Line Bundles and Vector Bundles

This post was machine-translated from the Korean original by Marvin (via Kimi). It may contain errors or awkward phrasing — the Korean original is the source of truth.

In the previous post we defined divisors on a variety \(X\) and saw that their linear equivalence classes form \(\Cl(X)\). However, not every divisor arises as the zero or pole locus of some rational function. For instance, since \(\Cl(\mathbb{P}^n) \cong \mathbb{Z}\) (§Divisors, ⁋Example 11), a general divisor \(dH\) on \(\mathbb{P}^n\) is the zero set of a homogeneous polynomial only when \(d \ge 0\).

To overcome this restriction we introduce line bundles. A line bundle \(\mathcal{L}\) is a geometric object that assigns a one-dimensional vector space to each point \(p \in X\), and a section \(s\) of \(\mathcal{L}\) naturally defines a divisor \(\divisor(s)\). From this viewpoint, for any divisor \(D\) we can construct a line bundle \(\mathcal{O}_X(D)\) whose sections correspond to divisors greater than or equal to \(D\). In other words, line bundles allow us to treat divisors independently of the constraint that they be zeros or poles of a function.

Definition of Line Bundle

Line bundles, and more generally vector bundles, which we will define later in this post, are defined in the same way as in differential geometry and other fields. ([Manifolds] §Tangent and Cotangent Bundles, ⁋Definition 1 or [Algebraic Topology] §Stiefel-Whitney Characteristic Classes, ⁋Definition 2, etc.)

Definition 1 A line bundle \(\mathcal{L}\) on a variety \(X\) consists of the following data.

1. A projection \(\pi: \mathcal{L} \rightarrow X\).

2. An open cover \(\{U_i\}\) of \(X\) and, for each \(i\), a local trivialization \(\phi_i: \pi^{-1}(U_i) \overset{\sim}{\longrightarrow} U_i \times \mathbb{A}^1\). The maps

   \[\phi_j \circ \phi_i^{-1}: (U_i \cap U_j) \times \mathbb{A}^1 \rightarrow (U_i \cap U_j) \times \mathbb{A}^1\]

   induced by these have the form \((p, t) \mapsto (p, g_{ij}(p)t)\) for suitable transition functions \(g_{ij} \in \mathcal{O}_X(U_i \cap U_j)^\times\).

A sheaf \(\mathcal{F}\) is an \(\mathcal{O}_X\)-module if for each open set \(U\) the group \(\mathcal{F}(U)\) is an \(\mathcal{O}_X(U)\)-module and this multiplication is compatible with restriction maps; that is, we can multiply a section by a locally defined regular function. A morphism of \(\mathcal{O}_X\)-modules is a sheaf morphism that preserves this multiplication on each \(U\), i.e. an \(\mathcal{O}_X(U)\)-module homomorphism.

A morphism \(\varphi \colon \mathcal{L} \rightarrow \mathcal{M}\) between two line bundles \(\mathcal{L}, \mathcal{M} \rightarrow X\) is given by a \(\mathbb{K}\)-linear map \(\varphi_p \colon \mathcal{L}_p \rightarrow \mathcal{M}_p\) between fibers at each point \(p \in X\), which can be expressed over a suitable open cover \(\{U_k\}\) as an \(\mathcal{O}_X(U_k)\)-module homomorphism

\[\varphi_k \colon \mathcal{O}_{U_k} \rightarrow \mathcal{O}_{U_k}\]

satisfying

\[g^{\mathcal{L}}_{kl} \circ \varphi_l = \varphi_k \circ g^{\mathcal{M}}_{kl}.\]

Since the fiber of a line bundle is one-dimensional, each \(\varphi_k\) is given by multiplication by some \(h_k \in \mathcal{O}_X(U_k)\), i.e. \(s \mapsto h_k s\). When \(\varphi\) is bijective on each fiber, we call it an isomorphism and write \(\mathcal{L} \cong \mathcal{M}\). Because the fiber is one-dimensional, this is equivalent to giving a nonzero scalar at each point, i.e. choosing \(h_k \in \mathcal{O}_X(U_k)^\ast\) compatibly.

The following proposition is verified directly from the definition of transition functions.

Proposition 2 (Cocycle condition) Transition functions \(\{g_{ij}\}\) satisfy the following cocycle condition.

1. \(g_{ii} = 1\) for all \(i\).
2. \(g_{ij} = g_{ji}^{-1}\) for all \(i, j\).
3. \(g_{ij} g_{jk} = g_{ik}\) on \(U_i \cap U_j \cap U_k\) for all \(i, j, k\).

Example 3 The trivial line bundle \(X \times \mathbb{A}^1\) is the line bundle all of whose transition functions are \(g_{ij} = 1\). This is the simplest line bundle, with no twist.

Thus the second condition in Definition 1 means that the line bundle \(\mathcal{L}\) is isomorphic to the trivial line bundle when restricted to a suitable open set \(U \subseteq X\).

Proposition 2 (Cocycle condition) is the familiar gluing condition, and by this condition a line bundle can be thought of as a kind of sheaf. ([Topology] §Sheaves, ⁋Definition 1) Concretely, given a line bundle \(\mathcal{L}\), we define its sheaf of sections by

\[U\mapsto \mathcal{O}_X(\mathcal{L})(U)=\{s: U \rightarrow \mathcal{L} \mid \pi \circ s = \id_U\}.\]

That is, \(\mathcal{O}_X(\mathcal{L})\) is the sheaf of sections of the surjection \(\pi\). ([Topology] §Sheaves, ⁋Example 9)

Then by the local trivialization \(\phi_i: \pi^{-1}(U_i) \rightarrow U_i \times \mathbb{A}^1\) we have \(\mathcal{O}_X(\mathcal{L})\vert_{U_i} \cong \mathcal{O}_{U_i}\). This allows us to think of these sections locally on \(U_i\) as ordinary \(\mathbb{K}\)-valued functions.

