Course Notes for Math 737 (Algebraic Geometry)

0.2 Pre-Introduction. Sheaves, Manifolds and Vector Bundles. We define sheaves, and investigate the two most important examples from geometry: the sheaf (of commutative rings with 1) of differentiable or holomorphic functions on a manifold and the sheaf (of abelian groups) of differentiable or holomorphic sections of a vector bundle.

What is a sheaf? Suppose that T is a topological space and that F is a presheaf of rings (or abelian groups) on T. The we say that F is a sheaf if it satisfies F(∅) = {0} and the following additional two properties:

Suppose ∪_{λ∈Λ}U_{λ} = U is an open cover of an open subset of T. Then:

(i) If s ∈ F(U) and ρ_{U,U_{λ}}(s) = 0 for all λ ∈ Λ, then s = 0.

(ii) If there exist s_{λ} ∈ F(U_{λ}) for all λ ∈ Λ such that for all λ, μ ∈ Λ, we have: ρ_{U_{λ},Uλ∩U_{μ}}(s_{λ}) = ρ_{U_{μ},U_{λ}∩U_{μ}}(s_{μ}), then there exists an s ∈ F(U) (necessarily unique by (i)!) such that ρ_{U,Uλ}(s) = s_{λ} for all λ ∈ Λ.

Example: The presheaf C_{T,R} of continuous functions from T to a continuous ring R (or continuous abelian group) is a sheaf. That’s because:

(i) If s : U → R and s|_{U_{λ}} ≡ 0 for all λ, then s is the zero function!

(ii) If s_{λ} : U_{λ} → R are functions for all λ, and if two such functions restrict to the same function on the overlaps, then they glue together to define a function s : U → R. (The value of s at a point is well-defined, as it doesn’t depend upon which open set U_{λ} contains that point!)

Lest you think every presheaf is a sheaf:

Nonexamples: (i) Suppose T is a topological space with the property that there exists an open cover T = ∪U_{λ} by proper open subsets of T. (Most topological spaces have this property!) Suppose that R is a nonzero ring, and let F be the presheaf defined by

F (U) = \{ \begin{matrix}0 if U ≠ T \\ R if U = T \end{matrix}

and all the ρ_{U,V} are the zero homomorphism (except for ρ_{T,T}, of course!) Then F violates (i) for the cover T = ∪U_{λ}, so it isn’t a sheaf.

(ii) Suppose T is a topological space which contains disjoint (nonempty!) open subsets U_{1} and U_{2} (some interesting topological spaces do not have this property!) and R is a nonzero ring. Let F be the presheaf defined by:

F (U) = \{ \begin{matrix}R if U ≠ ∅ \\ 0 if U = ∅ \end{matrix}

and let the ρ_{U,V} all be the identity, except for ρ_{U,∅}, which is forced to be zero. Then if we choose s_{1} = 0 ∈ F (U_{1}) and s_{2} ≠ 0 ∈ F (U_{2}), we cannot find an s ∈ F (U_{1} ∪ U_{2}) satisfying (ii). (Do note, though, that (i) is satisfied!)

On to some more interesting examples:

What is a differentiable manifold? An n-dimensional differentiable manifold is built out of open subsets of R^{n} and “gluing” diffeomorphisms:

(The Gluing construction) Given open subsets B_{1}, B_{2}, ... ⊆ R^{n} and open subsets U_{i,j} ⊂ B_{i} such that each U_{i,i} = B_{i}, and further, that each U_{i,j} is related to U_{j,i} by a diffeomorphism: ϕ_{i,j} : U_{i,j} →^{∼} U_{j,i} with the following properties:

(i) ϕ_{i,i} is the identity map for all i.

(ii) ϕ_{j,i} = ϕ_{i,j}^{−1} for all i, j and

(iii) ϕ_{i,j}(U_{i,j} ∩ U_{i,k}) = U_{j,i} ∩ U_{j,k} for all i, j, k and:

ϕ_{j,k} ∘ ϕ_{i,j} = ϕ_{i,k} as functions from U_{i,j} ∩ U_{i,k} to B_{k}.

Then we build a topological space M as follows:

M = ∐_{i=1}^{∞} B_{i}/ ∼ where p ∼ q ⇔ p ∈ U_{i,j}, q ∈ U_{j,i} and ϕ_{i,j}(p) = q.

