DIVISORS AND LINE BUNDLES


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1 DIVISORS AND LINE BUNDLES TONY PERKINS 1. Cartier divisors An analytic hypersurface of M is a subset V M such that for each point x V there exists an open set U x M containing x and a holomorphic function f x defined on U x such that V U x is the zeroset of f x. Such an f x is called a local defining function for V near x. The quotient of any two local defining functions around x is a nonvanishing holomorphic function around x. An analytic hypersurface V is called irreducible if it can not be written as the union of two smaller analytic hypersurfaces. Every analytic hypersurface is a finite union of its irreducible components. Definition 1. A divisor D on M is a locally finite formal linear combination of irreducible analytic hypersurfaces on M. D = a i V i Remark 1. Locally finite here means that for any p M, there exists a neighborhood of p meeting only finite number of the V i s appearing in D; of course, if M is compact, this just means that the sum is finite. The set of divisors in M is naturally an additive group denoted Div(M). Definition 2. A divisor is effective if a i 0 for all i; we write D 0 for D effective. An analytic hypersurface V will usually be identified with V i where the V i s are just the irreducible components of V. Let V M be an irreducible analytic hypersurface, p V any point, and f a local defining function for V near p. For any holomorphic function g defined near p, we define the order ord V,p (g) of g along V at p to be the largest integer a such that in the local ring O M,p g = f a h. Well this is according to [1], but they unfortunately don t tell me what h is! I think the correct definition of order is the largest positive integer a such that g/fx a is holomorphic around x. Evidently relatively prime elements of O M,p stay relatively prime in nearby local rings, thusly for a holomorphic function g, the function ord V,p (g) is independent of p. Thus we can define the order of g along V, ord V, to be the order of g along V at any point p V. Note that for g, h any holomorphic functions, V any irreducible hypersurface, Date: April 8, ord V (gh) = ord V (g) + ord V (h).
2 Now let f be a meromorphic function on M, written locally as f = g/h with g, h holomorphic and relatively prime. For V an irreducible hypersurface, we define ord V (f) = ord V (g) ord V (h). We usually say that f has a zero of order a along V if ord V (f) = a > 0, and that f has a pole of order a along V if ord V (f) = a < 0. We define the divisor (f) of the meromorphic function f by (f) = V ord V (f) V. If f is written locally as g/h, we take the divisor of zeros (f) 0 of f to be (f) 0 = V ord V (g) V and the divisor of poles (f) to be (f) = V ord V (h) V. Clearly these are welldefined as long as we require g and h to be relatively prime, and (f) = (f) 0 (f). 2. Weil divisors and meromorphic functions Divisors can also be described in sheaftheoretic terms, as follows: Recall that a meromorphic function on an open set U M is given locally as the quotient of two holomorphic functions, i.e. for some covering {U i } of U, f Ui = g i /h i, where g i, h i are relatively prime in O(U i ) and g i h j = g j h i in O(U i U j ). A meromorphic function is not, strictly speaking, a function even if we consider a value: at points where g i = h i = 0, it is not defined. The sheaf of meromorphic functions is denoted M and the multiplicative sheaf of meromorphic functions not identically zero is denoted M. Also recall that O denotes the multiplicative sheaf of not identically zero holomorphic functions. Clearly O M. So we have the following short exact sequence 0 O M M /O 0. The quotient sheaf M /O is denoted called the sheaf of divisors on M. A global section of M /O is just a divisor on M. On the one hand, a global section {f} of M /O is given by an open cover {U α } of M and a meromorphic functions f α 0 in U α with for any V M, then and we can associate to {f} the divisor f α O (U α U β ); ord V (f α ) = ord V ( ), D = V ord V (f α ) V, 2
