Chapter 1 Manifolds and Varieties via Sheaves In rough terms, a manifold is a topological space along with a distinguished collection of functions, which looks locally like Euclidean space. Although it is rarely presented this way in introductory texts (e. g. [Spv, Wa]), sheaf theory is a natural language in which to make such a notion precise. An algebraic variety can be de¯ned similarly as a space which looks locally like the zero set of a collection of polynomials. The sheaf theoretic approach to varieties was introduced by Serre in the early 1950's, this approach was solidi¯ed with the work of Grothendieck shortly thereafter, and algebraic geometry has never been the same since. 1.1 Sheaves of functions In many parts of mathematics, we encounter spaces with distinguished classes of functions on them. When these classes are closed under restriction, as they often are, then they give rise to presheaves. More precisely, let X be a topological space, and T a set. For each open set U X, let MapT (U) be the set of maps from U to T . De¯nition 1.1.1. A collection of subsets P (U) ½ MapT (U), with U ½ X nonempty open, is called a presheaf of ( T -valued) functions on X, if it is closed under restriction, i. e. if f 2 P (U) and V ½ U then fjV 2 P (V ). If the de¯ning conditions for P (U) are local, which means that they can be checked in a neighbourhood of a point, then the presheaf is called sheaf. Or to put it another way: De¯nition 1.1.2. A presheaf of functions P is called a sheaf if f 2 P (U) whenever there is an open cover fUig of U such that fjUi 2 P (Ui). 7 Example 1.1.3. Let PT (U) be the set of constant functions from U to T . This is a presheaf but not a sheaf in general. Example 1.1.4. A function is locally constant if it is constant in a neigh- bourhood of a point. The set of locally constant functions, denoted by T (U) or TX (U), is a sheaf. It is called the constant sheaf. Example 1.1.5. Let T be another topological space, then the set of continuous functions ContX;T (U) from U X to T is a sheaf. When T is discrete, this coincides with the previous example. Example 1.1.6. Let X = Rn, the sets C1(U) of C1 real valued functions form a sheaf. Example 1.1.7. Let X = C (or Cn), the sets O(U) of holomorphic functions on U form a sheaf. Example 1.1.8. Let L be a linear di®erential operator on Rn with C1 coef- 2 2 1 ¯cients (e. g. @ =@xi ). Let S(U) denote the space of C solutions in U. This is a sheaf. P Example 1.1.9. Let X = Rn, the sets L1(U) of L1-functions forms a presheaf which is not a sheaf. We can always force a presheaf to be a sheaf by the following construction. Example 1.1.10. Given a presheaf P of functions to T . De¯ne the s P (U) = ff : U ! T j 8x 2 U; 9 a neighbourhood Ux of x, such that fjUx 2 P (Ux)g This is a sheaf called the shea¯¯cation of P . When P is a presheaf of constant functions, P s is exactly the sheaf of locally constant functions. When this construction is applied to the presheaf L1, we obtain the sheaf of locally L1 functions. Exercise 1.1.11. 1. Check that P s is a sheaf. 2. Let ¼ : B ! X be a surjective continuous map of topological spaces. Prove that the presheaf of sections B(U) = f : U ! B j continuous, 8x 2 U; ¼ ± (x) = xg is a sheaf. 3. Let F : X ! Y be surjective continuous map. Suppose that P is a sheaf of T -valued functions on X. De¯ne f 2 Q(U) ½ MapT (U) if and only if ¤ ¡1 its pullback F f = f ± F jf ¡1U 2 P (F (U)). Show that Q is a sheaf on Y . 8 4. Let Y ½ X be a closed subset of a topological space. Let P be a sheaf of T -valued functions on X. For each open U ½ Y , let PY (U) be the set of functions f : U ! T locally extendible to an element of P , i.e. f 2 PY (U) if and only there for each y 2 U, there exists a neighbourhood V ½ X and an element of P (V ) restricting to fjV \U . Show that PY is a sheaf. 