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 ${\mathcal {I}}$. The conormal exact sequence for i is

$0\to {\mathcal {I}}/{\mathcal {I}}^{2}\to i^{*}\Omega _{X}\to \Omega _{Y}\to 0,$

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

$\omega _{Y}=i^{*}\omega _{X}\otimes \operatorname {det}({\mathcal {I}}/{\mathcal {I}}^{2})^{\vee },$

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 ${\mathcal {O}}(D)$ on X, and the ideal sheaf of D corresponds to its dual ${\mathcal {O}}(-D)$. The conormal bundle ${\mathcal {I}}/{\mathcal {I}}^{2}$ is $i^{*}{\mathcal {O}}(-D)$, which, combined with the formula above, gives

$\omega _{D}=i^{*}(\omega _{X}\otimes {\mathcal {O}}(D))$

In terms of canonical classes, this says that

$K_{D}=(K_{X}+D)|_{D}$

Both of these two formulas are called the adjunction formula.

Poincaré residue

The restriction map $\omega _{X}\otimes {\mathcal {O}}(D)\to \omega _{D}$ 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 ${\mathcal {O}}(D)$ 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

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

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

${\frac {g(z)\,dz_{1}\wedge \dotsb \wedge dz_{n}}{f(z)}}\mapsto (-1)^{{i-1}}{\frac {g(z)\,dz_{1}\wedge \dotsb \wedge \widehat {dz_{i}}\wedge \dotsb \wedge dz_{n}}{\partial f/\partial z_{i}}}{\bigg |}_{{f=0}}.$

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

$\omega _{D}\otimes i^{*}{\mathcal {O}}(-D)=i^{*}\omega _{X}.$

On an open set U as before, a section of $i^{*}{\mathcal {O}}(-D)$ 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 $i^{*}{\mathcal {O}}(-D)$.

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.

Applications to curves

The genus-degree formula for plane curves can be deduced from the adjunction formula.^[1] Let C ⊂ P² be a smooth plane curve of degree d and genus g. Let H be the class of a hypersurface 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

$g=(d-1)(d-2)/2.$

Similarly,^[2] 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

${\displaystyle g=(d_{1}-1)(d_{2}-1)=d_{1}d_{2}-d_{1}-d_{2}+1.}$

This gives a simple proof of the existence of curves of any genus as the graph of a function of degree ${\displaystyle g+1}$.

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 ${\displaystyle (d_{1}+\cdots +d_{n-1}-n-1)d_{1}\cdots d_{n-1}H^{n-1}}$. The Riemann–Roch theorem implies that the genus of this curve is

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

See also

Logarithmic form

References

1. Hartshorne, chapter V, example 1.5.1
2. Hartshorne, chapter V, example 1.5.2

• Phillip Griffiths; Joe Harris (1994). Principles of Algebraic Geometry. Wiley-Interscience. ISBN 0-471-05059-8.
• 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.