The conditions (i)-(iii) imply that ∼ is an equivalence relation and that the natural inclusions B_{i} ↪ ∐ B_{i} induce inclusions B_{i} ↪ M. We will be sloppy and identify each B_{i} with its image. We put the topology on M generated by the open subsets of the B_{i}. The fact that the ϕ_{i,j} are homeomorphisms implies that U ⊂ M is open if and only if U ∩ B_{i} is open in B_{i} for all i. Finally, observe that U_{i,j} = B_{i} ∩ B_{j}, when the intersection is thought of as a subset of B_{i}.

Forming a topological space M in this way allows us to define C^{∞} functions on each open subset of M. This information is packaged as a sheaf:

Definition: The presheaf of C^{∞} functions on a topological space M constructed as above is denoted by C_{M}^{∞} and is defined as follows:

(a) C_{M}^{∞}(U) = {f : U → R | f|_{U∩B_{i}} is C^{∞} on U ∩ B_{i} ⊂ B_{i} for each i.}

(b) ρ_{U,V} is the restriction of functions.

C_{M}^{∞} is a sheaf of commutative rings with 1. This is proved exactly as in the earlier example of the sheaf C_{T,R} of continuous functions.

Definition: A differentiable premanifold is a topological space M together with a sheaf C_{M}^{∞} which is constructed as above.

So why don’t we call these things manifolds? It’s because of the:

Bad Example: Suppose B_{1} = R = B_{2}, U_{1,2} = U_{2,1} = R − {0}, and ϕ_{1,2} = id = ϕ_{2,1}. Then the gluing construction gives an M which looks like R with two origins. This isn’t Hausdorff!

A manifold is a Hausdorff premanifold. More precisely, suppose:

(i) N is a Hausdorff topological space,

(ii) C_{N}^{∞} is a sheaf of commutative rings with 1 which is a subsheaf of C_{N,R}. (By this I mean that each s ∈ C_{N}^{∞}(U) is a continuous function s : U → R and the restriction map on C_{N}^{∞} is just restriction of functions.)

Then:

Defintion: The pair (N, C_{N}^{∞}) is a differentiable manifold if there is a premanifold M and a homeomorphism h : N → M such that for every open subset U ⊂ M, the natural “pull-backs” h^{∗} : C_{M,R}(U) → C_{N,R}(h^{−1}(U)) restrict to isomorphisms:

h^{∗} : C_{M}^{∞}(U) →^{∼} C_{N}^{∞}(h^{−1}(U))

Note:: It is certainly not a given that the image of C_{M}^{∞}(U) lies in C_{N}^{∞}(h^{−1}(U)) under h^{∗}, though it is a given that the restriction of h^{∗} to C_{M}^{∞}(U) is injective (but not that it is surjective!).

Notice also that the essential data in defining a manifold consists of a topological space and a sheaf.

Definition: If (N, C_{N}^{∞}) and (N′, C_{N′}^{∞}) are differentiable manifolds, then a map f : N → N′ is called a differentiable map if

(i) f is continuous, and

(ii) the image of each C_{N′}^{∞}(U) under the pull-back f^{∗} is in C_{N}^{∞}(f^{−1}(U)) (informally: “f^{∗} pulls back C^{∞} functions to C^{∞} functions”.)

Remark: One needs to check, of course, that this is well-defined. That is, if (N, C_{N}^{∞}) is a differentiable manifold in more than one way, for example, via homeomorphisms h : N → M and h′ : N → M′ to premanifolds, then the definition of a differentiable map is unaffected, and similarly for N′. But this is easily checked! Also notice that in spite of its importance, we will often leave the sheaf out of our notation, usually referring to a manifold as “M”.

Extension: If the subsets B_{i} are in R^{2n} = C^{n} and the ϕ_{i,j} are holomorphic, then one defines M in the same way, and the sheaf O_{M} of holomorphic functions on M analogously. Complex premanifolds, complex manifolds and holomorphic maps of complex manifolds are all defined analogously.

Definition: (a) The collection of differentiable manifolds M together with differentiable maps is often called “the differentiable (or C^{∞}) category”.

(b) The collection of complex manifolds M together with holomorphic maps is often called “the holomorphic category”.

Observation: If M is a differentiable manifold, then each open U ⊂ M admits a sheaf of C^{∞} functions which makes it a manifold. Its topology is the induced topology, and its sheaf of C^{∞} functions is the sheaf C_{U}^{∞} defined by:

C_{U}^{∞}(V) := C_{M}^{∞}(V) for all open subsets V ⊂ U.

