Survey

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Survey

Document related concepts

Fundamental theorem of algebra wikipedia, lookup

Linear algebra wikipedia, lookup

Euclidean vector wikipedia, lookup

Cartesian tensor wikipedia, lookup

Tensor operator wikipedia, lookup

Matrix calculus wikipedia, lookup

Vector space wikipedia, lookup

Laplace–Runge–Lenz vector wikipedia, lookup

Algebraic K-theory wikipedia, lookup

Covariance and contravariance of vectors wikipedia, lookup

Bra–ket notation wikipedia, lookup

Transcript

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