Inmathematics, especially inalgebraic geometry and the theory ofcomplex manifolds, theadjunction formula relates thecanonical bundle of a variety and ahypersurface inside that variety. It is often used to deduce facts about varieties embedded in well-behaved spaces such asprojective space or to prove theorems by induction.

LetX be asmooth algebraic variety or smooth complex manifold andY be a smooth subvariety ofX. Denote the inclusion mapY →X byi and theideal sheaf ofY inX by${\displaystyle \mathcal{I}}$. Theconormal exact sequence fori is

${\displaystyle 0\to \mathcal{I}/{\mathcal{I}}^2\to i^{\ast }{\mathrm{\Omega }}_X\to {\mathrm{\Omega }}_Y\to 0,}$

where Ω denotes acotangent bundle. The determinant of this exact sequence is a natural isomorphism

${\displaystyle {\omega }_Y=i^{\ast }{\omega }_X\otimes \det (\mathcal{I}/{\mathcal{I}}^2{)}^{\vee },}$

where${\displaystyle \vee }$ denotes the dual of a line bundle.

Suppose thatD is a smoothdivisor onX. Itsnormal bundle extends to aline bundle${\displaystyle \mathcal{O}(D)}$ onX, and the ideal sheaf ofD corresponds to its dual${\displaystyle \mathcal{O}(-D)}$. The conormal bundle${\displaystyle \mathcal{I}/{\mathcal{I}}^2}$ is${\displaystyle i^{\ast }\mathcal{O}(-D)}$, which, combined with the formula above, gives

${\displaystyle {\omega }_D=i^{\ast }({\omega }_X\otimes \mathcal{O}(D)).}$

In terms of canonical classes, this says that

${\displaystyle K_D=(K_X+D){|}_D.}$

Both of these two formulas are called theadjunction formula.

Given a smooth degree${\displaystyle d}$ hypersurface${\displaystyle i:X\hookrightarrow {\mathbb{P}}_S^n}$ we can compute its canonical and anti-canonical bundles using the adjunction formula. This reads as

${\displaystyle {\omega }_X\cong i^{\ast }{\omega }_{{\mathbb{P}}^n}\otimes {\mathcal{O}}_X(d)}$

which is isomorphic to${\displaystyle {\mathcal{O}}_X(-n-1+d)}$.

For a smooth complete intersection${\displaystyle i:X\hookrightarrow {\mathbb{P}}_S^n}$ of degrees${\displaystyle (d_1,d_2)}$, the conormal bundle${\displaystyle \mathcal{I}/{\mathcal{I}}^2}$ is isomorphic to${\displaystyle \mathcal{O}(-d_1)\oplus \mathcal{O}(-d_2)}$, so the determinant bundle is${\displaystyle \mathcal{O}(-d_1-d_2)}$ and its dual is${\displaystyle \mathcal{O}(d_1+d_2)}$, showing

${\displaystyle {\omega }_X\cong {\mathcal{O}}_X(-n-1)\otimes {\mathcal{O}}_X(d_1+d_2)\cong {\mathcal{O}}_X(-n-1+d_1+d_2).}$

This generalizes in the same fashion for all complete intersections.

${\displaystyle {\mathbb{P}}^1\times {\mathbb{P}}^1}$ embeds into${\displaystyle {\mathbb{P}}^3}$ as a quadric surface given by the vanishing locus of a quadratic polynomial coming from a non-singular symmetric matrix.^[1] We can then restrict our attention to curves on${\displaystyle Y={\mathbb{P}}^1\times {\mathbb{P}}^1}$. We can compute the cotangent bundle of${\displaystyle Y}$ using the direct sum of the cotangent bundles on each${\displaystyle {\mathbb{P}}^1}$, so it is${\displaystyle \mathcal{O}(-2,0)\oplus \mathcal{O}(0,-2)}$. Then, the canonical sheaf is given by${\displaystyle \mathcal{O}(-2,-2)}$, which can be found using the decomposition of wedges of direct sums of vector bundles. Then, using the adjunction formula, a curve defined by the vanishing locus of a section${\displaystyle f\in \mathrm{\Gamma }(\mathcal{O}(a,b))}$, can be computed as

${\displaystyle {\omega }_C\cong \mathcal{O}(-2,-2)\otimes {\mathcal{O}}_C(a,b)\cong {\mathcal{O}}_C(a-2,b-2).}$