The inclusion map U ↪ M becomes a differentiable map, of course.

The same method works in the holomorphic category. Please do notice (or remember) that there is no analogous method for making closed subsets of M into manifolds (even though they do inherit an induced topology).

We can define “germs ” of functions on a manifold as a special case of:

Definition: Suppose F is a presheaf of rings (or abelian groups) on T and that p ∈ T. Then the stalk of F at p is the direct limit:

F_{p} := lim_{→} F (U)

over all open sets U containing p.

That is, a pair (U, s ∈ F(U)) determines an element of F_{p}, and two pairs (U, s) and (V, t) determine the same element of the stalk iff there is an open subset W ⊂ U ∩ V such that ρ_{U,W} (s) = ρ_{V,W} (t).

Germs: If M is a differentiable (resp. complex) manifold, then an element of a stalk C_{M,p}^{∞} (resp. O_{M,p}) is called a germ of functions at p.

Observations: (1) The evaluation at p of a germ of functions at p is welldefined. In the C^{∞} case, this is a real number, while in the holomorphic case, it is a complex number. This is obvious from the definition.

(2) The rings of germs: C_{M,p}^{∞} or O_{M,p} are local rings. The maximal ideal is the set of germs that evaluate to 0 at p. Any other germ has a “germ” reciprocal, because a function which is nonzero at p is nonzero in a neighborhood of p (continuity!).

What is a Vector Bundle? Suppose M is a C^{∞} manifold of dimension n.

Suppose that M = ∪_{i=1}^{∞}U_{i} is an open cover of M by countably many open subsets, and that for each i, j, there are C^{∞} “transition functions”:

g_{i,j} : U_{i} ∩ U_{j} → GL(r, R).

(To be C^{∞} means that each coordinate of the matrix is in C_{M}^{∞}(U_{i} ∩ U_{j}).) Suppose that the maps satisfy:

(i) g_{i,i} is the constant function mapping to the identity matrix for all i,

(ii) g_{i,j}(p) = g_{j,i}^{−1}(p) for all i, j and p ∈ U_{i} ∩ U_{j} and

(iii) For all i, j, k and p ∈ U_{i} ∩ U_{j} ∩ U_{k}, we have:

g_{j,k}(p) ∗ g_{i,j}(p) = g_{i,k}(p).

Then we build a space E and map π : E → M as follows. As a set,

E = ∐_{i=1}^{∞} U_{i} × R^{r}/ ∼

where (p, v) ∈ U_{i} × R^{r} and (q, w) ∈ U_{j} × R^{r} are equivalent if and only if p = q ∈ U_{i} ∩ U_{j}, and g_{i,j}(p)(v) = w. Notice that ∼ is an equivalence relation because of (i),(ii) and (iii). Moreover, notice that the map:

π : E → M, given by π\overline{(p, v)} = p

is well-defined. As in the gluing construction of M, the sets U_{i} × R^{r} are naturally subsets of E, and E is a topological space, with U ⊂ E open if and only if U ∩ (U_{i} × R^{r}) is open for all i. You check that E is Hausdorff.

If V ⊂ U_{i} × R^{r}, we’ll say f : V → R is a C^{∞} function if f|_{V ∩(B_{j}×R^{r})} is C^{∞} for all B_{j}. Then define the “sheaf of C^{∞} functions on E” by letting:

C_{E}^{∞}(U) = {f : U → R | f|_{U∩(U_{i}×R^{r})} is a C^{∞} function for all i}.

Lemma: E together with its sheaf C_{E}^{∞} of C^{∞} functions is a differentiable manifold and π : E → M is a differentiable map.

Proof: Exercise.

Definition: Suppose M and F are differentiable manifolds and σ : F → M is a C^{∞} map. Then we say that the pair (F, σ) is a (geometric) vector bundle over M if there is a pair (E, π) built out of transition functions as above, and a diffeomorhism h : F → E such that π ∘ h = σ.

The Tangent Bundle. (The most important example!) Let M be a manifold constructed out of the gluing functions:

ϕ_{i,j} : U_{i,j} → U_{j,i}

and let ϕ_{i,j}^{k} denote the k-th component of ϕ_{i,j}, thought of as a map to R^{n}. Let M = ∪_{i=1}^{∞}U_{i} be the open cover by subsets U_{i} = B_{i} (identified with their images in M), so that U_{i} ∩ U_{j} = U_{i,j}, and let:

g_{i,j} = (\frac{∂ϕ_{i,j}^{k}}{∂x_{l}} ) : U_{i} ∩ U_{j} → GL(n, R).

Then the functions g_{i,j} satisfy all the desired properties for transition functions. This is because of the chain rule. The vector bundle built out of these transition functions is called the tangent bundle and is denoted by T_{M} (the map to M is almost always denoted by π).

Suppose M is a differentiable manifold and π : E → M is a vector bundle.

Definition: If U ⊂ M is an open set, then a C^{∞} section of E over U is a differentiable map:

s : U → π^{−1}(U)

with the property that π|_{π^{−1}(U)} ∘ s : U → U is the identity.

Let E(U) be the set of C^{∞} sections of E over U. Define operations:

+ : E(U) × E(U) → E(U) and ∗ : C_{M}^{∞}(U) × E(U) → E(U)

as follows. For each p ∈ U, choose a some U_{i} such that p ∈ U_{i}, and consider U_{i} × R^{r} ↪ E. It follows that for each s, t ∈ E(U), we can write s(p) = (p, v) and t(p) = (p, w) for some vectors v, w ∈ R^{n}. If in addition, f ∈ C_{M}^{∞}(U), then define:

(s + t)(p) := (p, v + w) and fs(p) = (p, f(p) · v)

Of course, this definition involved a choice of U_{i}. But since the transition functions g_{i,j} take values in GL(r, R), it follows that this definition is independent of the choice of U_{i}. Moreover, it clearly takes sections to sections and preserves the C^{∞}-ness of the section since that is locally defined, anyway(!)

Remark: Notice that there is a well-defined zero section of E over each open U ⊂ M, but there is no a priori reason why, for instance, the group E(M) ≠ {0} (and it often is in the holomorphic category!).

Propostion: The assignments:

U ↦ E(U) together with “+” and ordinary restriction of sections

defines a sheaf of abelian groups (which we denote by E.) This is called the sheaf of C^{∞} sections of E.

Proof: Everything is nearly obvious. The sheaf property follows just as in the earlier examples of sheaves of functions, because as in the case of functions, a section is zero if it is zero locally, and it is defined if it is defined locally in such a way that the local definitions agree on the overlaps!!

But we can do a little better than this!

Definition: Suppose T is a topological space, O is a sheaf of commutative rings with 1 on T, and F is a sheaf of abelian groups on T. Then F is called a sheaf of O-modules if:

(i) Each F(U) has the structure of a O(U)-module such that:

(ii) For each V ⊂ U and s ∈ F(U), f ∈ O(U), we have:

ρ_{U,V} (f) · ρ_{U,V} (s) = ρ_{U,V} (f · s).

Proposition: Given E, then the structure given above for E(U) as a C_{M}^{∞}(U)- module makes E into a sheaf of C_{M}^{∞}-modules.

Proof: Obvious from the definitions.

A final remark/definition. If U ⊂ U_{i} for some U_{i}, then E(U) is isomorphic to the free module (C_{M}^{∞}(U_{i}))^{⊕r} of rank r. That’s because when we indentify U_{i} × R^{r} with π^{−1}(U_{i}), then the C^{∞} sections can be identified with the C^{∞} maps:

s = (s_{1}, ..., s_{r}) : U_{i} → R^{r}

In general, if F is a sheaf of O-modules with this property (i.e. that there is a cover of T by open subsets U_{i} such that each F(U_{i}) is free), then we will say that F is a locally free sheaf of O-modules.

Definition: Θ_{M} is the sheaf of C^{∞} sections of T_{M}.

And all this can be similarly defined in the holomorphic category!

Some Manifold and Vector Bundle Examples

Example 1: Projective Space. (Compare with Property 1 in §1) We are given B_{0}, ..., B_{n} ⊂ R^{n+1} defined as: B_{i} := {(a_{0}, ..., a_{i−1}, 1, a_{i+1}, ..., a_{n})}. Of course, these are not open in R^{n+1}, but they are each equal to R^{n}, when we just forget the ith coordinate. The open subsets are:

U_{i,j} := {(a_{0}, ..., a_{n}) ∈ B_{i} |a_{j} ≠ 0}

and the gluing functions are:

ϕ_{i,j}(a_{0}, ..., a_{n}) = (\frac{a_{0}}{a_{j}} , ..., \frac{a_{n}}{a_{j}})

which do indeed map U_{i,j} to U_{j,i} are are quite definitely C^{∞} mappings (and holomorphic if we started with B_{i} ⊂ C^{n+1} with the same definition).