3 where for each V we choose α such that V U α. On the other hand, given D = V i α i V i, we can find an open cover {U α } of M such that in each U α, every V i appearing in D has a local defining function g iα O(U α ). We can then set f α = i g a i iα M (U α ) to obtain a global section of M /O. The f α s are called local defining functions for D. It follows immediately from the definitions that the identification H 0 (M, M /O ) = Div(M) is in fact a homomorphism. Given a holomorphic map π : M N of complex manifolds, we define the map π : Div(N) Div(M) by associating to every divisor D = ({U α }, {f}) on N the pullback divisor π D = ({π 1 U α }, {π f α }) on M; this is welldefined as long as π(m) D. Note that for a divisor on N given by an analytic hypersurface V N, π V on M lies over V but need not coincide with the analytic hypersurface π (V ) M  multiplicities may occur. We want to make one more remark before going on to consider bundles. On a Riemann surface M, any point is an irreducible analytic hypersurface, and so clearly Div(M) is always large. This is, in a sense, misleading; a complex manifold M of dimension greater than one need not have any nonzero divisors on it at all. If, however M is embedded in projective space P N, the intersections of M with hyperplanes in P N generate a large number of divisors. In fact among all compact complex manifolds those which are embeddable in a projective space can be characterized by having sufficiently many divisors, in a sense that we shall make more precise in later sections. 3. Line bundles All line bundles discussed in this section are taken to be holomorphic. Recall that for any holomorphic line bundle L π M on the complex manifold M, we can find an open cover {U α } of M and trivializations φ α : L Uα U α C of L Uα = π 1 (U α ). We derive the tranistion function g αβ : U α U β C for L relative to the trivializations {φ α } by g αβ (z) = (φ α φ 1 β ) L z C. The functions g αβ are cleary holomorphic, nonvanishing and satisfy { gαβ g (1) βα = 1, g αβ g βγ g γα = 1; 3
4 conversely, given a collection of functions {g αβ O (U α U β )} satisfying these identities, we can construct a line bundle L with transition functions {g αβ } by taking the union of U α C over all α identifying {z} C in U α C and U β C via multiplication by g αβ (z). Now, given L as above, for any collection of nonzero holomorphic functions f α O (U α ) we can define alternate trivializations of L over {U α } by φ α = f α φ α ; transition functions g αβ for L relative to {φ α} will then be given by (2) g αβ = f α g αβ One the other hand, any other trivializations of L over {U α } can be obtained in this way, and so we see that collections {g αβ } and {g αβ } of transition functions define the same line bundle if and only if there exist functions f α O (U α ) satisfying (2). The description of the line bundles by transition functions lends itself well to a sheaftheoretic interpretation. First, the transition functions {g αβ O (U α U β )} for a line bundle L M represent a Čech 1cochain on M with coefficients in O ; the relation (1) simply asserts that {g αβ } and {g αβ } define the same line bundle if and only if their difference is {g αβ g 1 αβ } is a Čech coboundary; consequently the set of line bundles on M is just H 1 (M, O ). We can give the set of line bundles on M the structure of a group, multiplication being given by a tensor product and inverses by dual bundles. If L is given by data {g αβ }, L by {g αβ }, we have seen that L L {gαβg αβ}, L {g 1 αβ } so the group structure on the set of line bundles on M is the same as the group structure on H 1 (M, O ). The group H 1 (M, O ) is then called the Picard group of M, denoted Pic(M). 4. Divisors and line bundles We now describe the basic correspondence between divisors and line bundles. Essentially what we will see is that do any divisor D we can associate a line bundle [D] on M. Let D be a divisor on M, with local defining functions f α M (U α ) over some open cover {U α } of M. Then the functions g αβ = f α are holomorphic and nonzero in U α U β, and in U α U β U γ we have g αβ g βγ g γα = f α fβ f γ fγ f α = 1 The line bundle given by the transition function {g αβ = f α / } is called the associated line bundle of D, and written [D]. We check that it is welldefined: if {f α} are alternate 4
