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  by i and the ideal sheaf of Y in X by . The conormal exact sequence for i is

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

where  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  on X, and the ideal sheaf of D corresponds to its dual . The conormal bundle  is , which, combined with the formula above, gives

In terms of canonical classes, this says that

Both of these two formulas are called the adjunction formula. Examples

Degree d hypersurfaces 
Given a smooth degree  hypersurface  we can compute its canonical and anti-canonical bundles using the adjunction formula. This reads aswhich is isomorphic to . Complete intersections 
For a smooth complete intersection  of degrees , the conormal bundle  is isomorphic to , so the determinant bundle is  and its dual is , showingThis generalizes in the same fashion for all complete intersections. Curves in a quadric surface 
 embeds into  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 . We can compute the cotangent bundle of  using the direct sum of the cotangent bundles on each , so it is . Then, the canonical sheaf is given by , 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 , can be computed as

Poincaré residue 

The restriction map  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  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

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

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

On an open set U as before, a section of  is the product of a holomorphic function s with the form . The Poincaré residue is the map that takes the wedge product of a section of ωD and a section of . 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  be a smooth plane curve cut out by a degree  homogeneous polynomial . We claim that the canonical divisor is  where  is the hyperplane divisor. First work in the affine chart . The equation becomes  where  and . We will explicitly compute the divisor of the differential

At any point  either  so  is a local parameter or
  so  is a local parameter. In both cases the order of vanishing of  at the point is zero. Thus all contributions to the divisor  are at the line at infinity, . Now look on the line . Assume that  so it suffices to look in the chart  with coordinates  and . The equation of the curve becomes 

 

Hence 

so 

with order of vanishing . Hence  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  to C equals the canonical class of C. This restriction is the same as the intersection product  restricted to C, and so the degree of the canonical class of C is . By the Riemann–Roch theorem, g − 1 = (d−3)d − g + 1, which implies the formula

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  by definition of the bidegree and by bilinearity, so applying Riemann–Roch gives  or

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 , which is the intersection product of  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 , that is, it has degree . By the Riemann–Roch theorem, this implies that the genus of C is

More generally, if C is the complete intersection of  hypersurfaces  of degrees  in Pn, then an inductive computation shows that the canonical class of C is . The Riemann–Roch theorem implies that the genus of this curve is

In low dimensional topology 

Let S be a complex surface (in particular a 4-dimensional manifold) and let  be a smooth (non-singular) connected complex curve. Then 

where  is the genus of C,  denotes the self-intersections and  denotes the Kronecker pairing . See also 
 Logarithmic form
 Poincare residue
 Thom conjecture

References

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