Holomorphic sections of a line bundle Let π : L → X be a holomorphic line bunlde over a complex manifold X. A holomorphic section of L is a holomorphic map s : X → L such that π ◦s = Id. We denote by Γ(X, L), or H0(X, L), the set of all holomorphic sections of L over X. Let L be given by local data {Uα,gαβ}. −1 θα≃ L ⊃ π (Uα) −−→ Uα × C∋ (z,sα(z)) ↓ π ↑ s ր X ⊃ Uα∋ z −1 On each Uα, we find a unique holomorphic function sα ∈ O(Uα) so that s(z)= θα (z,sα(z)). −1 −1 Then on any Uα ∩ Uβ = ∅, we have θα (z,sα(z)) = θβ (z,sβ(z)). By the linearity, we have −1 −1 sα(z)θα (z, 1) = sβ(z)θβ (z, 1), (10) i.e., −1 (z,sβ(z)) = sα(z)θβ ◦ θα (z, 1), which implies sβ(z)= gβαsα(z), i.e., sα = gαβsβ, onUα ∩ Uβ. Conversely, any collection {sα}α∈I satisfying the above identity defines a holomorphic section −1 s ∈ Γ(X, L) by setting s := sαeα, where eα(z) := θα (z, 1). Every bundle has a trivial section, given by ζi = 0; the graph of this section is often called the zero section. If there are no other sections, we say that the bundle is said to have no sections. n n n Line bundles over CP Recall CP = {[z0 : z1 : ... : zn]}. Here CP , the set of all complex lines passing through the origin, is defined to a quotient space Cn+1 −{0}/ ∼ where n+1 (z0, z1, ..., zn), (z0, z1, ..., zn) ∈ C −{0} z , z , ..., z λ z , z , ..., z λ ∈ C −{ } are equivalent if and only if ( 0 1 n)= ( 0 1 n) with 0 . 1 (1) The holomorphic line bundle OCP1 (k) of degree k ∈ Z over CP is constructed 1 explicitly as follows. Let CP = U0 ∪ U1 be the standard open covering. Using the standard 0 0 1 1 atlas, take two “trivial” pieces U0 × C and U1 × C with coordinates (z ,ζ ) and (z ,ζ ) 21 i i where z is the base coordinate of Ui and ζ is the fibre coordinate over Ui - and “glue them together” over the set C∗ = CP1\{0, ∞} by the identification 1 ζ1 z = , ζ0 = . (11) 0 z1 (z1)k This line bundle is usually denoted OCP1 (k), or simply O(k) for brevity. The total space k k H of O(k) is the identification space; thus H consists of the disjoint union U0 U1 of two copies of C2 = C × C, with respective coordinates (z0,ζ0) and (z1,ζ1), which have been glued together along C∗ × C according to (11). The transition function from z0to z1 is 1 g (z0)= ; 01 (z0)k 1 1 the other is g10(z )= (z1)k . n CPn In general for higher dimensional case, let {U} = {Uα}α=0 be the standard atlas on . The line bundle OCPn (k) is defined by the transition functions z k g ([z]) = β . (12) αβ z α 0 1 0 1 z z k 1 In particular, when n = 1, g01([z]) = g01([z , z ]) = g01([ z1 : 1]) = z0 , i.e., g01(λ) = λk which is the same formula above. 1 2 (2) For each point [Z] in CP , let ℓ[Z] be the line in C represented by this point. Show 1 that the family {ℓ[Z] | [Z] ∈ CP } may be regarded as a holomorphic line bundle in the 1 2 following manner: Set L = {([Z],ζ) ∈ CP × C | ζ ∈ ℓ[Z]}, with projection to the first factor. This holomorphic line bundle is called the tautological bundle. Since there is a natural holomorphic map from L ∩ (CP1 × C) to C, by using the fact that a holomorphic map s : CP1 → C2 must be constant to show that L has no sections. In general for higher dimensional case, let us define a line bundle over CPn by assigning a one dimensional vector space at each point [ℓ] ∈ CPn: {([ℓ], v) ∈ CPn × Cn+1 | v ∈ ℓ}. Here ℓ is a complex line in Cn+1 passing through the origin. This defines a line bundle L, which is called the tautological line bundle over CPn. As above, such line bundle L has no section. 22 CPn n Let us find out the transition functions of the line bundle L. Let = ∪α=0Uα be the standard open covering. A canonical trivialization of L over Uα is given by −1 θα : π (Uα) ≃ Uα × C ([ℓ], z) → ([ℓ], zα) −1 C C where z = (z0, ..., zn). The transition maps gαβ([ℓ]) = θα ◦ θβ ([ℓ]) : → are given by zα w → · w, where [ℓ] = [z0 : ... : zn]. Then by (12), it concludes that k = −1 and hence the zβ tautological line bundle L = OCPn (−1). (3) When k < 0, the bundle O(k) has no sections. In fact, since the tautological line k bundle is O(−1), then O(k) = O(−1) ⊗| |. As in (2), any holomorphic section of O(−1) induces a holomorphic map from CPn → C so that such map must be constant and hence zero. Similarly, any holomorphic section of O(−1)⊗|k| alsi induces a holomorphic map from CPn to C so that it is a zero section. When k = 0, O(k) is trivial bundle so that any holomorphic section is a holomorphic map from CPn to C. By the maximum principle, the trivial bundle has zero section only. d When k > 0, O(k) always has holomorphic sections. Let Symn denote the space of homogeneous polynomials of degree d on Cn. Let f be any homogeneous polynomial of d d z0, ..., zn of degree d. Then f = (zα) fα = (zβ) fβ holds on Uα ∩ Uβ where fα is the non- d homogeneous polynomial obtained by dividing f with (zα) . In other words, fα = gαβfβ zβ d holds on U ∩ U , where g = is the transition function for OCPn (k). α β αβ zα k Therefore, the space of holomorphic sections of OCPn (k) is equal to Symn+1 for k > 0 and is trivial otherwise. In particular, the homogeneneous coordinate functions are sections of the line bundle O(−1). Here we use the notation O(1) := O(−1)∗, O(k) := O(1)⊗k = O(1) ⊗ ... ⊗ O(1) (k − times) and O(−k) := O(−1)⊗k. As a consequence, we have Proposition 3.2 ([Demailly09], corollary (15.6), p. 278) n + k if k ≥ 0, 0 CPn dim H ( , O(k)) = n 0, ifk< 0. 23 Theorem 3.3 (Grothendieck lemma, [H05], p.244) Every holomorphic vector bundle E on 1 CP is isomorphic to a holomorphic vector bundle of the form ⊕O(aj ). The ordered sequence a1 ≥ a2 ≥ ... ≥ ar is uniquely determined. Grothendieck lemma no longer holds for n > 1. A vector bunlde on CPn for n ≥ 2 is a direct sum of line bundle is equivalent to some geometric and topological condition [K90]. For more information on vector bundles over CPn, see the book [OSS80]. 24.