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

  $g = d_1 d_2 - d_1 - d_2 + 1$

• 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, 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

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

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.