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.
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.
on X, and the ideal sheaf of D corresponds to its dual
we can compute its canonical and anti-canonical bundles using the adjunction formula.
This generalizes in the same fashion for all complete intersections.
as a quadric surface given by the vanishing locus of a quadratic polynomial coming from a non-singular symmetric matrix.
using the direct sum of the cotangent bundles on each
, 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
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
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 df/f.
The Poincaré residue is the map that takes the wedge product of a section of ωD and a section of
The adjunction formula is false when the conormal exact sequence is not a short exact sequence.
They are an important tool in modern birational geometry.
be a smooth plane curve cut out by a degree
We will explicitly compute the divisor of the differential At any point
The equation of the curve becomes Hence so with order of vanishing
Consequently, the adjunction formula says that the restriction of (d − 3)H to C equals the canonical class of C. This restriction is the same as the intersection product (d − 3)H ⋅ dH restricted to C, and so the degree of the canonical class of C is d(d−3).
By the Riemann–Roch theorem, g − 1 = (d−3)d − g + 1, which implies the formula Similarly,[3] 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).
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.
Applying the adjunction formula to D shows that its canonical divisor is (d − 4)H|D, which is the intersection product of (d − 4)H 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 (d + e − 4)H ⋅ dH ⋅ eH, that is, it has degree de(d + e − 4).
By the Riemann–Roch theorem, this implies that the genus of C is More generally, if C is the complete intersection of n − 1 hypersurfaces D1, ..., Dn − 1 of degrees d1, ..., dn − 1 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 Let S be a complex surface (in particular a 4-dimensional manifold) and let
be a smooth (non-singular) connected complex curve.