Adjunction formula

In mathematics, especially in algebraic geometry and the theory of complex manifolds, the adjunction formula relates the canonical bundle of a variety and a hypersurface inside that variety. It is often used to deduce facts about varieties embedded in well-behaved spaces such as projective space or to prove theorems by induction.

Adjunction for smooth varieties

Formula for a smooth subvariety

Let X be a smooth algebraic variety or smooth complex manifold and Y be a smooth subvariety of X. Denote the inclusion map Y → X by i and the ideal sheaf of Y in X by ${ℐ}$. The conormal exact sequence for i is

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

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

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

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

The particular case of a smooth divisor

Suppose that D is a smooth divisor on X. Its normal bundle extends to a line bundle ${𝒪}(D)$ on X, and the ideal sheaf of D corresponds to its dual ${𝒪}(-D)$. The conormal bundle ${ℐ}/{{ℐ}}^2$ is $i^{*}{𝒪}(-D)$, which, combined with the formula above, gives

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

In terms of canonical classes, this says that

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

Both of these two formulas are called the adjunction formula.

Examples

Degree d hypersurfaces

Given a smooth degree

$d$

hypersurface

$i:X\hookrightarrow {\mathbb{\mathbb{P}}}_S^n$

we can compute its canonical and anti-canonical bundles using the adjunction formula. This reads as

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

which is isomorphic to

${{𝒪}}_X(-n-1+d)$

.

Complete intersections

For a smooth complete intersection

$i:X\hookrightarrow {\mathbb{\mathbb{P}}}_S^n$

of degrees

$(d_1,d_2)$

, the conormal bundle

${ℐ}/{{ℐ}}^2$

is isomorphic to

${𝒪}(-d_1)\oplus {𝒪}(-d_2)$

, so the determinant bundle is

${𝒪}(-d_1-d_2)$

and its dual is

${𝒪}(d_1+d_2)$

, showing

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

This generalizes in the same fashion for all complete intersections.

Curves in a quadric surface

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

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

Poincaré residue

The restriction map ${\omega }_X\otimes {𝒪}(D)\to {\omega }_D$ is called the Poincaré residue. Suppose that X is a complex manifold. Then on sections, the Poincaré residue can be expressed as follows. Fix an open set U on which D is given by the vanishing of a function f. Any section over U of ${𝒪}(D)$ can be written as s/f, where s is a holomorphic function on U. Let η be a section over U of ω_X. The Poincaré residue is the map

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

that is, it is formed by applying the vector field ∂/∂f to the volume form η, then multiplying by the holomorphic function s. If U admits local coordinates z₁, ..., z_n such that for some i, ∂f/∂z_i ≠ 0, then this can also be expressed as

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

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

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

On an open set U as before, a section of $i^{*}{𝒪}(-D)$ is the product of a holomorphic function s with the form df/f. The Poincaré residue is the map that takes the wedge product of a section of ω_D and a section of $i^{*}{𝒪}(-D)$.

Inversion of adjunction

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 of X with the singularities of D. Theorems of this type are called inversion of adjunction. They are an important tool in modern birational geometry.

The Canonical Divisor of a Plane Curve

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

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

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

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

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

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

Hence

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

so

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

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

Applications to curves

The genus-degree formula for plane curves can be deduced from the adjunction formula. Let C ⊂ P² be a smooth plane curve of degree d and genus g. Let H be the class of a hyperplane in P², that is, the class of a line. The canonical class of P² is −3H. Consequently, the adjunction formula says that the restriction of (d − 3)H to C equals the canonical class of C. This restriction is the same as the intersection product (d − 3)H ⋅ dH restricted to C, and so the degree of the canonical class of C is d(d−3). By the Riemann–Roch theorem, g − 1 = (d−3)d − g + 1, which implies the formula

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

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

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

The genus of a curve C which is the complete intersection of two surfaces D and E in P³ can also be computed using the adjunction formula. Suppose that d and e are the degrees of D and E, respectively. Applying the adjunction formula to D shows that its canonical divisor is (d − 4)H|_D, which is the intersection product of (d − 4)H and D. Doing this again with E, which is possible because C is a complete intersection, shows that the canonical divisor C is the product (d + e − 4)H ⋅ dH ⋅ eH, that is, it has degree de(d + e − 4). By the Riemann–Roch theorem, this implies that the genus of C is

$g=de(d+e-4)/2+1.$

More generally, if C is the complete intersection of n − 1 hypersurfaces D₁, ..., D_{n − 1} of degrees d₁, ..., d_{n − 1} in Pⁿ, then an inductive computation shows that the canonical class of C is $(d_1+\cdots +d_{n-1}-n-1)d_1\cdots d_{n-1}{H}^{n-1}$. The Riemann–Roch theorem implies that the genus of this curve is

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

In low dimensional topology

Let S be a complex surface (in particular a 4-dimensional manifold) and let $C\to S$ be a smooth (non-singular) connected complex curve. Then

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

where $g(C)$ is the genus of C, $[C{]}^2$ denotes the self-intersections and $c_1(S)[C]$ denotes the Kronecker pairing $<c_1(S),[C]>$.

See also

• Logarithmic form
• Poincare residue
• Thom conjecture

References

• 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.