Form P_{R}^{n} by the gluing construction.

Exercises: (a) P_{R}^{n} is Hausdorff. Hence a C^{∞} manifold.

(b) The map π : R^{n+1} − (0, ..., 0) → P_{R}^{n} defined by:

π(a_{0}, ..., a_{n}) = (\frac{a_{0}}{a_{i}} , ..., \frac{a_{n}}{a_{i}}) ∈ B_{i} if a_{i} ≠ 0

is well-defined and differentiable, as is the induced map from S^{n} to P_{R}^{n}, which proves that P_{R}^{n} is compact. (Property 3 from §1).

(c) Convince yourself that all this works for R replaced by C, giving rise to the complex manifold P_{C}^{n} which is likewise compact.

Specific Example: The Riemann Sphere. You may have learned about the Riemann Sphere in another context, namely, as C ∪ {∞}, where:

(i) neighborhoods of ∞ are complements of closed, bounded sets in C,

(ii) f(z) is holomorphic at ∞ if f(\frac{1}{z}) is holomorphic at 0, etc.

Notice that this is the same information provided by P_{C}^{1} with its sheaf O_{P_{C}^{1}} of holomorphic functions. In this case, P_{C}^{1} is formed from B_{0}, B_{1} = C with gluing functions ϕ_{0,1}(z) = \frac{1}{z} and ϕ_{1,0}(w) = \frac{1}{w}. Notice also:

(a) O_{P_{C}^{1}}(P_{C}^{1}) = C, the constant functions. Suppose s ∈ O_{P_{C}^{1}}. Then the two restrictions s_{0} := ρ_{P^{1},B_{0}}(s) and s_{1} := ρ_{P^{1},B_{1}}(s) must be holomorphic functions of z and w, respectively. Thus,

s_{0}(z) = a_{0} + a_{1}z + a_{2}z^{2} + .... and s_{1}(w) = b_{0} + b_{1}w + b_{2}w^{2} + ...

But because they “glue together” to the global section s, it follows that s_{1}(\frac{1}{z}) = s_{0}(z) on the overlap, i.e. a_{0} + a_{1}z + ... = b_{0} + b_{1}z^{−1} + ... so that a_{0} = b_{0} and all the other terms are zero(!)

(b) Θ_{P_{C}^{1}}(P_{C}^{1}) ≅ C^{3}, as a vector space over the constant functions(!) (Recall that Θ_{M} is a sheaf of O_{M}-modules, so that when O_{M}(M) is a field, then Θ_{M}(M) is a vector space.) Suppose that t ∈ Θ_{P_{C}^{1}}(P_{C}^{1}). Then the restrictions t_{0}, t_{1} to B_{0} and B_{1}, repsectively, may be thought of as functions t_{0} : B_{0} → C and t_{1} : B_{1} → C that satisfy the following on the overlap:

t_{1}(\frac{1}{z}) = (\frac{∂w}{∂z} ) t_{0}(z) = −\frac{1}{z^{2}} t_{0}(z)

or, if t_{0}(z) = a_{0} + a_{1}z + ... and t_{1}(w) = b_{0} + b_{1}w + ..., then:

a_{0}z^{−2} + a_{1}z^{−1} + a_{2} + a_{3}z + ... = b_{0} + b_{1}z^{−1} + b_{2}z^{−2} + b^{3}z^{−3} + ...

so that a_{0} = b_{2}, a_{1} = b_{1}, a_{2} = b_{0} and all other terms are zero(!)

This is the first case of the following important “rigidity” properties of compact complex manifolds (which we will state without proof). Namely, let M be a compact complex manifold. Then:

Rigidity 1: O_{M}(M) = C, the constant functions.

Rigidity 2: Let π : E → M be any holomorphic vector bundle on M. Then E(M) is a finite-dimensional vector space over C.

Note that this rigidity does NOT apply to differentiable manifolds M. (Hint: Think about bump functions!)

Final Exercise: Prove by hand (as above) that:

(a) O_{P_{C}^{n}}(P_{C}^{n}) = C and (b) Θ_{P_{C}^{n}}(P_{C}^{n}) ≅ C^{n^{2}+2n}