This means the following.

Definition 4 An \(\mathcal{O}_X\)-module \(\mathcal{F}\) is called invertible if for every point \(p \in X\) there is a neighborhood \(U\) such that \(\mathcal{F}\vert_U \cong \mathcal{O}_U\) as \(\mathcal{O}_U\)-modules.

What we showed above is that the sheaf of sections of a line bundle is invertible. The next proposition shows that the converse also holds.

Proposition 5 The sheaf of sections \(\mathcal{O}_X(\mathcal{L})\) of a line bundle \(\mathcal{L}\) is an invertible sheaf. Conversely, every invertible sheaf comes from a line bundle that is unique up to isomorphism.

Proof

For an invertible sheaf \(\mathcal{F}\), one can define transition functions from the local isomorphisms \(\mathcal{F}\vert_{U_i} \cong \mathcal{O}_{U_i}\), and from these reconstruct the line bundle.

By this proposition we know that line bundles and invertible sheaves are the same concept. For this reason, when denoting a line bundle we use \(\mathcal{L}\) rather than the upright letter \(L\) used to denote a space.

Operations on Line Bundles

In the world of differential geometry it is natural to construct new bundles by performing fiberwise the operations of linear algebra. The same is true in algebraic geometry; since we are currently looking at the case of line bundles, what we need to examine are \(\otimes\) and \(\Hom\), and in particular the dual \((-)^\vee\).

Proposition 6 The tensor product \(\mathcal{L} \otimes \mathcal{M}\) of two line bundles \(\mathcal{L}, \mathcal{M}\) is also a line bundle. Its transition functions are \(\{g_{ij} h_{ij}\}\), where \(\{g_{ij}\}, \{h_{ij}\}\) are the transition functions of \(\mathcal{L}, \mathcal{M}\) respectively.

Proof

The fiber of the tensor product is \(\mathcal{L}_p \otimes_{\mathbb{K}} \mathcal{M}_p\), which is again one-dimensional since it is the tensor product of two one-dimensional vector spaces. The transition function is the product of \(\phi_j \circ \phi_i^{-1}\) and \(\psi_j \circ \psi_i^{-1}\), hence \(g_{ij} h_{ij}\).

For any line bundle \(\mathcal{L}\), the dual bundle \(\mathcal{L}^\vee\) is the bundle whose fibers are given by

\[\mathcal{L}_x^\vee=\Hom_\mathbb{K}(\mathcal{L}_x, \mathbb{K}).\]

If we think of line bundles as (invertible) sheaves following Proposition 5, then \(\mathcal{L}^\vee\) corresponds to the sheaf Hom \(\sHom_{\mathcal{O}_X}(\mathcal{L}, \mathcal{O}_X)\).

Proposition 7 The dual bundle \(\mathcal{L}^\vee\) of a line bundle \(\mathcal{L}\) is also a line bundle, and its transition functions are \(\{g_{ij}^{-1}\}\).

Proof

The fiber of the dual bundle is \(\mathcal{L}_p^\vee = \Hom_{\mathbb{K}}(\mathcal{L}_p, \mathbb{K})\), which is again one-dimensional since it is the dual of a one-dimensional vector space. The transition function is the inverse of \(g_{ij}\).

The next proposition shows the relationship between \(\otimes\) and \((-)^\vee\), which plays an important role in defining the Picard group.

Proposition 8 For any line bundle \(\mathcal{L}\) we have \(\mathcal{L} \otimes \mathcal{L}^\vee \cong \mathcal{O}_X\).

Proof

The transition functions of \(\mathcal{L} \otimes \mathcal{L}^\vee\) are \(g_{ij} \cdot g_{ij}^{-1} = 1\), so it is the trivial bundle.

As always, we can understand the structure of a line bundle by examining it over a sufficiently small affine open set. Consider a line bundle \(\mathcal{L}\) and choose an affine open subset \(U_i\) over which \(\mathcal{L}\) is trivial. Then the projection map

\[\pi\vert_{\pi^{-1}(U_i)}:\pi^{-1}(U_i) \rightarrow U_i\]

is a map between affine varieties, and hence by §Affine Varieties, ⁋Proposition 16 induces a ring homomorphism between coordinate rings. This ring homomorphism makes the coordinate ring of \(\pi^{-1}(U_i)\) into an algebra over the coordinate ring \(A\) of \(U_i\), and the \(A\)-module \(\mathcal{O}_X(\mathcal{L})(U_i)\) consisting of sections of \(\pi\) is identified with \(A\) via the trivialization \(\phi_i\), so it is a free module of rank \(1\). Since \(\mathcal{L}\) is trivial over any open subset of \(U_i\) as well, we can verify that a line bundle becomes an invertible module over the coordinate ring affine-locally. ([Commutative Algebra] §Fractional Ideals, ⁋Definition 1) Then the operations \(\otimes\) and \(\vee\) defined on line bundles come from the operations of [Commutative Algebra] §Fractional Ideals, ⁋Theorem 3, and therefore it is not unnatural to adopt the following name following [Commutative Algebra] §Fractional Ideals, ⁋Definition 5.

Definition 9 The Picard group \(\Pic(X)\) of a variety \(X\) is the group obtained by taking the set of isomorphism classes of line bundles on \(X\) with tensor product as the operation. The identity element is the trivial bundle \(\mathcal{O}_X\), and the inverse of \(\mathcal{L}\) is \(\mathcal{L}^\vee\).

That the trivial bundle actually serves as the identity element is verified directly from Example 3 and Proposition 6. Moreover, by the properties of tensor product the following holds.

Proposition 10 \(\Pic(X)\) is an abelian group.

Proof

By Proposition 6 the tensor product is a binary operation on line bundles, and by Proposition 8 the element \(\mathcal{O}_X\) is the identity and \(\mathcal{L}^\vee\) is the inverse of \(\mathcal{L}\). The commutativity \(\mathcal{L} \otimes \mathcal{M} \cong \mathcal{M} \otimes \mathcal{L}\) and associativity \((\mathcal{L} \otimes \mathcal{M}) \otimes \mathcal{N} \cong \mathcal{L} \otimes (\mathcal{M} \otimes \mathcal{N})\) of tensor product follow directly at the level of transition functions from \(g_{ij}h_{ij} = h_{ij}g_{ij}\) and \((g_{ij}h_{ij})k_{ij} = g_{ij}(h_{ij}k_{ij})\).

As in the previous post, our toy examples are \(\mathbb{A}^n\) and \(\mathbb{P}^n\).

Example 11 The coordinate ring \(R = \mathbb{K}[\x_1, \ldots, \x_n]\) of \(\mathbb{A}^n\) is a Noetherian UFD, and by the above discussion line bundles on \(\mathbb{A}^n\) correspond to invertible modules over \(R\). By [Commutative Algebra] §Fractional Ideals, ⁋Theorem 4 invertible modules over a Noetherian UFD are free, so \(\Pic(\mathbb{A}^n) = 0\).

Example 12 We define the line bundle \(\mathcal{O}_{\mathbb{P}^n}(d)\) on \(\mathbb{P}^n\) as follows. First, each standard open set

\[U_i = \{[x_0 : \cdots : x_n] \mid x_i \ne 0\}\]

is a trivializing open set for this bundle. We explicitly define the trivialization over each of these by

\[\phi_i\colon \mathcal{O}(d)\vert_{U_i} \xrightarrow{\sim} \mathcal{O}_{U_i}, \qquad \phi_i(s) = s \cdot \x_i^{-d}.\]

From this we know that the space of sections has the form

\[\mathcal{O}(d)(U_i) = \x_i^d \cdot \mathcal{O}(U_i) = \x_i^d\mathbb{K}[\x_0/\x_i, \ldots, \widehat{\x_i/\x_i}, \ldots, \x_n/\x_i].\]

Now comparing the two trivializations on \(U_i \cap U_j\) we can derive the transition function. That is, the transition function \(\phi_j \circ \phi_i^{-1}\colon \mathcal{O}_{U_i}\vert_{U_i \cap U_j} \rightarrow \mathcal{O}_{U_j}\vert_{U_i \cap U_j}\) is

\[\phi_j \circ \phi_i^{-1}(f) = (\x_i/\x_j)^d \cdot f,\]

so we obtain \(g_{ij} = (\x_i/\x_j)^d\). More concretely, for each point \(x \in U_i \cap U_j\) and fiber element \(v \in \mathcal{O}_{\mathbb{P}^n}(d)_x \cong \mathbb{A}^1\) at that point,

\[g_{ij}(x)\colon v \mapsto (\x_i/\x_j)^d(x) \cdot v.\]

Now we can define a group homomorphism

\[\mathbb{Z}\rightarrow \Pic(\mathbb{P}^n);\qquad d\mapsto [\mathcal{O}_{\mathbb{P}^n}(d)].\]

Our claim is that this is an isomorphism. First, for any line bundle \(\mathcal{L}\), the restriction \(\mathcal{L}\vert_{U_i}\) is isomorphic to the trivial line bundle by Example 11, so the transition functions \(h_{ij}\) on each \(U_i\cap U_j\) completely determine \(\mathcal{L}\). But by definition \(h_{ij}\in \mathcal{O}_{\mathbb{P}^n}(U_i\cap U_j)^\ast\) on \(U_i\cap U_j\), so \(h_{ij}\) must necessarily be of the form \(c_{ij}(\x_i/\x_j)^d\). Since a line bundle whose transition functions differ by a constant factor is trivial, we see from this that the above group homomorphism is surjective. Similarly, assuming \(\mathcal{O}_{\mathbb{P}^n}(d)\cong \mathcal{O}_{\mathbb{P}^n}(d')\) and comparing transition functions, we obtain

\[\mathcal{O}_{\mathbb{P}^n}(d-d')\cong \mathcal{O}_{\mathbb{P}^n}(d)\otimes \mathcal{O}_{\mathbb{P}^n}(-d')\cong \mathcal{O}_{\mathbb{P}^n}(d)\otimes \mathcal{O}_{\mathbb{P}^n}(d')^\vee\cong \mathcal{O}_{\mathbb{P}^n}.\]

Now setting \(e:=d-d'\), the statement \(\mathcal{O}_{\mathbb{P}^n}(e)\cong \mathcal{O}_{\mathbb{P}^n}\) means that there exist \(u_i\in \mathcal{O}(U_i)^\ast\) satisfying \((\x_i/\x_j)^e=u_i/u_j\), but since \(U_i\cong \mathbb{A}^n\) we have \(\mathcal{O}(U_i)^\ast=\mathbb{K}^\ast\) and therefore \((\x_i/\x_j)^e\) must be constant, i.e. \(e=0\). Hence it is also injective.

