INTRODUCTION TO ALGEBRAIC GEOMETRY, CLASS 21

RAVI VAKIL

Contents

1. Line bundles and invertible sheaves 1 1.1. Line bundles, classically 1 1.2. Line bundles on varieties (and schemes) 2 1.3. Sheaf of sections of a line bundle, and the correspondence with line

bundles 2

1.4. Examples 3

Problem set info! PS8 back Tuesday November 30. PS9 due Tuesday November 30. PS10 due Thursday December 2. (PS11 due Thurs. Dec. 9. PS12 due Monday Dec. 13.)

Brand new topic:

1. Line bundles and invertible sheaves

Invertible sheaves are fundamental objects. They are essentially the same thing as line bundles, which are geometrically easy to visualize. So I’ll tell you intuitively about line bundles first, and then introduce invertible sheaves. (I won’t discuss motivation just yet, so I hope you believe me when I say how important this concept is!)

1.1. Line bundles, classically. Line bundle on a complex manifold (with classical topology). Draw a picture. Precisely, given a complex manifold X, a line bundle is given by the data of a cover U_{i} (i ∈ I some index set), and trivializations U_{i} × ℂ, and transition functions f_{ij} on U_{i} ∩ U_{j} that is an analytic function, nowhere zero, satisfying f_{ij}f_{jk} = f_{ik} (the cocyle condition), from which f_{ii} = 1 and f_{ij} = f_{ji}^{−1}. What this means: if u ∈ U_{i} ∩ U_{j}, then the point (u, z) ∈ U_{j} × ℂ is identified with the point (u, f_{ij}z) ∈ U_{i} × ℂ. (I think I said this incorrectly in class.) If you’ve seen Cech cohomology before, you can check that the data of a line bundle is the same as an element of H^{1}(X, O^{∗}), where O^{∗} is the sheaf of invertible (=nowhere zero) analytic functions.

Date: Tuesday, November 23, 1999.

In particular, the total space L of a line bundle is also a complex manifold (of dimension one higher than that of X), with a morphism L → X.

A section of a line bundle is the data of maps g_{i} : U_{i} → ℂ (or if you prefer, U_{i} → U_{i} × ℂ), satisfying g_{i}(p) = f_{ij}(p)g_{j}(p) for points p ∈ U_{i} ∩ U_{j}. (Draw a picture of a section of L → X.)

Note that there is always a zero-section given by g_{i}(p) = 0 for all i, p ∈ U_{i}.

Two line bundles L_{1} → X, L_{2} → X are isomorphic if there is an isomorphism L_{1} → L_{2} commuting with the morphisms to X.

Example. The trivial line bundle X × ℂ → X. Global sections are just analytic functions.

At this point you might think that all line bundles are trivial (which isn’t true). Indeed there are line bundles that seem nontrivial that are trivial. Exercise. Suppose U_{i} is a finite open cover of X, and for each i, h_{i} is an invertible function on U_{i}. Prove that the data of the open cover U_{i} and the transition functions f_{ij} = h_{i}/h_{j} defines a line bundle, and prove that this is isomorphic to the trivial line bundle. (In fancy language, this shows that a zero-element of H^{1}(X, ℂ^{∗}) indeed induces a trivial line bundle.)

Example. Inside ℂ^{n+1}, we know that ℙ^{n} parametrizes lines through the origin. This will give a line bundle on ℙ^{n} (as we’ll soon see rigorously): “above” each point of ℙ^{n}, we have the line corresponding to that point. (We will later see that this is not trivial, if n > 0.)

Example. Let X be a one-dimensional complex manifold. Then there is a tangent line bundle (and similarly a cotangent line bundle). (Describe it pictorially.) This will be tricky to describe precisely.

1.2. Line bundles on varieties (and schemes). We can use the same definition for varieties, except this time we use the Zariski topology, and the functions should be invertible regular functions. (Experts in Cech cohomology can later check that line bundles are parametrized be H^{1}(X, O_{X}^{∗}), where O_{X}^{∗} is the sheaf of invertible functions and the topology on X is of course the Zariski topology; in fact, if you’ve seen Cech cohomology of sheaves in any other setting, you know a whole lot about cohomology of sheaves in algebraic geometry.)

1.3. Sheaf of sections of a line bundle, and the correspondence with line bundles. Note that the sections actually form a sheaf. (Sketch intuitively first.) Here, explicitly, is how. Suppose the line bundle L on our variety X is defined by an open cover U_{i} and transition functions f_{ij} (on U_{ij} = U_{i} ∩ U_{j}). Then the sheaf of sections L is defined by:

Elements of L(U) are functions h_{i} on U ∩U_{i} such that h_{i} = f_{ij}h_{j} on U ∩U_{i} ∩U_{j}.

Notice that this sheaf L is an O_{X}-module. What this means is: for each open U, L(U) is an O_{X}(U)-module (and this action commutes with restriction: if U ⊂ V, and a ∈ L(V) and f ∈ O_{X}(V), then the restriction of (fa) to U is the same as the restriction of f to U acting on the restriction of a to U.

We can now define an invertible sheaf.

