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Vector bundles 2 Paul Hacking 2/21/10 1 Sections of line bundles on Pn Last time we defined the tautological line bundle L on projective space Pn and proved that there are no global sections of L. Let H := L∗ be the dual of L. We compute the global sections of H ⊗d . (Some notation: If L is a line bundle on a complex manifold X then L ⊗ L∗ is isomorphic to the trivial line bundle C × X. This follows from the analogous fact for vector spaces: if V is a 1-dimensional vector space then V ⊗ V ∗ = Hom(V, V ) is naturally isomorphic to C via idV 7→ 1. So sometimes we write L−1 for L∗ , etc. The line bundle H ⊗d on Pn is often denoted O(d).) The line bundle H ⊗d over Pn can be obtained as the quotient of the trivial line bundle (Cn+1 \ {0}) × C over Cn+1 \ {0} by the following action of C× : C× 3 λ : (x0 , . . . , xn , y) 7→ (λx0 , . . . , λxn , λd y). For example, in case d = −1 (the tautological line bundle L = H −1 ) the quotient map (Cn+1 \ {0}) × C → L ⊂ Pn × Cn+1 is given by (x0 , . . . , xn , y) 7→ ((x0 : · · · : xn ), y · (x0 , . . . , xn )). So a global section s of H ⊗d corresponds to a holomorphic function F : Cn+1 \ {0} → C such that F (λx0 , . . . , λxn ) = λd F (x0 , . . . , xn ) 1 (1) for λ ∈ C× . By Hartogs’ theorem [GH78, p. 7], F extends to a holomorphic function F : Cn+1 → C with the same property (1). We can expand F in a power series about 0 ∈ Cn+1 : X F = ai0 ,...,in xi00 · · · xinn . i0 ,...,in P The property (1) holds iff ai0 ,...,in is zero for ij 6= d, that is, iff F is a homogeneous polynomial of degree d. We deduce that the global sections Γ(Pn , H ⊗d ) of H ⊗d are the homogeneous polynomials of degree d in the homogeneous coordinates X0 , . . . , Xn on Pn . (In particular, there are no global sections if d < 0). 2 Line bundles and divisors There is a correspondence between line bundles and divisors on complex manifolds. To explain this we first consider the case of Riemann surfaces. Suppose X is a Riemann surface and P ∈ X a point. We can define a line bundle L = O(P ) associated to P as follows. Let U1 = X \ {P } and U2 = {|z| < δ} be a small disc in X centered at P . (That is, z is a local coordinate at P and δ > 0 is sufficiently small so that z defines an isomorphism of U2 onto the disc.) We glue L from the trivial line bundles over U1 and U2 via the transition function g12 = z : U1 ∩ U2 = U2 \ {P } → C× . Notice that the sections of L over an open set U ⊂ X can be identified with meromorphic functions f on U with at worst a simple pole at P (if P ∈ U ). Indeed f is given by holomorphic functions f1 , f2 on U ∩ U1 , U ∩ U2 , such that f2 = g12 f1 = zf1 on the overlap. So f := f1 is meromorphic on X, has at worst a simple polePat P , and is holomorphic elsewhere. Similarly, if D = ri=1 ni Pi , ni ∈ Z, is a formal sum of points of X (a divisor on X) we can define a line bundle O(D) by O(D) = O(P1 )⊗n1 ⊗ · · · ⊗ O(Pr )⊗nr . If f is a meromorphic function at a point P ∈ X, let z be a local coordinate at P and write f = aν z ν + aν+1 z ν+1 + · · · near P , where ν ∈ Z, aν 6= 0. Then we say the order of vanishing of f at P equals ν and write ordP (f ) = ν. Then for U ⊂ X the sections Γ(U, O(D)) 2 of O(D) over U are identified with meromorphic functions f on U such that the order of vanishing ordPi (f ) of f at Pi is at least −ni (if Pi ∈ U ) for each i. The general construction is as follows (see [GH78,P p. 129–136] for more details). Let X be a complex manifold and D = ni Yi a formal sum of codimension 1 complex subvarieties (a divisor ). For f a meromorphic function on X, the associated principal divisor is X (f ) := ordY (f ) · Y Y Then there is an open covering {Ui } of X such that for each i the restriction D|Ui is a principal divisor (fi ). We can define a line bundle O(D) associated to D by specifying transition functions gij = fj /fi : Uij → C× . For open U ⊂ X, we can identify Γ(U, O(D)) = {f meromorphic on U | (f ) + D|U ≥ 0}. In particular, a meromorphic function f on X determines a meromorphic section s of O(D), with divisor of zeroes minus poles (s) = (f ) + D. Note also that O(D0 ) ' O(D) iff D0 = D + (g) for some meromorphic function g, the isomorphism being given by multiplication by g. Conversely, if L is a holomorphic line bundle with meromorphic section s, then L ' O(D) where D = (s). 