Abstract Manifolds and Varieties

11 Abstract Manifolds and Varieties

Abstract ∞ C -manifolds

An abstract ∞ C -manifold of dimension n consists of a topological space X together with an R-algebra ∞ C (V ) of continuous functions f : V → R for each open subset

V ⊂ X, such that the following conditions are satisfied.

(1) If U ∞ ⊂ V ⊂ X are open and f ∈ C (V ), then the restriction f | U is in C (U ). (2) If (U ∞

i ) is an open cover of an open subset V ⊂ X, and if f i ∈C (U i ) such that f i | U i ∩ U j =f j | U i ∩ U j for all i, j, then there exists a unique f

∈C ∞ (V ) such that

f i =f | U i for all i. (3) Each point P ∈ X has an open neighbourhood V together with a homeomor-

phism φ : V n →R , such that for any open subset U ⊂ V , a function f : U → R is

− 1 in ∞ C (U ) if and only if the function f ◦φ : φ(U ) → R is a C function on φ(U ). The ∞ C -functions on an open subset V ⊂ X are by definition the elements of

C ∞ (V ). A C -morphism φ : X → Y of C -manifolds is the same as a continuous map φ : X

→ Y such that for any open W ⊂ Y and any g ∈ C ∞ (W ), the composite

− g 1 ◦φ:φ (W ) → R is in C (φ W ).

11.1 ∞ Exercise Relate this definition of C -manifolds, C -functions and C -mor- phisms to the definition in terms of overlapping patches that you may have encoun-

tered previously.

Abstract holomorphic manifolds

An abstract holomorphic manifold of dimension n consists of a topological space X together with an C-algebra H(V ) of continuous functions f : V → C for each open subset V ⊂ X, such that the following conditions are satisfied.

(1) If U ⊂ V ⊂ X are open and f ∈ H(V ), then the restriction f| U is in H(U). (2) If (U i ) is an open cover of an open subset V ⊂ X, and if f i ∈ H(U i ) such

that f i | U i ∩ U j =f j | U i ∩ U j for all i, j, then there exists a unique f ∈ H(V ) such that

f i =f | U i for all i. (3) Each point P ∈ X has an open neighbourhood V together with a homeo-

morphism φ : V n → B where B is an open subset of C , such that for any open subset U ⊂ V , a function f : U → C is in H(U) if and only if the function

− f 1 ◦φ : φ(U ) → C is a holomorphic function on φ(U). The holomorphic functions on an open subset V ⊂ X are by definition the elements

of H(V ). A holomorphic morphism φ : X → Y of holomorphic manifolds is the same as a continuous map φ : X → Y such that for any open W ⊂ Y and any

− 1 − g 1 ∈ H(W ), the composite g ◦ φ : φ (W ) → C is in H(φ W ).

11.2 Exercise Relate this definition of holomorphic manifolds, holomorphic func- tions and holomorphic morphisms to the definition in terms of overlapping patches that you may have encountered previously.

Abstract varieties of Serre

It is an important property of quasi-affine and quasi-projective varieties that they admit open covers by affine subvarieties. This property was made the key point of the definition of an abstract variety by Serre.

An abstract variety in the sense of Serre is a topological space X together with a

C -algebra O(V ) of continuous functions f : V → C for each open V ⊂ X such that the following conditions are satisfied.

(1) If U ⊂ V ⊂ X are open and f ∈ O(V ), then the restriction f| U is in O(U). (2) If (U i ) is an open cover of an open subset V ⊂ X, and if f i ∈ O(U i ) such

that f i | U i ∩ U j =f j | U i ∩ U j for all i, j, then there exists a unique f ∈ O(V ) such that

f i =f | U i for all i. (3) Each point P ∈ X has an open neighbourhood V together with a homeomor-

phism φ : V → Y where Y is an affine variety, such that for any open subset U ⊂ V , − a function f : U 1 → C is in O(U) if and only if the function f ◦ φ : φ(U ) → C is a

regular function on the quasi-affine variety φ(U ).

The regular functions on an open subset V ⊂ X are by definition the elements of O(V ). A regular morphism φ : X → Y of varieties is the same as a continuous map φ : X → Y such that for any open W ