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.

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 &rarr; 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.

Degree d hypersurfaces
Given a smooth degree $$d$$ hypersurface $$i: X \hookrightarrow \mathbb{P}^n_S$$ we can compute its canonical and anti-canonical bundles using the adjunction formula. This reads as"$\omega_X \cong i^*\omega_{\mathbb{P}^n}\otimes \mathcal{O}_X(d)$"which is isomorphic to $$\mathcal{O}_X(-n{-}1{+}d)$$.

Complete intersections
For a smooth complete intersection $$i: X \hookrightarrow \mathbb{P}^n_S$$ of degrees $$(d_1, d_2)$$, the conormal bundle $$\mathcal{I}/\mathcal{I}^2$$ is isomorphic to $$\mathcal{O}(-d_1)\oplus \mathcal{O}(-d_2)$$, so the determinant bundle is $$\mathcal{O}(-d_1{-}d_2)$$ and its dual is $$\mathcal{O}(d_1{+}d_2)$$, showing $$\omega_X \,\cong\, \mathcal{O}_X(-n{-}1)\otimes \mathcal{O}_X(d_1{+}d_2) \,\cong\, \mathcal{O}_X(-n{-}1 {+} d_1 {+} d_2).$$ This generalizes in the same fashion for all complete intersections.

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

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

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 ωX. 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, ..., zn such that for some i, ∂f/∂zi ≠ 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 ωD 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.

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 $$Z \neq 0$$. The equation becomes $$f(x, y) = F(x, y, 1) = 0$$ where $$x = X/Z$$ 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 $$\partial f / \partial y \neq 0$$ so $$x - x_0$$ is a local parameter or $$\partial f / \partial x \neq 0$$ 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 $$[1, 0, 0] \not\in C$$ so it suffices to look in the chart $$Y \neq 0$$ 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 &minus;3H. Consequently, the adjunction formula says that the restriction of (d &minus; 3)H to C equals the canonical class of C. This restriction is the same as the intersection product (d &minus; 3)H &sdot; dH restricted to C, and so the degree of the canonical class of C is d(d&minus;3). By the Riemann–Roch theorem, g &minus; 1 = (d&minus;3)d &minus; g + 1, which implies the formula
 * $$g = \tfrac12(d{-} 1)(d {-} 2).$$

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 (&minus;2,&minus;2), the adjunction formula shows that the canonical class of C is the intersection product of divisors of bidegrees (d1,d2) and (d1&minus;2,d2&minus;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 $(d &minus; 4)H|_{D}$, which is the intersection product of (d &minus; 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 $(d + e &minus; 4)H &sdot; dH &sdot; eH$, that is, it has degree $de(d + e &minus; 4)$. 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 $n &minus; 1$ hypersurfaces $D_{1}, ..., D_{n &minus; 1}$ of degrees $d_{1}, ..., d_{n &minus; 1}$ 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
 * $$g = 1 + \tfrac{1}{2}(d_1 + \cdots + d_{n-1} - n - 1)d_1 \cdots d_{n-1}.$$

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 $$$$.