Adjunction formula (algebraic geometry)
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In mathematics, the adjunction formula of algebraic geometry and complex manifold theory relates, for a hypersurface, its normal bundle, its canonical bundle, and the canonical bundle of the ambient variety or manifold.
Let H be a hypersurface in a complex manifold M. Then firstly the canonical bundle of H is
- KM|HL
where KM is the canonical bundle of M, and the notation denotes its restriction to H, and L is the line bundle associated to H as Cartier divisor, restricted to H. (The product of line bundles is here implied, i.e. tensor product of line bundles.). Secondly, L more explicitly can be taken as the normal bundle NH in M. This gives
- KH = KM|HNH,
the adjunction formula.
For example, the formula for projective n-space is
- K = O(−n − 1)
where O(1) is the Serre twist sheaf. This is compatible with the formula and the basic case n = 1, which states that
- d(1/z) = −z-2
on the Riemann sphere.