Intuitively, the integer \(d\) in the line bundle \(\mathcal{O}_{\mathbb{P}^n}(d)\) on \(\mathbb{P}^n\) can be understood as a measure of how many times the fiber twists as it moves over the base. When \(d=0\) the bundle \(\mathcal{O}(0)\) is trivial so there is no twist, when \(d>0\) it twists \(d\) times in one direction, and when \(d<0\) it twists \(\lvert d\rvert\) times in the opposite direction. This means that \(d\) in the transition function \(g_{ij}(x) = (x_i/x_j)^d(x)\) directly represents the amount of twisting. However, this intuition may be somewhat imprecise, so some additional explanation will be needed after Example 16.

On the other hand, on projective space \(\mathbb{P}^n\) there is a special line bundle that arises naturally from its very definition. This tautological bundle is the bundle that assigns to each point of \(\mathbb{P}^n\) the line represented by that point, and it plays a fundamental role in understanding the geometry of projective space.

Definition 13 For each point \(x = [x_0 : \cdots : x_n]\) of \(\mathbb{P}^n\), consider the space obtained by attaching to each point the line \(\ell_x = \{(\lambda x_0, \ldots, \lambda x_n) \mid \lambda \in \mathbb{K}\}\) through the origin of \(\mathbb{A}^{n+1}\):

\[\mathcal{O}_{\mathbb{P}^n}(-1) = \{(x, v) \in \mathbb{P}^n \times \mathbb{A}^{n+1} \mid v \in \ell_x\}.\]

Then the line bundle over \(\mathbb{P}^n\) defined by the projection map \(\pi=\pr_1\) from \(\mathcal{O}_{\mathbb{P}^n}(-1)\) to \(\mathbb{P}^n\) is called the tautological line bundle.

That is, in this definition each fiber \(\mathcal{O}_{\mathbb{P}^n}(-1)_x\) is the very line represented by the point \(x\). As the notation suggests, the following holds. For distinction, in the next proposition only, let us regard \(\mathcal{O}_{\mathbb{P}^n}(-1)\) as the bundle from Definition 13, not from Example 12.

Proposition 14 The tautological bundle \(\mathcal{O}_{\mathbb{P}^n}(-1)\) is the dual of \(\mathcal{O}_{\mathbb{P}^n}(1)\) defined in Example 12 above. That is, \(\mathcal{O}_{\mathbb{P}^n}(-1) \cong \mathcal{O}_{\mathbb{P}^n}(1)^\vee\).

Proof

Let us construct a local trivialization of \(\mathcal{O}_{\mathbb{P}^n}(-1)\) over the standard open cover \(U_i = \{x \mid x_i \ne 0\}\). For any \((x, v) \in \mathcal{O}_{\mathbb{P}^n}(-1)\) we can write \(v = \lambda x\) (\(\lambda \in \mathbb{K}\)), so defining \(\phi_i(x, v) = (x, v_i)\) gives \(\phi_i: \pi^{-1}(U_i) \rightarrow U_i \times \mathbb{A}^1\). The inverse is \(\phi_i^{-1}(x, t) = (x, (t/x_i)x)\). The transition function on \(U_i \cap U_j\) is obtained from \(\phi_j \circ \phi_i^{-1}(x, t) = (x, t x_j / x_i)\) as \(g_{ij}(x) = x_j/x_i\). This is the inverse of the transition function \(x_i/x_j\) of \(\mathcal{O}_{\mathbb{P}^n}(1)\).

In particular, examining \(\mathcal{O}(-1)\) on \(\mathbb{P}^1\) makes the meaning of the twist described intuitively above much clearer. The process of forming \(\mathbb{P}^1\) from \(\mathbb{A}^2\setminus \{0\}\) can be thought of as first collapsing \(\mathbb{A}^2\setminus\{0\}\) onto the unit circle via radial projection and then identifying antipodal points on the unit circle; in this process, two vectors in opposite directions become identified, that is, the fibers become twisted. One way to see this twist is to look at sections of a line bundle \(\mathcal{L}\).

Definition 15 We denote the space of global sections of a line bundle \(\mathcal{L}\) by \(\Gamma(X, \mathcal{L})\). That is, \(\Gamma(X, \mathcal{L})\) is the set of regular maps assigning to each point \(x\in X\) an element in the fiber \(\pi^{-1}(x)\subseteq \mathcal{L}\).

Another popular notation for the global section space is \(H^0(X, \mathcal{L})\). This notation will be justified in §Sheaf Cohomology, ⁋Definition 1, but until then we shall use \(\Gamma(X, \mathcal{L})\).

Example 16 The only global section of \(\mathcal{O}_{\mathbb{P}^n}(-1)\) is \(0\). That is,

\[\Gamma(\mathbb{P}^n, \mathcal{O}_{\mathbb{P}^n}(-1)) = 0.\]

To verify this, by Example 12 we have \(\mathcal{O}(-1)(U_i) = \x_i^{-1} \cdot \mathbb{K}[\x_0/\x_i, \ldots, \widehat{\x_i/\x_i}, \ldots, \x_n/\x_i]\), and the trivialization is given by \(\phi_i(s) = s \cdot \x_i\). Thus the trivialized section \(\phi_i(s) \in \mathcal{O}(U_i) = \mathbb{K}[\x_0/\x_i, \ldots, \x_n/\x_i]\), and on \(U_i \cap U_j\) the cocycle condition requires

\[\phi_j(s) = (\x_j/\x_i)\phi_i(s).\]

However, since \(\phi_i(s) \in \mathbb{K}[\x_0/\x_i, \ldots, \widehat{\x_i/\x_i}, \ldots, \x_n/\x_i]\) cannot contain a term \(\x_i/\x_j\), for \((\x_j/\x_i)\phi_i(s)\) to lie in \(\mathcal{O}(U_j) = \mathbb{K}[\x_0/\x_j, \ldots, \widehat{\x_j/\x_j}, \ldots, \x_n/\x_j]\) we must have \(\phi_i(s) = 0\). Hence \(s = 0\).