The restriction map${\displaystyle {\omega }_X\otimes \mathcal{O}(D)\to {\omega }_D}$ is called thePoincaré residue. Suppose thatX is a complex manifold. Then on sections, the Poincaré residue can be expressed as follows. Fix an open setU on whichD is given by the vanishing of a functionf. Any section overU of${\displaystyle \mathcal{O}(D)}$ can be written ass/f, wheres is a holomorphic function onU. Let η be a section overU of ω_X. The Poincaré residue is the map

${\displaystyle \eta \otimes \frac{s}{f}\mapsto s\frac{\mathrm{\partial }\eta }{\mathrm{\partial }f}{|}_{f=0},}$

that is, it is formed by applying the vector field ∂/∂f to the volume form η, then multiplying by the holomorphic functions. IfU admits local coordinatesz₁, ...,z_n such that for somei,∂f/∂z_i ≠ 0, then this can also be expressed as

${\displaystyle \frac{g(z)d{z}_1\wedge \cdots \wedge d{z}_n}{f(z)}\mapsto (-1{)}^{i-1}\frac{g(z)d{z}_1\wedge \cdots \wedge \widehat{d{z}_i}\wedge \cdots \wedge d{z}_n}{\mathrm{\partial }f/\mathrm{\partial }{z}_i}{|}_{f=0}.}$

Another way of viewing Poincaré residue first reinterprets the adjunction formula as an isomorphism

${\displaystyle {\omega }_D\otimes i^{\ast }\mathcal{O}(-D)=i^{\ast }{\omega }_X.}$

On an open setU as before, a section of${\displaystyle i^{\ast }\mathcal{O}(-D)}$ is the product of a holomorphic functions with the formdf/f. The Poincaré residue is the map that takes the wedge product of a section of ω_D and a section of${\displaystyle i^{\ast }\mathcal{O}(-D)}$.

The adjunction formula is false when the conormal exact sequence is not a short exact sequence. However, it is possible to use this failure to relate the singularities ofX with the singularities ofD. Theorems of this type are calledinversion of adjunction. They are an important tool in modern birational geometry.

Let${\displaystyle C\subset {\mathbf{P}}^2}$ be a smooth plane curve cut out by a degree${\displaystyle d}$ homogeneous polynomial${\displaystyle F(X,Y,Z)}$. We claim that the canonical divisor is${\displaystyle K=(d-3)[C\cap H]}$ where${\displaystyle H}$ is the hyperplane divisor.

First work in the affine chart${\displaystyle Z\ne 0}$. The equation becomes${\displaystyle f(x,y)=F(x,y,1)=0}$ where${\displaystyle x=X/Z}$ and${\displaystyle y=Y/Z}$.We will explicitly compute the divisor of the differential

${\displaystyle \omega :=\frac{dx}{\mathrm{\partial }f/\mathrm{\partial }y}=\frac{-dy}{\mathrm{\partial }f/\mathrm{\partial }x}.}$

At any point${\displaystyle (x_0,y_0)}$ either${\displaystyle \mathrm{\partial }f/\mathrm{\partial }y\ne 0}$ so${\displaystyle x-x_0}$ is a local parameter or${\displaystyle \mathrm{\partial }f/\mathrm{\partial }x\ne 0}$ so${\displaystyle y-y_0}$ is a local parameter.In both cases the order of vanishing of${\displaystyle \omega }$ at the point is zero. Thus all contributions to the divisor${\displaystyle \text{div}(\omega )}$ are at the line at infinity,${\displaystyle Z=0}$.

Now look on the line${\displaystyle Z=0}$. Assume that${\displaystyle [1,0,0]\notin C}$ so it suffices to look in the chart${\displaystyle Y\ne 0}$ with coordinates${\displaystyle u=1/y}$ and${\displaystyle v=x/y}$. The equation of the curve becomes

${\displaystyle g(u,v)=F(v,1,u)=F(x/y,1,1/y)=y^{-d}F(x,y,1)=y^{-d}f(x,y).}$

Hence

${\displaystyle \mathrm{\partial }f/\mathrm{\partial }x=y^d\frac{\mathrm{\partial }g}{\mathrm{\partial }v}\frac{\mathrm{\partial }v}{\mathrm{\partial }x}=y^{d-1}\frac{\mathrm{\partial }g}{\mathrm{\partial }v}}$

so

${\displaystyle \omega =\frac{-dy}{\mathrm{\partial }f/\mathrm{\partial }x}=\frac{1}{u^2}\frac{du}{y^{d-1}\mathrm{\partial }g/\mathrm{\partial }v}=u^{d-3}\frac{du}{\mathrm{\partial }g/\mathrm{\partial }v}}$

with order of vanishing${\displaystyle {\nu }_p(\omega )=(d-3){\nu }_p(u)}$. Hence${\displaystyle \text{div}(\omega )=(d-3)[C\cap \{Z=0\}]}$ which agrees with the adjunction formula.

