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 [math]\displaystyle{ \mathcal{I} }[/math]. The conormal exact sequence for i is

[math]\displaystyle{ 0 \to \mathcal{I}/\mathcal{I}^2 \to i^*\Omega_X \to \Omega_Y \to 0, }[/math]

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

[math]\displaystyle{ \omega_Y = i^*\omega_X \otimes \operatorname{det}(\mathcal{I}/\mathcal{I}^2)^\vee, }[/math]

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

[math]\displaystyle{ \omega_D = i^*(\omega_X \otimes \mathcal{O}(D)). }[/math]

In terms of canonical classes, this says that

[math]\displaystyle{ K_D = (K_X + D)|_D. }[/math]

Both of these two formulas are called the adjunction formula.

Examples

Degree d hypersurfaces

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

[math]\displaystyle{ \omega_X \cong i^*\omega_{\mathbb{P}^n}\otimes \mathcal{O}_X(d) }[/math]

which is isomorphic to [math]\displaystyle{ \mathcal{O}_X(-n{-}1{+}d) }[/math].

Complete intersections

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

[math]\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). }[/math]

This generalizes in the same fashion for all complete intersections.

Curves in a quadric surface

[math]\displaystyle{ \mathbb{P}^1\times\mathbb{P}^1 }[/math] embeds into [math]\displaystyle{ \mathbb{P}^3 }[/math] 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 [math]\displaystyle{ Y= \mathbb{P}^1\times\mathbb{P}^1 }[/math]. We can compute the cotangent bundle of [math]\displaystyle{ Y }[/math] using the direct sum of the cotangent bundles on each [math]\displaystyle{ \mathbb{P}^1 }[/math], so it is [math]\displaystyle{ \mathcal{O}(-2,0)\oplus\mathcal{O}(0,-2) }[/math]. Then, the canonical sheaf is given by [math]\displaystyle{ \mathcal{O}(-2,-2) }[/math], 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 [math]\displaystyle{ f \in \Gamma(\mathcal{O}(a,b)) }[/math], can be computed as

[math]\displaystyle{ \omega_C \,\cong\, \mathcal{O}(-2,-2)\otimes \mathcal{O}_C(a,b) \,\cong\, \mathcal{O}_C(a{-}2, b{-}2). }[/math]

Poincaré residue

The restriction map [math]\displaystyle{ \omega_X \otimes \mathcal{O}(D) \to \omega_D }[/math] 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 [math]\displaystyle{ \mathcal{O}(D) }[/math] 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

[math]\displaystyle{ \eta \otimes \frac{s}{f} \mapsto s\frac{\partial\eta}{\partial f}\bigg|_{f = 0}, }[/math]

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

[math]\displaystyle{ \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}. }[/math]

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

[math]\displaystyle{ \omega_D \otimes i^*\mathcal{O}(-D) = i^*\omega_X. }[/math]

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

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

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

[math]\displaystyle{ \omega := \frac{dx}{\partial f / \partial y} = \frac{-dy}{\partial f / \partial x}. }[/math]

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

Now look on the line [math]\displaystyle{ {Z = 0} }[/math]. Assume that [math]\displaystyle{ [1, 0, 0] \not\in C }[/math] so it suffices to look in the chart [math]\displaystyle{ Y \neq 0 }[/math] with coordinates [math]\displaystyle{ u = 1/y }[/math] and [math]\displaystyle{ v = x/y }[/math]. The equation of the curve becomes

[math]\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). }[/math]

Hence

[math]\displaystyle{ \partial f/\partial x = y^d \frac{\partial g}{\partial v} \frac{\partial v}{\partial x} = y^{d-1}\frac{\partial g}{\partial v} }[/math]

so

[math]\displaystyle{ \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} }[/math]

with order of vanishing [math]\displaystyle{ \nu_p(\omega) = (d-3)\nu_p(u) }[/math]. Hence [math]\displaystyle{ \text{div}(\omega) = (d-3)[C \cap \{Z = 0\}] }[/math] 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

[math]\displaystyle{ g = \tfrac12(d{-} 1)(d {-} 2). }[/math]

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 [math]\displaystyle{ ((d_1,d_2),(e_1,e_2))\mapsto d_1 e_2 + d_2 e_1 }[/math] by definition of the bidegree and by bilinearity, so applying Riemann–Roch gives [math]\displaystyle{ 2g-2 = d_1(d_2{-}2) + d_2(d_1{-}2) }[/math] or

[math]\displaystyle{ g = (d_1 {-} 1)(d_2 {-} 1) \,=\, d_1 d_2 - d_1 - d_2 + 1. }[/math]

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

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

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 [math]\displaystyle{ (d_1 + \cdots + d_{n-1} - n - 1)d_1 \cdots d_{n-1} H^{n-1} }[/math]. The Riemann–Roch theorem implies that the genus of this curve is

[math]\displaystyle{ g = 1 + \tfrac{1}{2}(d_1 + \cdots + d_{n-1} - n - 1)d_1 \cdots d_{n-1}. }[/math]

In low dimensional topology

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

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

where [math]\displaystyle{ g(C) }[/math] is the genus of C, [math]\displaystyle{ [C]^2 }[/math] denotes the self-intersections and [math]\displaystyle{ c_1(S)[C] }[/math] denotes the Kronecker pairing [math]\displaystyle{ \lt c_1(S),[C]\gt }[/math].

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.