This proposition shows the twist of the tautological bundle from the viewpoint of sections. For instance, the fact that \(\Gamma(\mathbb{P}^1, \mathcal{O}(-1))=0\) means in particular that there is not even a “constant function” assigning \(1\) in the fiber for every \(x\in \mathbb{P}^1\). From the geometric viewpoint above, this is because after going around \(\mathbb{P}^1\) once, the original \(1\) has become (for example) \(-1\).

Meanwhile, the computation in Example 16 can be extended to arbitrary \(d\); in particular, for any \(d<0\) one can show by the same logic that \(\Gamma(\mathbb{P}^1, \mathcal{O}(d))=0\), and for \(d=0\), that is, for \(\mathcal{O}_{\mathbb{P}^n}(0)=\mathcal{O}_{\mathbb{P}^n}\), the sections are homogeneous polynomials of degree \(0\), i.e., constant functions, so the computation in §Quasi-Projective Varieties, ⁋Example 6 is confirmed again.

The case to pay attention to is \(d>0\). In this case, by exactly the same computation as in Example 16, one can verify that the sections are homogeneous polynomials of degree \(d\). In particular \(\Gamma(\mathbb{P}^n, \mathcal{O}(d))\neq 0\), which can be regarded as a calculation showing that the intuition after Example 12 was somewhat overly simplistic.

A more precise explanation of this phenomenon is as follows. For convenience, let us look at the example on \(\mathbb{P}^1\). The sections of \(\mathcal{O}(-1)\) are homogeneous of degree \(-1\), so they have the form, for instance,

\[s([x_0:x_1])=\frac{a}{x_0}+\frac{b}{x_1},\]

and for this function to be defined on all of \(\mathbb{P}^1\) we must have \(a=b=0\). On the other hand, the sections of \(\mathcal{O}(1)\) are homogeneous polynomials of degree \(1\), so they are functions of the form

\[s([x_0:x_1])=ax_0+bx_1,\]

and unlike above there is no restriction on \(a,b\). Intuitively, the sections of \(\mathcal{O}(-1)\) cannot cross the zero section because of the denominators, and therefore every section cannot avoid the problem created by the twist of “\(1\) attaching to \(-1\)”. This twist creates the same problem in \(\mathcal{O}(1)\) as well: namely, the “constant section” \(s([x_0:x_1])=1\) is likewise not a section in \(\mathcal{O}(1)\). However, this time the sections of \(\mathcal{O}(1)\) can cross the zero section, so we obtain \(\Gamma(\mathbb{P}^1, \mathcal{O}(1))\neq 0\). From the viewpoint of transition functions or trivializations, one can understand this as follows: since \(\mathcal{O}(d)\) multiplies by the degree \(d\) polynomial \(\x_i^d\), a pole of degree at most \(d\) can be erased by this trivialization.

Divisor – Line Bundle correspondence

We now establish the essential connection between divisors and line bundles. First we show that one can construct a line bundle from a Cartier divisor.

Definition 17 For a Cartier divisor \(D = \{(U_i, f_i)\}\), we define the line bundle \(\mathcal{O}_X(D)\) by the transition functions \(g_{ij} = f_j/f_i\).

That is, we take the trivial bundle over each \(U_i\) and glue them over the overlaps using exactly the information contained in the Cartier divisor. If we regard \(\mathcal{O}_X(D)\) as a sheaf, that is, if we consider the sheaf of sections of the line bundle defined above, then for each open set \(U\) the sheaf \(\mathcal{O}_X(D)(U)\) is (as a sheaf) the sheaf of functions satisfying

\[\divisor(f)+D\geq 0.\]

That is, \(\mathcal{O}_X(D)\) is the sheaf of rational functions that may have a pole of order at most \(1\) along \(D\), if we regard \(D\) as a codimension \(1\) subvariety of \(X\). Conversely, when \(D\) is effective, \(\mathcal{O}_X(-D)\) is given by the condition

\[\divisor(f)-D\geq 0,\]

which is exactly the sheaf of functions vanishing on \(D\). That is,

\[\mathcal{O}_X(-D)(U)=\{f\in \mathcal{O}_X(U)\mid \text{$f$ vanishes on $D\cap U$}\},\]

and from this we obtain the short exact sequence

\[0\rightarrow \mathcal{O}_X(-D)\rightarrow \mathcal{O}_X\rightarrow \mathcal{O}_D\rightarrow 0.\]

Then \(\mathcal{O}_X(-D)\) is the sheaf of ideals defining \(D\), and for this reason we write it as \(\mathcal{I}_D\) and call it the ideal sheaf.

Proposition 18 The above definition is well-defined. That is, equivalent Cartier divisors define isomorphic line bundles.

Proof

If two Cartier divisors \(\{(U_i, f_i)\}\) and \(\{(V_j, g_j)\}\) are equivalent, then \(f_i/g_j \in \mathcal{O}_X(U_i \cap V_j)^\ast\). Let us compare the two line bundles over the common refinement \(\{U_i \cap V_j\}\) and set \(u_{ij} := g_j/f_i \in \mathcal{O}_X(U_i \cap V_j)^\ast\). Then on \((U_i \cap V_j) \cap (U_k \cap V_l)\) the transition functions of the two line bundles are \(f_k/f_i\) and

\[\frac{g_l}{g_j} = \frac{u_{kl} f_k}{u_{ij} f_i} = \frac{u_{kl}}{u_{ij}} \cdot \frac{f_k}{f_i}\]

respectively. Thus they differ only by the units \(\{u_{ij}\}\), and hence the two line bundles are identified by the isomorphism they define.

