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 Template:Nowrap by i and the ideal sheaf of Y in X by Failed to parse (⧼mw_math_mathml⧽: Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{I}} . The conormal exact sequence for i is

Failed to parse (⧼mw_math_mathml⧽: Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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

Failed to parse (⧼mw_math_mathml⧽: Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (⧼mw_math_mathml⧽: Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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

Template:See also 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, Template:Nowrap begin∂f/∂z_i ≠ 0Template:Nowrap end, 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 Template:Nowrap. 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. 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 Template:Nowrap to C equals the canonical class of C. This restriction is the same as the intersection product Template:Nowrap restricted to C, and so the degree of the canonical class of C is Template:Nowrap. By the Riemann–Roch theorem, Template:Nowrap beging − 1 = (d − 3)d − g + 1Template:Nowrap end, which implies the formula

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

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 (−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 Failed to parse (⧼mw_math_mathml⧽: Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (⧼mw_math_mathml⧽: Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2g-2 = d_1(d_2-2) + d_2(d_1-2)} or

Failed to parse (⧼mw_math_mathml⧽: Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (⧼mw_math_mathml⧽: Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Template:Math, which is the intersection product of Template:Nowrap 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 Template:Math, that is, it has degree Template:Math. By the Riemann–Roch theorem, this implies that the genus of C is

Failed to parse (⧼mw_math_mathml⧽: Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g = de(d + e - 4) / 2 + 1.}

More generally, if C is the complete intersection of Template:Math hypersurfaces Template:Math of degrees Template:Math 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

Failed to parse (⧼mw_math_mathml⧽: Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g = 1 + \frac{1}{2}(d_1 + \cdots + d_{n-1} - n - 1)d_1 \cdots d_{n-1}.}

See also

• Logarithmic form
• Poincare residue

References

• Intersection theory 2nd edition, William Fulton, Springer, Template:ISBN, Example 3.2.12.
• Principles of algebraic geometry, Griffiths and Harris, Wiley classics library, Template:ISBN pp 146–147.
• Algebraic geometry, Robin Hartshorne, Springer GTM 52, Template:ISBN, Proposition II.8.20.