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

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

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

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

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

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

In terms of canonical classes, this says that

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

Both of these two formulas are called the adjunction formula.

Examples

Degree d hypersurfaces

Given a smooth degree ${\displaystyle d}$ hypersurface ${\displaystyle i:X\hookrightarrow \mathbb {P} _{S}^{n}}$ we can compute its canonical and anti-canonical bundles using the adjunction formula. This reads as

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

which is isomorphic to ${\displaystyle {\mathcal {O}}_{X}(-n{-}1{+}d)}$.

Complete intersections

For a smooth complete intersection ${\displaystyle i:X\hookrightarrow \mathbb {P} _{S}^{n}}$ of degrees ${\displaystyle (d_{1},d_{2})}$, the conormal bundle ${\displaystyle {\mathcal {I}}/{\mathcal {I}}^{2}}$ is isomorphic to ${\displaystyle {\mathcal {O}}(-d_{1})\oplus {\mathcal {O}}(-d_{2})}$, so the determinant bundle is ${\displaystyle {\mathcal {O}}(-d_{1}{-}d_{2})}$ and its dual is ${\displaystyle {\mathcal {O}}(d_{1}{+}d_{2})}$, showing

${\displaystyle \omega _{X}\,\cong \,{\mathcal {O}}_{X}(-n{-}1)\otimes {\mathcal {O}}_{X}(d_{1}{+}d_{2})\,\cong \,{\mathcal {O}}_{X}(-n{-}1{+}d_{1}{+}d_{2}).}$

This generalizes in the same fashion for all complete intersections.

Curves in a quadric surface

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

${\displaystyle \omega _{C}\,\cong \,{\mathcal {O}}(-2,-2)\otimes {\mathcal {O}}_{C}(a,b)\,\cong \,{\mathcal {O}}_{C}(a{-}2,b{-}2).}$

Poincaré residue

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

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

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

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

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

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

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

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

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

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

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

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

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

Hence

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

so

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

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

Applications to curves

The genus-degree formula for plane curves can be deduced from the adjunction formula.^[2] 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

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

Similarly,^[3] 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 ${\displaystyle ((d_{1},d_{2}),(e_{1},e_{2}))\mapsto d_{1}e_{2}+d_{2}e_{1}}$ by definition of the bidegree and by bilinearity, so applying Riemann–Roch gives ${\displaystyle 2g-2=d_{1}(d_{2}{-}2)+d_{2}(d_{1}{-}2)}$ or

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

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

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

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

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

In low dimensional topology

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

${\displaystyle 2g(C)-2=[C]^{2}-c_{1}(S)[C]}$

where ${\displaystyle g(C)}$ is the genus of C, ${\displaystyle [C]^{2}}$ denotes the self-intersections and ${\displaystyle c_{1}(S)[C]}$ denotes the Kronecker pairing ${\displaystyle <c_{1}(S),[C]>}$.

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.