5 local data for D, then h α = f α /f α O (U α ), and for each α, β. g αβ = f α f β = g αβ hα h β Example 1. The tautological line bundle on CP n is defined as the complex line bundle π : L CP n whose fibre L [z] over some point [z] CP n is the complex line z C n+1. Let H denote the dual of L, called the hyperplane line bundle. In other words, the fibre of H over some point [z] CP n is the set of Clinear maps zc C. Let H denote the hyperplane {z 0 = 0} in CP n and consider the usual open covering U i = {z i 0} of CP n. Then 1 is a local defining function for H on U 0 and z 0 /z i are local defining functions on U i. Then line bundle [H] has thus transition functions g αβ = z β /z α, which are exactly the transition functions of the hyperplane line bundle above. Proposition 1. The correspondence [ ] has these immediate properties: First, if D and D are two divisors given by local data {f α } and {f α}, resp. then D + D is given by {f α f α}; if follows that so the map [D + D ] = [D] [D ] [ ]: Div(M) Pic(M) is a homomorphism. Second, if D = (f) for some meromorphic function f on M, we may take as local data for D over any cover {U α } the functions f α = f Uα ; then f α / = 1 and so [D] is trivial. Conversely, if D is given by local data {f α } and the line bundle [D] is trivial, then there exist function h α O (U α ) such that f α = g αβ = h α h β ; f = f α h 1 α = h 1 β is then a global meromorphic function on M with divisor D. The the line [D] associated to a divisor D on M is trivial if and only if D is the divisor of a meromorphic function. We say that two divisors D, D on M are linearly equivalent and write D D if D = D + (f) for some f M (M), or equivalently if [D] = [D ]. Remark 2. In about 30 more pages of [1], in particular on page 161, we will get an interesting characterization of the kernel associated to the homomorphism above. Essentially L(D), to be defined shortly, forms and equivalence class, such that Div(M) = Pic(M) Also, note that [ ] is functorial: that is, if f : M N is a holomorphic map of complex manifolds, it is easy to check that for any D Div(N), π ([D]) = [π (D)], 5
6 All these assertions are implicit in the following cohomological interpretation of the correspondence [ ]. The exact sheaf sequence 0 O M j M /O 0 on M gives us, in part, the exact sequence the exact sequence H 0 (M, M ) j H 0 (M, M /O ) δ H 1 (M, O ) of cohomology groups. The reader may easily verify that under the natural identifications Div(M) = H 0 (M, M /O ) and Pic(M) = H 1 (M, O ) for any meromorphic function f on M, j f = (f), and for any divisor D on M, δd = [D]. Indeed, we will generally violate the previous multiplicative notation and write L + L for the tensor production of two line bundles or ml for the m th tensor power L m of L. 5. Sections of line bundles Now we wish to discuss holomorphic and meromorphic sections of line bundles. Let L M be a holomorphic line bundle, with trivializations φ α : L Uα U α C over an open cover {U α } of M and transition functions {g αβ } relative to {φ α }. As we have seen, the trivializations φ α induce isomorphism φ α : O(L)(U α ) O(U α ) we see via the correspondence s O(L)(U) {s α = φ α(s) O(U U α ) that a section of L over U M is given exactly by a collection of functions s α O(U U α ) satisfying s α = g αβ s β in U U α U β. In the same way a meromorphic section s of L over U  defined to be a section of the sheaf O(L) O M  is given by a collection of meromorphic functions s α inm(u U α ) satisfying s α = g αβ s β in U U α U β. Note that the quotient of two meromorphic sections s, s 0 of L is a welldefined meromorphic function. If s is a global meromorphic section of L, s α /s β O (U α U β ), and so for any irreducible hypersurface V M, ord V (s α ) = ord V (s β ). Thus we can define the order of s along V by ord V (s) = ord V (s α ) for any α such that U α V ; we take the divisor (s) of the meromorphic section s to be given by (s) = ord V (s) V. V With this convention s is holomorphic if and only if s is effective. Now if D Div(M) is given by local data f α M(U α ), then the functions f α clearly give a meromorphic section s f of D with (s f ) = D. 6
7 conversely, if L is given by trivializations φ α with transition functions g αβ and s is any global meromorphic section of L, we see that s α s β = g αβ, i.e., L = [(s)]. Thus if D is any divisor such that [D] = L, there exists a meromorphic section s of L with (s) = D, and for any meromorphic section s of L, L = [(s)]. In particular, we see that L is the line bundle associated to some divisor D on M if and only if it has a global meromorphic section not identically zero; it is the line bundle of an effective divisor if and only if it has a nontrivial global holomorphic section. Definition 3. L:et L(D) denote the space of meromorphic functions f on M such that D + (f) 0, i.e. that are holomorphic on M \ V i with ord Vi (f) a i. In other words, we only allow f L(D) if f has poles no worse that D. We denote by D Div(M) the set of all effective divisors linearly equivalent to D; if L = [D], we write L for D. Given this notation, we can also view this correspondence as follows: Proposition 2. Proof. For some divisor L(D) = H 0 (M, O([D])) D = a i V i on M, set s 0 be a global meromorphic section [D] with (s 0 ) = D. holomorphic sections s of [D], the quotient Then for any global is a meromorphic function on M with f s = s s 0 (f s ) = (s) (s 0 ) D, i.e. f s L(D) and (s) = D + (f s ) D. On the other hand, for any f L(D) the section s = f s 0 of [D] is holomorphic. Thus multiplication by s 0 gives an identification L(D) s 0 H 0 (M, O([D])). Proposition 3. Now suppose that M is compact. D = P(L(D)) = P(H 0 (M, O([D]))) Proof. On the the hand, if M is compact, for every D D. There exists f L(D) such that D = D + (f), and conversely any two such functions f, f differ by a nonzero constant, i.e. note that (s) = D + (f s ) D. Thus we have the additional correspondence Definition 4. In general, the family of effective divisors on M corresponding to a linear subspace of P(H 0 (M, O(L))) for some L M is called a linear system of divisors. A linear system is called complete if it is of the form D, i.e. if it contains every effective divisor linearly equivalent to any of its members. When we speak of dimension of a linear system, 7
8 we will refer to the dimension of the projective space parametrizing it; thus we write dim D for the dimension of the complete linear system associated to a divisor D, we have dim D = h 0 (M, O(D)) 1. A linear system of dimension 1 is called a pencil, of dimension 2 a net, and of dimension 3 a web. We will mention here two special properties of linear systems. the first is elementary; if E = {D λ } λ P n is a linear system, then for any λ 0,..., λ n linearly independent in P n, D λ0 D λn = λ P n D λ. The common intersection of the divisors in a linear system is called the base locus of the system; in particular, a divisor F in the base locus that is, such that D λ F 0 for all λ is called a fixed component of E. The second property is more remarkable; like the first, it is peculiar to linear systems and is not the case for general families of divisors, even general families of linearly equivalent divisors. Theorem 1 (Bertini s Theorem). The generic element of a linear system is smooth away from the base locus of the system. Proof. If the generic element of a linear system is singular away from the base locus, one could find a pencil with the same property. Thus we may assume that we have a pencil. Suppose that {D λ ) λ P 1 is a pencil, given in local coordinates on M by D λ = (f(z 1,..., z n ) + λg(z 1, z n ) = 0), and suppose that p λ is a singular point of the divisor D λ (λ 0, ), but not in the base locus of the pencil. We have f(p λ ) + λg(p λ ) = 0, and f (p λ ) + λ g (p λ ) = 0, i = 1,..., n. z i z i As p λ B, at least one of f and g does not vanish at p λ and so both don t vanish there. Thus λ = f(p λ )/g(p λ ), and so f (p λ ) f(p λ) g (p λ ) = 0, i = 1,..., n. z i g(p λ ) z i Therefore ( ) f = 0. z i g Let V be the singular locus of the divisors belonging to the pencil. Then V is defined by the equations above, and so V is analytic. Now f/g is constant on any component of V, away from the base locus, by the calculations above. Hence {λ: V D λ } B <. References [1] Griffiths, P., Harris, J., Principles of Algebraic Geometry,
Proof. The map. G n i. where d is the degree of D.
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