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Home , Line bundle, Section (fiber bundle), Tautological bundle

ALGEBRAIC I, FALL 2016.

VECTOR BUNDLES.

1. The definition of a vector The definitions below can be applied to any geometric that contains vector spaces (topological spaces, smooth varieties, algebraic varieties, etc). We need to be able to say what is a variety and a morphism of varieties, we need on our varieties, and we need direct product. A over a variety X is a variety Y with a map p : Y → X, plus the structure of a on every p−1(x) for x ∈ X. We say that two vector bundles p1 : Y1 → X and p2 : Y2 → X are isomorphic, if there is an isomorphism Y1 → Y2 that commutes with p1 and p2, and induces a −1 −1 linear map on every fiber: p1 (x) → p2 (x). Define a trivial vector bundle as X ×V with the direct product projection for some vector space V . We say that a vector bundle is locally trivial, if for every x ∈ X there is an open neighborhood x ∈ U ⊂ X such that p : p−1(U) → U is isomorphic to a trivial vector bundle on U. On a connected component of X all fibers of a locally trivial vector bundle have the same dimension, which is called the rank of this vector bundle. From now on, we will assume that all our vector bundles are locally trivial. n+1 n Example 1. Let V = k and let X = P = P(V ). For x ∈ X denote by lx ⊂ V the line in V that corresponds to x. Then the set of pairs (x, v) where x ∈ X and n v ∈ lx forms a on X = P . This line bundle is called the tautological line n bundle on P . It is a subbundle of the trivial bundle X × V .

Example 2. For a smooth variety X, the set of pairs (x, v) with x ∈ X and v ∈ TxX forms a vector bundle that is called the of X and denoted by TX.

2. Transition functions Let p : Y → X be a vector bundle over X with fiber V of dimension n. Choose S ∼ −1 a covering X = Ui of X by open sets with trivializations φi : Ui × V −→ p (Ui). Then on each intersection Uij = Ui ∩ Uj we have a composition φ−1 φi −1 j Uij × V −→ p (Uij) −−→ Uij × V. Since we required trivializations to be linear maps on every fiber, these maps are also linear on every fiber. These compositions may be viewed either as n × n matrices whose entries are functions on Uij, or as functions gij : Uij → GLn(k). The functions gij that are obtained in this manner from a vector bundle satisfy two conditions: first, gii = Id, then, gjkgij = gik on Ui ∩ Uj ∩ Uk for all i, j, k. They are not defined uniquely by the vector bundle and the covering: we can choose ∼ −1 −1 different trivializations ψi : Ui×V −→ p (Ui) and get a different system fij = ψjψi . 1 2 VECTOR BUNDLES.

However since both φi and ψi are isomorphisms, we can assume ψi = φiAi, where Ai : Ui × V → Ui × V is another function with values in GLn(k), this time on Ui. Then −1 −1 −1 −1 fij = ψj ψi = Aj φj φiAi = Aj gijAi. The above conditions are enough to define a vector bundle: whenever we have a S covering X = Ui of X by open subsets, and functions gij : Ui ∩ Uj → GLn(k), such that gii = Id and gjkgij = gik on Ui ∩ Uj ∩ Uk for all i, j, k, there is a vector bundle that produces gij as transition functions. We can define this vector bundle S by factorizing the union (Ui × V ) by an obvious equivalence relation.

n 3. Tautological line bundle on P (See the definition and the notation for the in Example 1). n Let us find a set of transition functions for the tautological line bundle on P . Let n n+1 P = P(V ), where V = k with coordinates x0, . . . , xn. Let Ui be the affine chart xi 6= 0. Then over each Ui the tautological line bundle is trivial: we can write a map Ui × k → {(x, v)|x ∈ Ui, v ∈ lx} by sending each (x0 : ... : xn) × c to (x : ... : x ), c (x , . . . , x ). In other words, we choose a basis of k consisting of 0 n xi 0 n an element 1, and over each point (x0 : ... : xn) we send this basis vector to the point on the corresponding line where xi = 1. We can do that since for all points −1 in Ui we have xi 6= 0. The map φi then maps ((y0 : ... : yn), (x0, . . . , xn)) to (y0 : ... : yn) × xi. (here (x0 : ... : xn) = (y0 : ... : yn) when (x0 : ... : xn) is defined; the notation with y’s is only employed to treat the fact that (x0, . . . , xn) might be the origin). Using these trivializations, we find that g (c) = c xj , so g = xj ∈ GL (k) =∼ k. ij xi ij xi 1 n Definition 1. We denote by O(k) the line bundle on P that is trivial on each affine  k chart U and whose transition functions are g = xi . Thus, the tautological line i ij xj bundle is (isomorphic to) O(−1).

4. Operations on vector bundles For two vector bundles E and F over X, we can define a dual vector bundle E∨, a direct sum E ⊕ F , a E ⊗ F , and the bundle of homomorphisms Hom(E,F ) by performing the constructions for every fiber over every x ∈ X. An- other way to describe this is to find an open covering such that both E and F are trivial over each open subset, perform the operations over each set, and introduce transition functions using the transition functions for E and F .

