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Adjunction formula - Wikipedia

  • ️Tue Feb 11 2020

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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

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Formula for a smooth subvariety

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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

{\displaystyle 0\to {\mathcal {I}}/{\mathcal {I}}^{2}\to i^{*}\Omega _{X}\to \Omega _{Y}\to 0,}

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

{\displaystyle \omega _{Y}=i^{*}\omega _{X}\otimes \operatorname {det} ({\mathcal {I}}/{\mathcal {I}}^{2})^{\vee },}

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

The particular case of a smooth divisor

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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

{\displaystyle \omega _{D}=i^{*}(\omega _{X}\otimes {\mathcal {O}}(D)).}

In terms of canonical classes, this says that

{\displaystyle K_{D}=(K_{X}+D)|_{D}.}

Both of these two formulas are called the adjunction formula.

Degree d hypersurfaces

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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

{\displaystyle \omega _{X}\cong i^{*}\omega _{\mathbb {P} ^{n}}\otimes {\mathcal {O}}_{X}(d)}

which is isomorphic to {\displaystyle {\mathcal {O}}_{X}(-n{-}1{+}d)}.

Complete intersections

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For a smooth complete intersection {\displaystyle i:X\hookrightarrow \mathbb {P} _{S}^{n}} of degrees {\displaystyle (d_{1},d_{2})}, the conormal bundle {\displaystyle {\mathcal {I}}/{\mathcal {I}}^{2}} is isomorphic to {\displaystyle {\mathcal {O}}(-d_{1})\oplus {\mathcal {O}}(-d_{2})}, so the determinant bundle is {\displaystyle {\mathcal {O}}(-d_{1}{-}d_{2})} and its dual is {\displaystyle {\mathcal {O}}(d_{1}{+}d_{2})}, showing

{\displaystyle \omega _{X}\,\cong \,{\mathcal {O}}_{X}(-n{-}1)\otimes {\mathcal {O}}_{X}(d_{1}{+}d_{2})\,\cong \,{\mathcal {O}}_{X}(-n{-}1{+}d_{1}{+}d_{2}).}

This generalizes in the same fashion for all complete intersections.

Curves in a quadric surface

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{\displaystyle \mathbb {P} ^{1}\times \mathbb {P} ^{1}} embeds into {\displaystyle \mathbb {P} ^{3}} as a quadric surface given by the vanishing locus of a quadratic polynomial coming from a non-singular symmetric matrix.[1] We can then restrict our attention to curves on {\displaystyle Y=\mathbb {P} ^{1}\times \mathbb {P} ^{1}}. We can compute the cotangent bundle of {\displaystyle Y} using the direct sum of the cotangent bundles on each {\displaystyle \mathbb {P} ^{1}}, so it is {\displaystyle {\mathcal {O}}(-2,0)\oplus {\mathcal {O}}(0,-2)}. Then, the canonical sheaf is given by {\displaystyle {\mathcal {O}}(-2,-2)}, 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 {\displaystyle f\in \Gamma ({\mathcal {O}}(a,b))}, can be computed as

{\displaystyle \omega _{C}\,\cong \,{\mathcal {O}}(-2,-2)\otimes {\mathcal {O}}_{C}(a,b)\,\cong \,{\mathcal {O}}_{C}(a{-}2,b{-}2).}

The restriction map {\displaystyle \omega _{X}\otimes {\mathcal {O}}(D)\to \omega _{D}} 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 {\displaystyle {\mathcal {O}}(D)} 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

{\displaystyle \eta \otimes {\frac {s}{f}}\mapsto s{\frac {\partial \eta }{\partial f}}{\bigg |}_{f=0},}

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

{\displaystyle {\frac {g(z)\,dz_{1}\wedge \dotsb \wedge dz_{n}}{f(z)}}\mapsto (-1)^{i-1}{\frac {g(z)\,dz_{1}\wedge \dotsb \wedge {\widehat {dz_{i}}}\wedge \dotsb \wedge dz_{n}}{\partial f/\partial z_{i}}}{\bigg |}_{f=0}.}

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

{\displaystyle \omega _{D}\otimes i^{*}{\mathcal {O}}(-D)=i^{*}\omega _{X}.}

On an open set U as before, a section of {\displaystyle i^{*}{\mathcal {O}}(-D)} 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 {\displaystyle i^{*}{\mathcal {O}}(-D)}.