Thegenus-degree formula for plane curves can be deduced from the adjunction formula.^[2] LetC ⊂ P² be a smooth plane curve of degreed and genusg. LetH be the class of a hyperplane inP², that is, the class of a line. The canonical class ofP² is −3H. Consequently, the adjunction formula says that the restriction of(d − 3)H toC equals the canonical class ofC. This restriction is the same as the intersection product(d − 3)H ⋅dH restricted toC, and so the degree of the canonical class ofC isd(d−3). By theRiemann–Roch theorem,g − 1 = (d−3)d −g + 1, which implies the formula

${\displaystyle g=\frac{1}{2}(d-1)(d-2).}$

Similarly,^[3] ifC is a smooth curve on the quadric surfaceP¹×P¹ with bidegree (d₁,d₂) (meaningd₁,d₂ are its intersection degrees with a fiber of each projection toP¹), since the canonical class ofP¹×P¹ has bidegree (−2,−2), the adjunction formula shows that the canonical class ofC is the intersection product of divisors of bidegrees (d₁,d₂) and (d₁−2,d₂−2). The intersection form onP¹×P¹ is${\displaystyle ((d_1,d_2),(e_1,e_2))\mapsto d_1{e}_2+d_2{e}_1}$ by definition of the bidegree and by bilinearity, so applying Riemann–Roch gives${\displaystyle 2g-2=d_1(d_2-2)+d_2(d_1-2)}$ or

${\displaystyle g=(d_1-1)(d_2-1)=d_1{d}_2-d_1-d_2+1.}$

The genus of a curveC which is thecomplete intersection of two surfacesD andE inP³ can also be computed using the adjunction formula. Suppose thatd ande are the degrees ofD andE, respectively. Applying the adjunction formula toD shows that its canonical divisor is(d − 4)H|_D, which is the intersection product of(d − 4)H andD. Doing this again withE, which is possible becauseC is a complete intersection, shows that the canonical divisorC is the product(d +e − 4)H ⋅dH ⋅eH, that is, it has degreede(d +e − 4). By the Riemann–Roch theorem, this implies that the genus ofC is

${\displaystyle g=de(d+e-4)/2+1.}$

More generally, ifC is the complete intersection ofn − 1 hypersurfacesD₁, ...,D_{n − 1} of degreesd₁, ...,d_{n − 1} inPⁿ, then an inductive computation shows that the canonical class ofC is${\displaystyle (d_1+\cdots +d_{n-1}-n-1)d_1\cdots d_{n-1}{H}^{n-1}}$. The Riemann–Roch theorem implies that the genus of this curve is

${\displaystyle g=1+\frac{1}{2}(d_1+\cdots +d_{n-1}-n-1)d_1\cdots d_{n-1}.}$

LetS be a complex surface (in particular a 4-dimensional manifold) and let${\displaystyle C\to S}$ be a smooth (non-singular) connected complex curve. Then^[4]

${\displaystyle 2g(C)-2=[C{]}^2-c_1(S)[C]}$

where${\displaystyle g(C)}$ is the genus ofC,${\displaystyle [C{]}^2}$ denotes the self-intersections and${\displaystyle c_1(S)[C]}$ denotes theKronecker pairing.

• Logarithmic form
• Poincare residue
• Thom conjecture

• Intersection theory 2nd edition, William Fulton, Springer,ISBN 0-387-98549-2, Example 3.2.12.
• Principles of algebraic geometry, Griffiths and Harris, Wiley classics library,ISBN 0-471-05059-8 pp 146–147.
• Algebraic geometry,Robin Hartshorne, Springer GTM 52,ISBN 0-387-90244-9, Proposition II.8.20.

• Algebraic geometry

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