1.2 Manifolds Let k be a ¯eld. De¯nition 1.2.1. Let R be a sheaf of k-valued functions on X. We say that R is a sheaf of algebras if each R(U) Mapk(U) is a subalgebra. We call the pair (X; R) a concrete ringed space over k, or simply a k-space. n n 1 n (R ; CR), (R ; C ) and (C ; O) are examples of R and C-spaces. De¯nition 1.2.2. A morphism of k-spaces (X; R) ! (Y; S) is a continuous map F : X ! Y such that f 2 S(U) implies F ¤f 2 R(F ¡1U): This is good place to introduce, or perhaps remind the reader of, the notion of a category. A category C consists of a set (or class) of objects ObjC and for each pair A; B 2 C, a set HomC(A; B) of morphisms from A to B. There is a composition law ± : HomC(B; C) £ HomC(A; B) ! HomC(A; C); and distinguished elements idA 2 HomC(A; A) which satisfy 1. associativity: f ± (g ± h) = (f ± g) ± h, 2. identity: f ± idA = f and idA ± g = g, whenever these are de¯ned. Categories abound in mathematics. A basic example is the category of Sets consisting of the class of all sets, HomSets(A; B) is just the set of maps from A to B, and composition and idA have the usual meanings. Similarly, we can form the category of groups and group homomorphisms, the category of rings and rings homomorphisms, and the category of topological spaces and continuous maps. We have essentially constructed another example. We can take the objects to be k-spaces, and morphisms as above. These can be seen to constitute a category, once we observe that the identity is a morphism and the composition of morphisms is a morphism. The notion of an isomorphism makes sense in any category, we will spell in the above example. De¯nition 1.2.3. An isomorphism of k-spaces (X; R) =» (Y; S) is a homeo- morphism F : X ! Y such that f 2 S(U) if and only if F ¤f 2 R(F ¡1U). Given a sheaf S on X and open set U ½ X, let SjU denote the sheaf on U de¯ned by V 7! S(V ) for each V U. 9 1 1 De¯nition 1.2.4. An n-dimensional C manifold is an R-space (X; CX ) such that 1. The topology of X is given by a metric1. 1 2. X admits an open covering fUig such that each (Ui; CX jUi ) is isomorphic 1 n to (Bi; C jBi ) for some open ball B ½ R . 1 » 1 The isomorphisms (Ui; C jUi ) = (Bi; C jBi ) correspond to coordinate charts in more conventional treatments. The whole collection of data is called an atlas. There a number of variations on this idea: De¯nition 1.2.5. 1. An n-dimensional topological manifold is de¯ned as n 1 n above but with (R ; C ) replaced by (R ; ContRn;R). 2. An n-dimensional complex manifold can be de¯ned by replacing (Rn; C1) by (Cn; O). One dimensional complex manifolds are usually called Riemann surfaces. De¯nition 1.2.6. A C1 map from one C1 manifold to another is just a morphism of R-spaces. A holomorphic map between complex manifolds is de¯ned as a morphism of C-spaces. C1 (respectively complex) manifolds and maps form a category; an isomor- phism in this category is called a di®eomorphism (respectively biholomorphism). By de¯nition any point of manifold has neighbourhood di®eomorphic or biholo- morphic to a ball. Given a complex manifold (X; OX ), we say that f : X ! R is C1 if and only if f ± g is C1 for each holomorphic map g : B ! X from a ball in Cn. We state for the record: Lemma 1.2.7. An n-dimensional complex manifold together with its sheaf of C1 functions is a 2n-dimensional C1 manifold. Let us consider some examples of manifolds. Certainly any open subset of Rn (Cn) is a (complex) manifold in an obvious fashion. To get less trivial examples, we need one more de¯nition. De¯nition 1.2.8. Given an n-dimensional manifold X, a closed subset Y ½ X is called a closed m-dimensional closed submanifold if for any point x 2 Y , there exists a neighbourhood U of x in X and a di®eomorphism of to a ball B ½ Rn such that Y \ U maps to the intersection of B with an m-dimensional linear space. 1 Given a closed submanifold Y ½ X, de¯ne CY to be the sheaf of functions which are locally extendible to C1 functions on X. For a complex submanifold Y ½ X, we de¯ne OY to be the sheaf of functions which locally extend to holomorphic functions.