For example, for any principal divisor \(\divisor(f)\) the transition function is \(1\), so it becomes the trivial bundle. We now summarize the relationship between line bundles and Cartier divisors.

Proposition 19 For any variety \(X\), we have \(\Pic(X) \cong \CaCl(X)\).

Proof

First we verify that \(D \mapsto \mathcal{O}_X(D)\) is a group homomorphism from \(\CaDiv(X)\) to \(\Pic(X)\). For a Cartier divisor \(D = \{(U_i, f_i)\}\), the transition function of \(\mathcal{O}_X(D)\) is \(g_{ij} = f_j/f_i \in \mathcal{O}_X(U_i \cap U_j)^\times\), so it defines a line bundle. Moreover, writing two Cartier divisors \(D = \{(U_i, f_i)\}\) and \(D' = \{(U_i, f_i')\}\) with the same cover on a common refinement, we have \(D + D' = \{(U_i, f_i f_i')\}\) and its transition function is \((f_j f_j')/(f_i f_i') = g_{ij} g_{ij}'\), so by Proposition 6 we obtain \(\mathcal{O}_X(D + D') \cong \mathcal{O}_X(D) \otimes \mathcal{O}_X(D')\). That is, \(D \mapsto \mathcal{O}_X(D)\) is additive. A principal divisor \(\divisor(h)\) corresponds to the trivial bundle since its transition function is \(1\), and hence it induces a well-defined group homomorphism from \(\CaCl(X) = \CaDiv(X)/\Prin(X)\) to \(\Pic(X)\).

To show that this is an isomorphism, let an arbitrary line bundle \(\mathcal{L}\) be given. On a trivializing open \(U \subseteq X\) we have \(\mathcal{L}\vert_U \cong \mathcal{O}_U\), so we can take \(s \in \mathcal{L}(U)\) corresponding to the constant section \(1\) of \(\mathcal{O}_U\), and this \(s\) is a nonzero rational section. Now consider a trivializing cover \(\{U_i\}\) of \(\mathcal{L}\). Choose a trivialization \(\psi_i\colon \mathcal{L}\vert_{U_i} \cong \mathcal{O}_{U_i}\) on each \(U_i\) and define \(f_i := \psi_i(s\vert_{U_i \cap U}) \in \mathcal{O}_X(U_i \cap U) \subseteq \mathbb{K}(X)\). Then on \(U_i \cap U_j \cap U\) we have \(f_j = g_{ij} f_i\), and since \(X\) is irreducible, \(U_i \cap U_j \cap U\) is a dense open subset of \(U_i \cap U_j\), so this relation holds on all of \(U_i \cap U_j\). That is, \(f_j/f_i = g_{ij} \in \mathcal{O}_X(U_i \cap U_j)^\times\), so \(D = \{(U_i, f_i)\}\) is a Cartier divisor, and since the transition function of \(\mathcal{O}_X(D)\) is \(\{g_{ij}\}\), we have \(\mathcal{O}_X(D) \cong \mathcal{L}\).

Finally we show injectivity. Write two Cartier divisors \(D = \{(U_i, f_i)\}\) and \(D' = \{(U_i, f_i')\}\) with the same cover on a common refinement and suppose \(\mathcal{O}_X(D) \cong \mathcal{O}_X(D')\). The transition functions of two isomorphic line bundles differ by suitable units \(u_i \in \mathcal{O}_X(U_i)^\times\) as

\[\frac{f_j'}{f_i'} = \frac{u_j}{u_i} \cdot \frac{f_j}{f_i}.\]

Rewriting this relation, we have \(u_i f_i/f_i' = u_j f_j/f_j'\) on \(U_i \cap U_j\), so \(h := u_i f_i/f_i'\) defines a single rational function \(h \in \mathbb{K}(X)^\times\) independent of the choice of \(i\). Since \(u_i\) is a unit on each \(U_i\), we have \(\divisor(u_i) = 0\) and \(\divisor(h) = \divisor(f_i) - \divisor(f_i')\), so \(D - D' = \divisor(h)\), i.e., the two Cartier divisors are linearly equivalent.

If \(X\) is smooth, we already know that \(\CaCl(X)\cong \Cl(X)\).

Pullback of Line Bundles

Given a morphism \(\varphi: X \rightarrow Y\), the operation of “pulling back” a line bundle on \(Y\) to \(X\) is defined naturally. For example, if we pull back a hypersurface on \(Y\) to \(X\) via \(\varphi\), the corresponding line bundle should also be pulled back. This pullback operation induces a group homomorphism between Picard groups, and in the case of an embedding it can be understood as restricting a line bundle on the ambient space to the subvariety.

Proposition 20 For a morphism \(\varphi: X \rightarrow Y\) and a line bundle \(\mathcal{L}\) on \(Y\), the pullback \(\varphi^\ast \mathcal{L}\) is a line bundle on \(X\). Its transition functions are \(\{g_{ij} \circ \varphi\}\), where \(\{g_{ij}\}\) are the transition functions of \(\mathcal{L}\).