Inversion of adjunction

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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

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Let {\displaystyle C\subset \mathbf {P} ^{2}} be a smooth plane curve cut out by a degree {\displaystyle d} homogeneous polynomial {\displaystyle F(X,Y,Z)}. We claim that the canonical divisor is {\displaystyle K=(d-3)[C\cap H]} where {\displaystyle H} is the hyperplane divisor.

First work in the affine chart {\displaystyle Z\neq 0}. The equation becomes {\displaystyle f(x,y)=F(x,y,1)=0} where {\displaystyle x=X/Z} and {\displaystyle y=Y/Z}. We will explicitly compute the divisor of the differential

{\displaystyle \omega :={\frac {dx}{\partial f/\partial y}}={\frac {-dy}{\partial f/\partial x}}.}

At any point {\displaystyle (x_{0},y_{0})} either {\displaystyle \partial f/\partial y\neq 0} so {\displaystyle x-x_{0}} is a local parameter or {\displaystyle \partial f/\partial x\neq 0} so {\displaystyle y-y_{0}} is a local parameter. In both cases the order of vanishing of {\displaystyle \omega } at the point is zero. Thus all contributions to the divisor {\displaystyle {\text{div}}(\omega )} are at the line at infinity, {\displaystyle Z=0}.

Now look on the line {\displaystyle {Z=0}}. Assume that {\displaystyle [1,0,0]\not \in C} so it suffices to look in the chart {\displaystyle Y\neq 0} with coordinates {\displaystyle u=1/y} and {\displaystyle v=x/y}. The equation of the curve becomes

{\displaystyle g(u,v)=F(v,1,u)=F(x/y,1,1/y)=y^{-d}F(x,y,1)=y^{-d}f(x,y).}

Hence

{\displaystyle \partial f/\partial x=y^{d}{\frac {\partial g}{\partial v}}{\frac {\partial v}{\partial x}}=y^{d-1}{\frac {\partial g}{\partial v}}}

so

{\displaystyle \omega ={\frac {-dy}{\partial f/\partial x}}={\frac {1}{u^{2}}}{\frac {du}{y^{d-1}\partial g/\partial v}}=u^{d-3}{\frac {du}{\partial g/\partial v}}}

with order of vanishing {\displaystyle \nu _{p}(\omega )=(d-3)\nu _{p}(u)}. Hence {\displaystyle {\text{div}}(\omega )=(d-3)[C\cap \{Z=0\}]} which agrees with the adjunction formula.

Applications to curves

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The genus-degree formula for plane curves can be deduced from the adjunction formula.[2] 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 (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

{\displaystyle g={\tfrac {1}{2}}(d{-}1)(d{-}2).}

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). The intersection form on P1×P1 is {\displaystyle ((d_{1},d_{2}),(e_{1},e_{2}))\mapsto d_{1}e_{2}+d_{2}e_{1}} by definition of the bidegree and by bilinearity, so applying Riemann–Roch gives {\displaystyle 2g-2=d_{1}(d_{2}{-}2)+d_{2}(d_{1}{-}2)} or

{\displaystyle g=(d_{1}{-}1)(d_{2}{-}1)\,=\,d_{1}d_{2}-d_{1}-d_{2}+1.}

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 (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

{\displaystyle g=de(d+e-4)/2+1.}

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 {\displaystyle (d_{1}+\cdots +d_{n-1}-n-1)d_{1}\cdots d_{n-1}H^{n-1}}. The Riemann–Roch theorem implies that the genus of this curve is

{\displaystyle g=1+{\tfrac {1}{2}}(d_{1}+\cdots +d_{n-1}-n-1)d_{1}\cdots d_{n-1}.}

In low dimensional topology

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Let S be a complex surface (in particular a 4-dimensional manifold) and let {\displaystyle C\to S} be a smooth (non-singular) connected complex curve. Then[4]

{\displaystyle 2g(C)-2=[C]^{2}-c_{1}(S)[C]}

where {\displaystyle g(C)} is the genus of C, {\displaystyle [C]^{2}} denotes the self-intersections and {\displaystyle c_{1}(S)[C]} denotes the Kronecker pairing {\displaystyle <c_{1}(S),[C]>}.

  1. ^ Zhang, Ziyu. "10. Algebraic Surfaces" (PDF). Archived from the original (PDF) on 2020-02-11.
  2. ^ Hartshorne, chapter V, example 1.5.1
  3. ^ Hartshorne, chapter V, example 1.5.2
  4. ^ Gompf, Stipsicz, Theorem 1.4.17