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THE ADJUNCTION FORMULA FOR LINE BUNDLES C. BRUSCHEK Abstract. We give a short report on the adjunction formula for line bundles as it can be found in Griffiths-Harris “Principles of Algebraic Geometry”. Theorem 1. Let X be a smooth projective variety and Y ⊆ X a smooth subvariety of codimension 1 in X. Then KY = (KX ⊗ Y ) |Y (1) where KZ denotes the canonical class of a smooth variety Z. 1. Preliminaries First recall the notion of vector bundles: Definition 1.1. A smooth complex vector bundle of rank k over M is a smooth manifold E together with a surjective smooth map π : E → M of smooth manifolds such that: (1) for all x ∈ M the fiber over x, i.e., Ex := π −1 (x), is a k-dimensional complex vector space. (2) for all x ∈ M there is a neighbourhood U ⊆ M of x and a diffeomorphism ϕU : π −1 (U ) → U × Ck such that proj1 ◦ ϕU = π|U . Vector bundles of rank 1 are called line bundles. Example. A cylinder over S 1 is vector bundle of rank 1, thus a line bundle. Since it is of the form M × Ck , we call it a trivial bundle. The Moebius-stripe is a vector bundle, but not a trivial bundle. The restriction ϕU |Ex : Ex → Ck is an isomorphism of C-vector spaces. For any k two trivializations ϕU , ϕV of M at x, the composition ϕU ◦ ϕ−1 V : (U ∩ V ) × C → k (U ∩ V ) × C is of the form (x, z) 7→ (x, gU V (x) · z), gU V (x) ∈ Glk (C), i.e., it is C-linear in the second component. We call gU V the transition function of E at x relative to the trivializations ϕU and ϕV . Let M = ∪α Uα be a covering of M and denote by ϕα a trivialization on Uα for all α. This gives transition functions (gαβ )α,β . These fulfill the following properties: (∗) for all x ∈ Uα ∩ Uβ : gαβ (x) = gβα (x)−1 for all x ∈ Uα ∩ Uβ ∩ Uγ : gαβ (x)gβγ (x)gγα (x) = 1. Construction. Let M = ∪α Uα be a covering of M and let gαβ : Uα ∩ Uβ → Glk (C) be given such that the conditions (∗) are fulfilled, then this data defines (up to isomorphism) a unique vector bundle: Set 1 2 C. BRUSCHEK ! a E := Uα × C k /∼ , α where (x, z)β ∼ (x, gαβ (x)z)α for all x ∈ Uα ∩ Uβ and ϕα : [Uα × Ck ] → Uα × Ck , [(x, z)α ] 7→ (x, z). Here are some important constructions with fibre bundles. They are all given by fibre-wise operations: • Let E, F be vector bundles over M with trivializations ϕα , ψα (resp. transition functions gαβ ∈ Glk (C), hαβ ∈ Gll (C)). Then the tensor product of E and F is defined as E ⊗ F := a Ex ⊗C Fx x∈M with trivializations ϕα ⊗ ψα and transition functions gαβ ⊗ hαβ ∈ Gl(Ck ⊗ Cl ). Vr ` Vr Vr • The wedge-product of E: E := x∈M Ex with { ϕα } as trivializaVr tions and transition functions gαβ . ∗ • The dual bundle of E: E := > −1 tions gαβ . ` x∈M Ex∗ with {ϕ−1 α } and transition func- Definition 1.2. Let E be a vector bundle over M ; F ⊆ E is called a subbundle of E if (i) F ⊆ E is a submanifold and (ii) for all x ∈ M : Fx := F ∩ Ex ⊆ Ex is a C-subvectorspace. In this setting the quotient bundle E/F is defined by (E/F )x = Ex /Fx (transition functions are a bit more involved). 2. Divisors and Line Bundles The set of line bundles over a smooth manifold X has the structure of a group; multiplication is given by “⊗” and inverses by the dual bundle construction “(−)∗ ”, the unit element is the trivial bundle. Let L and L0 be two line bundles on X. Then −1 0 L ⊗ L0 is given by {gαβ · gαβ } and L∗ is given by {gαβ }. Indeed the inverse is given as stated above: −1 L ⊗ L∗ given by {gαβ gαβ } = {idαβ }. Thus L ⊗ L∗ is the trivial bundle. There is a fundamental correspondence between divisors on and linebundles over X. Let D be a divisor on X with local defining functions fα ∈ M∗ (Uα ), X = ∪α Uα . Then fα ∈ O∗ (Uα ∩ Uβ ), fβ are holomorphic and non-zero on Uα ∩ Uβ . Moreover, it’s easy to see gαβ := i.e., the gαβ that gαβ gβγ gγα = 1 and THE ADJUNCTION FORMULA FOR LINE BUNDLES 3 gαβ gβα = 1. In other words the gαβ define transition functions of a line bundle. We denote the line bundle associated to D by L(D). This constructions gives the following correspondencies: • L(D + D0 ) = L(D) ⊗ L(D0 ). • If D = (f ) for f ∈ M(X), then L(D) is trivial. • as an extension of the last statement: L(D) is trivial iff D = (f ) for some meromorphic function f on X. • D ∼ D0 , then L(D) = L(D0 ). Note that by the first statement we get a group homomorphism L(−) : DivX → PicX . 3. Proof of the Adjunction Formula Let X be a smooth projective variety over C, Y ⊆ X a smooth submanifold. The Normal bundle NY on Y is defined as NY := TX |Y /TY , i.e., for x ∈ Y we have NY,x = TX,x /TY,x . Thus we get an exact sequence of vector bundles 0 → NY → TX |Y → TY → 0 and further 0 → TY∗ → TX |∗Y → NY∗ → 0. On each fibre this gives us an exact sequence of C-vectorspaces of dimension n − 1, n and 1. Lemma 3.1. Let 0 → U → V → W → 0 be an exact sequence of C-vectorspaces of dimension a, a + b, b, then a+b ^ V = a ^ U⊗ b ^ W. In our situation the last lemma gives (2) n ^ TX |∗Y = n−1 ^ TY∗ ⊗ ^ NY∗ . By definition the left-hand side of equation (2) is KX |Y and the factor on the far right is just NY∗ , which can be multiplied to the left by means of (−)∗ . This gives (3) KY = KX |Y ⊗ NY . It remains to prove that NY = L(Y )|Y . For this note that the conormal bundle NY∗ is the subbundle of TX |∗Y of cotangent vectors, which vanish on TY . Since Y ⊆ X is of codimension 1, it is locally given by functions fα ∈ O(Uα ), X = ∪Xα . 4 C. BRUSCHEK Thus L(Y ) is given by transition functions gαβ = ffαβ . Moreover fα |Uα ∩Y = 0 and dfα ∈ Γ(Y ∩ Uα , NY∗ ). But Y is smooth, so dfα = 6 0 on Y ∩ Uα . Moreover on Uα ∩ Uβ ∩ Y we have (4) (5) (6) Therefore the dfα ∈ NY∗ ⊗ L(Y ), i.e., dfα = d(gαβ fβ ) = dgαβ · fβ + gαβ · dfβ = gαβ dfβ . Γ(Uα , NY∗ ) fit together to give a global non-zero section of NY∗ ⊗ L(Y )|Y = 1 the trivial bundle. Note that in our notation equations (4) show that the transition −1 functions for NY∗ are given by gβα that is by gαβ . Hence, NY∗ = L(−Y )|Y . Together with equation (3) this gives KY = (KX ⊗ L(Y )) |Y . Appendix A. hallo