Proof

Suppose the line bundle \(\mathcal{L}\) is given by transition functions \(\{g_{ij}\}\) on an open cover \(\{U_i\}\). The pullback \(\varphi^\ast \mathcal{L}\) is defined by transition functions \(\{g_{ij} \circ \varphi\}\) on the open cover \(\{\varphi^{-1}(U_i)\}\). To verify that \(\varphi^\ast \mathcal{L}\) is a line bundle on \(X\), it suffices to check that the transition functions satisfy the cocycle condition.

We check all three cocycle conditions.

1. \(g_{ii} \circ \varphi = 1 \circ \varphi = 1\) since \(g_{ii} = 1\).
2. \((g_{ij} \circ \varphi)(g_{ji} \circ \varphi) = (g_{ij} g_{ji}) \circ \varphi = 1 \circ \varphi = 1\) since \(g_{ij} g_{ji} = 1\).
3. \((g_{ij} \circ \varphi)(g_{jk} \circ \varphi) = (g_{ij} g_{jk}) \circ \varphi = g_{ik} \circ \varphi\) since \(g_{ij} g_{jk} = g_{ik}\).

Therefore \(\{g_{ij} \circ \varphi\}\) satisfies the cocycle condition.

Proposition 21 Pullback induces a group homomorphism \(\varphi^\ast: \operatorname{Pic}(Y) \rightarrow \operatorname{Pic}(X)\).

Proof

Since \(\varphi^\ast(\mathcal{L} \otimes \mathcal{M}) \cong \varphi^\ast \mathcal{L} \otimes \varphi^\ast \mathcal{M}\) and \(\varphi^\ast \mathcal{O}_Y \cong \mathcal{O}_X\), pullback is a group homomorphism.

To verify this, let us look from the viewpoint of transition functions. The transition function of \(\mathcal{L} \otimes \mathcal{M}\) is \(g_{ij}^{\mathcal{L}} g_{ij}^{\mathcal{M}}\), so the transition function of \(\varphi^\ast(\mathcal{L} \otimes \mathcal{M})\) is \((g_{ij}^{\mathcal{L}} g_{ij}^{\mathcal{M}}) \circ \varphi = (g_{ij}^{\mathcal{L}} \circ \varphi)(g_{ij}^{\mathcal{M}} \circ \varphi)\). These are respectively the transition functions of \(\varphi^\ast\mathcal{L}\) and \(\varphi^\ast\mathcal{M}\), so we obtain \(\varphi^\ast(\mathcal{L} \otimes \mathcal{M}) \cong \varphi^\ast\mathcal{L} \otimes \varphi^\ast\mathcal{M}\). Moreover, the transition functions of \(\mathcal{O}_Y\) are all \(1\), so the transition functions of \(\varphi^\ast\mathcal{O}_Y\) are also \(1\), i.e., \(\varphi^\ast\mathcal{O}_Y \cong \mathcal{O}_X\).

Example 22 For an embedding \(i: C \hookrightarrow \mathbb{P}^n\), the pullback \(i^\ast \mathcal{O}_{\mathbb{P}^n}(1)\) is a line bundle on the curve \(C\). We call this the hyperplane bundle on \(C\) and denote it by \(\mathcal{O}_C(1)\). In general \(\mathcal{O}_C(1)\) is nontrivial; for instance, when \(\mathbb{P}^1\) is embedded as a line in \(\mathbb{P}^n\), the bundle \(\mathcal{O}_C(1) = \mathcal{O}_{\mathbb{P}^1}(1)\) on \(C = \mathbb{P}^1\) is nontrivial, as we saw in Example 12. The name “hyperplane bundle” comes from the fact that it is obtained by pulling back to \(C\) the line bundle \(\mathcal{O}_{\mathbb{P}^n}(1)\) corresponding to a hyperplane \(H\), i.e. a hypersurface of degree \(1\) in \(\mathbb{P}^n\).

Vector Bundle

So far we have examined line bundles, whose fibers are one-dimensional vector spaces. We can generalize this notion to vector bundles, whose fibers are higher-dimensional vector spaces. A vector bundle captures geometric structures that arise naturally, such as the tangent space and normal space of a variety, and is the algebraic-geometric analogue of the tangent bundle and vector fields in differential geometry. A line bundle is precisely the special case of a rank 1 vector bundle, and from the perspective of vector bundle theory we can understand the properties of line bundles even more clearly.

Definition 23 A rank \(r\) vector bundle \(\mathcal{E}\) on a variety \(X\) consists of the following data.

1. A projection \(\pi: \mathcal{E} \rightarrow X\).

2. An open cover \(\{U_i\}\) of \(X\) and, for each \(i\), a local trivialization \(\phi_i: \pi^{-1}(U_i) \overset{\sim}{\longrightarrow} U_i \times \mathbb{A}^r\). The maps

   \[\phi_j \circ \phi_i^{-1}: (U_i \cap U_j) \times \mathbb{A}^r \rightarrow (U_i \cap U_j) \times \mathbb{A}^r\]

   are of the form \((p, v) \mapsto (p, g_{ij}(p)v)\) for suitable transition functions \(g_{ij} \in \GL_r(\mathcal{O}_X(U_i \cap U_j))\).

Comparing with the definition of a line bundle, the only differences are that the fiber is \(\mathbb{A}^r\) instead of \(\mathbb{A}^1\), and the transition functions take values in \(\GL_r(\mathcal{O}_X(U_i \cap U_j))\) rather than in \(\mathcal{O}_X(U_i \cap U_j)^\times = \GL_1(\mathcal{O}_X(U_i \cap U_j))\). Hence a line bundle is exactly a rank 1 vector bundle.

