Adjunction formula (algebraic geometry)

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.

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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 YX 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 %2B 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 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.

Applications to curves

See Also

Logarithmic form

References

  1. ^ Hartshorne, chapter V, example 1.5.1
  2. ^ Hartshorne, chapter V, example 1.5.2