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

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

where ${\displaystyle \vee }$ denotes the dual of a line bundle.

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

In terms of canonical classes, this says that

Both of these two formulas are called the adjunction formula.

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