Definition. An invertible sheaf on a variety X is an O_{X}-module L such that there is an open cover U_{i} of X satisfying:

• there are isomorphisms ϕ_{i} : L|_{U_{i}} → O_{U_{i}},

• there are invertible functions f_{ij} on U_{i} ∩ U_{j} such that for z ∈ L(U) for any U ⊂ U_{i} ∩ U_{j}, ϕ_{i}(z) (which is a function on U) equals f_{ij}ϕ_{j}(z) (the product of two functions on U).

Notice that the cocycle condition f_{ik} = f_{ij}f_{jk} on U_{i} ∩ U_{j} ∩ U_{k} follows from this definition.

Notice that if you have a line bundle, its sheaf of sections is an invertible sheaf. If you have an invertible sheaf, you can cook up a line bundle. And these constructions commute.

Philosophy: Invertible sheaves are easier to deal with (if less intuitive to picture) than line bundles, so we’ll deal with them in the future. Here’s why this fits into the the philosophy of algebraic geometry. Classically, people would study varieties by looking at them as topological spaces. In algebraic geometry, we study varieties by looking at functions on them (in some sense a dual point of view). In the same way, we will study line bundles by instead looking at their sheaf of sections.

1.4. Examples. (I’m going to discuss this example again later on.) Let’s bring this back down to earth. I’ll now give some examples. Try to think of them analytically at the same time as algebraically.

Define a meromorphic section (or “rational section”) to be a section of a line bundle over some non-empty open set.

Example. The trivial line bundle X × 𝔸^{1} → X. Global sections are just regular functions on X.

A specific case: let X = ℙ^{1}. Then the meromorphic sections of the trivial line bundle are just rational functions. They are (in affine coordinates) of the form (t − a_{1})^{n_{1}} … (t − a_{r})^{n_{r}}, where the n_{r} are integers, possibly negative. Define the order of vanishing of a meromorphic function (at a_{i} to be n_{i}). What’s the order of vanishing of this meromorphic function at ∞? Answer: − ∑ n_{i}. Hence the sum of the order of vanishings of any non-zero meromorphic section of the trivial line bundle on ℙ^{1} is 0. (Remark: whenever you are talking about orders of vanishing, keep in mind that there is DVR theory lurking in the background.)

Example: The “tautological” line bundle O_{ℙ^{n}}(−1) on ℙ^{n}. Let ℙ^{n} have coordinates (x_{0}; …; x_{n}). This will be the line bundle that assigns to a point of ℙ^{n} the corresponding line through the origin in 𝔸^{n+1}. Perhaps ask about section of O_{ℙ^{1}}(−1) corresponding to x_{0} = 1 here. We’ll define this on patches. On the patch U_{0} (i.e. the points (1; y_{1}; …; y_{n}), so we can take coordinates y_{i} on U_{0}), the line is of the form (s, sy_{1}, …, sy_{n}) in 𝔸^{n+1}, with coordinate s.

On U_{1} (i.e. the points (z_{0}; 1; z_{2}; …; z_{n})), the line is of the form (tz_{0}; t; tz_{2}; …; tz_{n}). How do we change coordinates from s to t on U_{0}∩U_{1}? s = (x_{0}/x_{1})t. The best way to see this is by example. Consider the point (1, 2, 3, ..., n + 1) ∈ 𝔸^{n+1}. This is a point on the line above (1; 2; …; n+1). Its s-coordinate is 1, and its t-coordinate is 2. Indeed s/t = x_{0}/x_{1}. Precisely: f_{10} = x_{0}/x_{1} = 1/y_{0} = z_{0}. f_{01} = x_{1}/x_{0} = y_{0} = 1/z_{0}.

In general, you can define coordinates s_{0} on U_{0}, …, s_{n} on U_{n}, and we have transition functions f_{ij} = x_{j}/x_{i}. Note that f_{ij} is indeed an invertible function on U_{i} ∩ U_{j}.

Proposition. O_{ℙ^{1}}(−1) on ℙ^{1} is not isomorphic to the trivial sheaf.

(I’ll restate this proof, and give a second proof next day.)

Proof. (Advantages: foreshadows idea of “degree of a line bundle”.) We construct a meromorphic section corresponding to the plane x_{0} = 1 in 𝔸^{n} (draw picture). It you think about it, you might convince yourself that it blows up at the point x_{0} = 0 in ℙ^{n}, and nowhere else (and it has no zeroes).

Cover ℙ^{1} with two standard opens, U_{0} and U_{1}. On the first standard open, we have the section s_{0} = 1; it has no zeroes. On the second standard open, with coordinates (z_{0}; 1), the coordinates on the line is s_{1}, and points on the line are of the form (z_{0}s_{1}, s_{1}). Thus the section (1, 0) gives (in this coordinate system) s_{1} = 1/z_{0}. This has a simple pole at z_{0} = 0.

In short, this nonzero meromorphic section of O_{ℙ^{1}}(−1) has a single simple pole, so the sum of its orders of vanishing is -1. But we earlier checked that any nonzero meromorphic section of the trivial sheaf has sum of orders of vanishing 0. So they are not the same.

Coming next: The line bundle O_{ℙ^{n}}(m). Maps to projective space correspond to a vector space of sections of a line bundle. The canonical invertible sheaf, genus. Riemann-Roch Theorem: statement (no proof) and applications. Riemann-Hurwitz.