tilde{Y}+E$. Combining this with the canonical bundle formula for a point blowup gives $[-\tilde{Y}]\cong K_{\tilde{X}}\otimes [E{]}^{\otimes (2-n)}$. The nontrivial powers of the exceptional line bundle of a point blowup prove the equivalence for $n\ge 2$. On a compact complex curve the premise cannot hold for an irreducible hypersurface: it is one point, so its negative divisor has degree $-1$, whereas the canonical bundle has even degree $2g-2$.

For a smooth complex analytic hypersurface $Y\subset X$, its holomorphic normal bundle is naturally the restriction of its holomorphic line bundle associated to a divisor:If reduced local defining functions satisfy $f_i=g_{ij}{f}_j$, then $d{f}_i{|}_Y=g_{ij}d{f}_j{|}_Y$. Pairing a normal vector with $d{f}_i$ and using the reciprocal transition functions of $\mathcal{O}(Y)$ gives the isomorphism.

At a generic smooth point of an irreducible complex analytic hypersurface $Y$, a nonzero meromorphic function has the form $f=t^{m}u$, where $t$ is a reduced transverse local equation and $u$ is a holomorphic unit. The integer $m$ is its order along $Y$: positive for a zero and negative for a pole. The unique factorization domain property of the local ring of a complex manifold makes this independent of the equation and propagates it along $Y$. These orders are the coefficients of the principal divisor on a complex manifold of $f$.

An irreducible analytic subvariety of a complex manifold is a closed subset locally defined by finitely many holomorphic functions which cannot be expressed as the union of two proper closed analytic subsets. In a complex $n$-manifold, codimension $k$ means pure complex dimension $n-k$. A divisor on a complex manifold is a locally finite formal integer combinationof irreducible analytic hypersurfaces. Compactness makes the set of components in this sum finite. Negative coefficients are permitted; a local equation for such a divisor on a complex manifold is generally meromorphic rather than holomorphic.

We use these precise local facts. In holomorphic coordinates at $x$, the local ring is ${\mathcal{O}}_{X,x}\cong \mathbb{C}\{z_1,\dots ,z_n\}$, a Noetherian regular local ring and a unique factorization domain. Each analytic hypersurface germ has finitely many irreducible local branches; their prime ideals have height one and are generated by irreducible holomorphic germs. A holomorphic germ is a unit exactly when its value at $x$ is nonzero. Consequently a nonzero meromorphic germ with order zero on every local hypersurface branch is a holomorphic unit, by cancelling its irreducible factors in numerator and denominator. These are the auxiliary local-ring properties required below.

After shrinking a neighbourhood $U_i$ around each point, choose local branch equations $h_{i\nu a}$, and putIts zero and pole orders give exactly the local divisor on a complex manifold $D{|}_{U_i}$. A globally irreducible hypersurface may have several branches locally; all of them must be included with the indicated multiplicity. On an overlap, $f_i/f_j$ has zero order along every branch, so the local-ring facts show that it extends to a nowhere-zero holomorphic function. These germwise extensions patch uniquely to a holomorphic unit on the overlap. This describes a local defining function of $D$ and proves why multiplicities, not just zero sets, matter.

Set $g_{ij}=f_i/f_j$ and glue line-bundle frames by $e_j=g_{ij}{e}_i$. The identities $g_{ij}{g}_{jk}=g_{ik}$ give a holomorphic line bundle associated to a divisor, denoted $[D]$. Equivalently, the frame $e_i$ can be represented by $1/f_i$ in the sheaf of meromorphic functions, so $[D]={\mathcal{O}}_X(D)$. If alternative equations are $f_i^{\prime }=u_i{f}_i$, where $u_i$ are holomorphic units, the corresponding frames are identified by $e_i^{\prime }\mapsto u_i^{-1}{e}_i$. This respects the changed transitions. Refining the cover gives the same identifications, proving that $[D]$ is well-defined up to holomorphic bundle isomorphism. The expressions $f_i{e}_i$ glue to its canonical meromorphic section, whose divisor is $D$.

For two divisors on a complex manifold use a common cover with equations $f_i$ and $h_i$. Their sum has equation $f_i{h}_i$, and the tensor product of vector bundles has frame transitionsSending the sum-divisor frame to the tensor product of the two frames givesThis also yields $[-D]\cong [D{]}^{\ast }$.

For a nonsingular complex analytic hypersurface $Y$, choose reduced local equations $f_i$ with $d{f}_i$ nonzero along $Y$. The holomorphic conormal bundle iswhose fibre is the annihilator of $T^{1,0}Y$. It has local frame $d{f}_i{|}_Y$. Here this restriction is as a covector in the ambient cotangent bundle along $Y$, not its pullback to ${\Omega }_Y^1$, where it would vanish. The line bundle $[-Y]$ has local frame $a_i=f_i$, and if $f_j=u_{ji}{f}_i$, thensince $f_{i}d{u}_{ji}$ vanishes along $Y$. The frame identifications $a_i{|}_Y\mapsto d{f}_i{|}_Y$ therefore patch and are fibrewise nonzero. This proves the conormal line of a smooth analytic hypersurface formulaEquivalently it identifies ${\mathcal{O}}_X(-Y){|}_Y={\mathcal{I}}_Y/{\mathcal{I}}_Y^2$ with the analytic conormal sheaf.

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.

This is the Poincaré residue of a meromorphic top form with a simple pole. Write $z=z_1$ and locally $\alpha =(dz/z)\wedge \beta$, with $\beta$ a holomorphic $(n-1)$-form. The proposed residue is the pullback of $\beta$ to $Y$.

First it does not depend on the choice of $\beta$: if $dz\wedge (\beta -{\beta }^{\prime })=0$, then at each point the difference has a factor $dz$, and its pullback to $Y$ is zero. Now change the local defining coordinate to $t=az$, where $a$ is a nowhere-zero holomorphic function. Such a factor exists because both coordinates define the same smooth complex analytic hypersurface. If $\alpha =(dt/t)\wedge \eta$, thenAt points of $Y$, this is an equality in the ambient exterior-power fibre, and the last term is zero. It follows that $dz\wedge (\beta -\eta )=0$ there, so their pullbacks to $Y$ agree. This also covers changes of the tangential coordinates.

The restrictions are holomorphic top forms on $Y$, and their agreement on overlaps makes them a global section of the canonical bundle. Thus the residue is independent of the adapted coordinates:The ambient-fibre comparison is important: pulling an ambient $n$-form directly back to the $(n-1)$-dimensional hypersurface would give zero and would not prove the required independence.

If a meromorphic top form has at most a simple pole along a smooth complex analytic hypersurface $Y=\{z=0\}$, write $\alpha =(dz/z)\wedge \beta$ with $\beta$ holomorphic. Its residue is $\beta {|}_Y$, a holomorphic section of the canonical bundle $K_Y$. A change of defining coordinate $t=az$, with $a$ holomorphic and nonzero, changes $dt/t$ by the holomorphic form $da/a$ and leaves this restriction invariant. Placing the normal factor first fixes the sign convention.