5. Sections of vector bundles f Let E −→ X be a vector bundle. A regular section of this vector bundle is a map s X −→ E such that f ◦ s = IdX . In other words, for every x ∈ X we choose a point −1 s(x) ∈ f (x). Then for every open U ⊂ X such that E|U is trivial, once we choose a trivialization φ : U × V ' f −1(U), the section s is given by regular functions φ−1(s(x)). A rational section is a section described likewise by rational functions: we can view it either as a rational function X → E such that f(s(x)) = x for all x for which s is well-defined, or as a set of maps si : Ui → V with rational components and I, FALL 2016. 3 S the condition gijsi = sj for some open cover X = Ui with trivializations over each Ui and transition functions gij. We can also define sections of a vector bundle over each open subset U ⊂ X. The set of all regular sections of E over U is denoted by Γ(U, E). The set of global sections of E is also denoted simply by Γ(E) = Γ(X,E). We will often omit the word ”regular” when talking about regular sections.

Example 3. The line bundle OX , or simply O, over X is the trivial line bundle X × k. Its sections over U ⊂ X are the regular functions over U. They form the ring k[U].

Note that the set Γ(U, E) is in bijection with the set of morphisms OU → E|U : for every s ∈ Γ(U, E) we can map (x, λ) 7→ λs(x) ∈ f −1(x), where x ∈ U, λ ∈ k. Similarly, for every Φ : OU → E|U we can define s(x) = Φ(x, 1) and get a section of E. 2 Example 4. Let X = P with coordinates (x : y), and let Ux = {x 6= 0} and Uy = {y 6= 0}. The coordinate on Ux is y/x, and the coordinate on Uy is x/y. The −1 line bundle O(1) has transition functions gxy = x/y and gyx = y/x = (gxy) . Then a section s is a pair of functions sx on Ux and sy on Uy such that gxysx = sy. One example of such a section will be sx = 1, sy = x/y. 6. The definition of a Let X be a . A presheaf F of abelian groups on X is the data of an abelian group F(U) for every open subset U ⊂ X, and a morphism F(U) → F(V ) whenever V ⊂ U, called the restriction morphism. Elements of the group F(U) are called sections of F on U. A presheaf F is a sheaf, if it satisfies the following property:

• For any collection (Ui, fi ∈ F(Ui)) such that for any i, j the sections fi and S fj coincide when restricted to Ui ∩ Uj, there is a unique section f ∈ F( Ui)

such that fi = f|Ui . One can also consider a presheaf, or a sheaf, of sets, rings, vector spaces, modules (over a ring, or a sheaf of rings or algebras) etc. We can also denote the set F(U) by Γ(U, F), similar to the sections of a vector bundle. Examples:

(1) The constant sheaf kX has k for sections over every open subset, with identity restriction maps. One can use any group, or object, instead of k. (2) The structure sheaf OX is a sheaf of functions on X. For an algebraic X, we take the sheaf of regular functions: for open U ⊂ X, set OX (U) = k[U]. It is a sheaf of algebras. (3) Sections of any vector bundle E over X form a over OX : for every open U ⊂ X the set Γ(U, E) is a module over OX (U). This sheaf is locally free: every x ∈ X has a neighborhood U such that Γ(U, E) is a free OX (U)-module (a neighborhood over which E is trivial will work). The reverse is true: a locally free sheaf of modules is a sheaf of sections for some vector bundle. (4) The skyscraper sheaf of a point p ∈ X is the sheaf F such that F(U) = k when x ∈ U and F(U) = 0 when x∈ / U. 4 VECTOR BUNDLES.

n 7. Line bundles on P For vector bundles over topological spaces with continuous transition functions, it is relatively easy to prove that any vector bundle over a contractible space such as n R is trivial. In the algebraic setting however this question stood for twenty years, being known as the Serre’s Conjecture, before finally being proven by Quillen and by Suslin in the 70’s: Theorem 1. Any vector bundle over the affine space kn is trivial. n n Let X be P with coordinates (x0 : ... : xn), covered by affine charts Ui ' k . Let L be a line bundle over X that is trivial over each affine chart. The transition functions gij must be rational functions that have neither zeroes nor poles over Ui ∩ Uj, which is the set where xi 6= 0 and xj 6= 0. Therefore, gij must be quotient of two monomials in xi, xj of the same degree, i.e. a power of xi/xj. Then since gijgjk = gik, the degree must be the same for all transition functions. Recall that we n k defined O(k) on P as the line bundle with transition functions gij = (xi/xj) for some k ∈ Z. n Proposition 1. Every line bundle on P is isomorphic to O(k) for some k. To describe all (global) regular sections of O(k), observe that if a section is given   by a polynomial P x1 ,..., xn in the chart U where x 6= 0, then it is given by x0 x0 0 0  k   x0 P x1 ,..., xn in the chart U . In order for that to be a regular function of xi x0 x0 i x0 ,..., xn we need P to be a homogeneous polynomial of degree k. This means that xi xi for k < 0 there are no global regular sections, and for k ≥ 0 the space of global regular sections is isomorphic to the space of all homogeneous polynomials of degree k. In particular, for k = 0 the space of all regular sections of the structure sheaf O is one-dimensional and consists only of constant functions. For each homogeneous P polynomial P (x0, . . . , xn) of degree k the corresponding section of O(k) is k on Ui. xi