The same cocycle condition as in Proposition 2 (Cocycle condition) holds. However, since the transition functions are matrix-valued, one must be careful about the order of multiplication.

Example 24 The simplest example is the rank \(r\) trivial vector bundle \(\mathcal{O}_X^{\oplus r}\) obtained from the line bundle \(\mathcal{O}_X\). This is constructed by taking the direct sum of the line bundle \(\mathcal{O}_X\) with itself \(r\) times.

Geometrically important objects are the tangent bundle and the cotangent bundle. The tangent bundle \(\mathcal{T}_X\) is the vector bundle whose fiber over each point \(p \in X\) is the tangent space \(T_p X\); if \(X\) is an \(n\)-dimensional smooth variety then it is a rank \(n\) vector bundle, and in local coordinates \(\x_1, \ldots, \x_n\) the partial derivatives \(\partial/\partial \x_1, \ldots, \partial/\partial \x_n\) form a local frame. The cotangent bundle \(\Omega_X^1 = \mathcal{T}_X^\vee\) is the dual of the tangent bundle, and in local coordinates \(\dd{\x_1}, \ldots, \dd{\x_n}\) form a local frame.

Intuitively, since \(\Omega_X^1\) is the bundle of differential \(1\)-forms on \(X\), we can take the \(r\)-th exterior power to obtain the bundle of \(r\)-forms. Among these, the most interesting is the top exterior power \(\omega_X = \bigwedge^n \Omega_X^1\), which is a rank \(1\) vector bundle, i.e. a line bundle; in differential geometry one would think of it as the bundle of volume forms. We call this the canonical line bundle.

As above, we can define operations on vector bundles analogous to those for line bundles. The tensor product \(\mathcal{E} \otimes \mathcal{F}\) of two vector bundles \(\mathcal{E}, \mathcal{F}\) is defined by the fiberwise tensor product, and its transition functions are \(g_{ij}^{\mathcal{E}} \otimes g_{ij}^{\mathcal{F}}\). The transition functions of the dual bundle \(\mathcal{E}^\vee\) are \(\left(g_{ij}^{\mathcal{E}}\right)^{-t}\) (inverse transpose). Moreover, the direct sum \(\mathcal{E} \oplus \mathcal{F}\) is defined by the fiberwise direct sum, and in this case the transition functions become the block diagonal matrix \(\begin{pmatrix} g_{ij}^{\mathcal{E}} & 0 \\ 0 & g_{ij}^{\mathcal{F}} \end{pmatrix}\).

Tautological Bundle on Grassmannian

The tautological bundle \(\mathcal{O}_{\mathbb{P}^n}(-1)\) on \(\mathbb{P}^n\) defined above generalizes naturally to Grassmannians. The Grassmannian \(\Gr(k, n)\) is the space of \(k\)-dimensional subspaces of an \(n\)-dimensional vector space, and in this generalization the tautological bundle becomes a rank \(k\) vector bundle; the quotient bundle dual to it is also defined naturally.

Definition 25 We define the following two vector bundles on the Grassmannian \(\Gr(k, n)\).

1. Tautological bundle \(S\): a rank \(k\) vector bundle that assigns to each point \([V] \in \Gr(k, n)\) (where \(V \subseteq \mathbb{A}^n\) is a \(k\)-dimensional subspace) the subspace \(V\) itself as its fiber. \(S = \{([V], v) \in \Gr(k, n) \times \mathbb{A}^n \mid v \in V\}\)

2. Quotient bundle \(Q\): a rank \(n-k\) vector bundle that assigns to each point \([V]\) the quotient space \(\mathbb{A}^n / V\) as its fiber. \(Q = \{([V], [w]) \in \Gr(k, n) \times (\mathbb{A}^n / S) \mid [w] \in \mathbb{A}^n / V\}\)

Between these there is a natural short exact sequence.

\[0 \rightarrow S \rightarrow \mathcal{O}_{\Gr(k,n)}^{\oplus n} \rightarrow Q \rightarrow 0\]

Here the middle term is \(\Gr(k, n) \times \mathbb{A}^n\), the trivial bundle of rank \(n\). The first morphism is the inclusion of each point \(([V], v) \in S\) into \(([V], v) \in \mathcal{O}^{\oplus n}\), and the second morphism is the quotient map sending \(([V], w) \in \mathcal{O}^{\oplus n}\) to \(([V], [w]) \in Q\).

Proposition 26 On \(\Gr(1, n+1) = \mathbb{P}^n\), the tautological bundle \(S\) is isomorphic to \(\mathcal{O}_{\mathbb{P}^n}(-1)\).

Proof

Each point of \(\Gr(1, n+1)\) is a one-dimensional subspace of \(\mathbb{A}^{n+1}\), i.e. a line through the origin. This corresponds exactly to a point of \(\mathbb{P}^n\). Since each fiber of the tautological bundle \(S\) is this line itself, it is identical to \(\mathcal{O}_{\mathbb{P}^n}(-1)\) defined in Definition 13.

This proposition shows that the tautological bundle on a Grassmannian reduces, in the case of projective space, to the familiar \(\mathcal{O}(-1)\). As for the quotient bundle \(Q\), on \(\Gr(1, n+1) = \mathbb{P}^n\) it has rank \(n\) and is closely related to the tangent bundle \(\mathcal{T}_{\mathbb{P}^n}\). Indeed, we have \(\mathcal{T}_{\mathbb{P}^n} \cong \Hom(S, Q) \cong S^\vee \otimes Q\).

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References

[Hart] R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, Springer, 1977.
[Sha] I. R. Shafarevich, Basic Algebraic Geometry I: Varieties in Projective Space, Springer, 2013.