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. 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 . 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. 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. Given a smooth degree hypersurface we can compute its canonical and anti-canonical bundles using the adjunction formula. This reads aswhich is isomorphic to . 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. 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.