3 The degree of a line bundle on a Riemann surface Let X be a compactPRiemann P surface and L a line bundle on X. The degree of a divisor D = ni Pi is ni . The degree of L is the degree of (s) for any nonzero meromorphic section s of L, that is, the number of zeroes minus the number of poles of s, counted with multiplicities. Note that if t is another meromorphic section then t/s = f is a meromorphic function, and deg(f ) = 0 (because f defines a map f : X → P1 and the fibers over 0, ∞ have the same size when counted with multiplicities), so deg(t) = deg(s). 3 4 Tangent bundle Let X be a complex manifold. The tangent bundle T X → X is the holomorphic vector bundle with fibres ∂ ∂ TP X = ,..., ∂z1 ∂zn C where z1 , . . . , zn are local coordinates at P ∈ X. Sections of T X are holomorphic vector fields. Sections of ∧p T ∗ X are holomorphic p-forms. The line bundle ∧n T ∗ X is called the canonical line bundle and denoted KX . The transition functions of KX are the Jacobian determinants gij = det D(φj ◦ φ−1 i ). Example 4.1. The canonical line bundle of projective space. The form Ω = dx1 ∧ · · · ∧ dxn is a nowhere zero holomorphic n-form on (X0 6= 0) ⊂ Pn , where xi = Xi /X0 . On (X1 6= 0) we have coordinates y0 , y2 , . . . , yn where yi = Xi /X1 . Now x1 = y0−1 , xi = yi /y0 for i ≥ 2, so −(n+1) Ω = −y0−2 dy0 ∧ y0−1 dy2 ∧ · · · ∧ y0−1 dyn = −y0 dy0 ∧ dy2 ∧ · · · ∧ dyn . So Ω has a pole of order n + 1 along the hyperplane (X0 = 0) ⊂ Pn . It follows that KPn ' H −(n+1) . 5 Riemann–Hurwitz, redux Let f : X → Y be a nonconstant map between compact Riemann surfaces. Then X KX = f ∗ KY ⊗ O( (eP − 1)P ). (Note: If X → Y is a morphism of complex manifolds and V is a line bundle on Y then f ∗ V is the vector bundle on X with fibres (f ∗ V )P = Vf (P ) .) This follows from a local calculation: if z is a local coordinate at P ∈ X and w is a local coordinate at f (P ) ∈ Y such that f is locally given by w = z e , then f ∗ (dw) = d(f ∗ w) = d(z e ) = ez e−1 dz. To relate this to the earlier (topological) form of Riemann–Hurwitz we need to show deg KX = 2g(X) − 2 4 There are many ways to prove this, perhaps the most basic is to reduce to the usual Poincaré–Hopf theorem for Riemann surfaces, cf. [G89, p. 22]. We give an argument using the topological version of Riemann–Hurwitz already proved. There exists a nonconstant meromorphic function f on the Riemann surface X (this is a hard theorem). The function f defines a map f : X → P1 of some degree d. We know deg KP1 = −2, so using the two forms of Riemann Hurwitz for f we obtain X X deg KX = deg(f ∗ KP1 ) + (eP − 1) = −2d + (eP − 1) = 2g(X) − 2 as required. 6 The adjunction formula Let X be a complex manifold and i : Y ⊂ X a complex submanifold. We have an exact sequence of vector bundles on Y 0 → TY → TX |Y → NY /X → 0 (2) where TX |Y = i∗ TX is the restriction of TX to Y and NY /X is the normal bundle of Y ⊂ X (which is defined by the above exact sequence). Lemma 6.1. If 0 → V1 → V → V2 → 0 is an exact sequence of vector spaces of dimensions r1 , r, r2 then there is a natural isomorphism ∧r V ' ∧r1 V1 ⊗ ∧r2 V2 Proof. Let v1 , . . . , vr1 be a basis of V1 , v r1 +1 , . . . , v r a basis of V2 , and vr1 +1 , . . . , vr lifts to V . Then the isomorphism is given by v1 ∧ · · · ∧ vr 7→ v1 ∧ · · · ∧ vr1 ⊗ v r1 +1 ∧ · · · ∧ v r . That it is independent of choices follows from A B det = det A · det D. 0 D 5 By the lemma, the exact sequence of vector bundles (2) implies ∧n TX |Y ' ∧m TY ⊗ ∧n−m NY /X where n = dim X and m = dim Y . Rearranging gives the adjunction formula KY = KX |Y ⊗ ∧n−m NY /X . If Y ⊂ X has codimension 1, then the ideal of holomorphic functions on X vanishing on Y are (the sections of) the line bundle IY = O(−Y ). The normal bundle of Y equals (IY |Y )∗ = O(Y )|Y . To see this, consider the dual of the sequence (2) 0 → NY∗ /X → TX∗ |Y → TY∗ → 0. The kernel of the map TX∗ |Y → TY∗ is locally generated by df where f is a local equation of Y , and this defines an isomorphism IY |Y → NY∗ /X , f 7→ df. So in this case the adjunction formula becomes KY = (KX ⊗ O(Y ))|Y . Example 6.2. (Genus formula for plane curve, redux.) Let X = P2 and Y = (F = 0) ⊂ X a plane curve of degree d. Then KY = (KX ⊗ O(Y ))|Y = (H −3 ⊗ H d )|Y = H d−3 |Y . In particular deg KY = (d − 3)d and using deg KY = 2g(Y ) − 2 we find 1 g(Y ) = (d − 1)(d − 2). 2 References [G89] P. Griffiths, Introduction to algebraic curves, AMS 1989. [GH78] P. Griffiths, J. Harris, Principles of algebraic geometry, Wiley 1978. 6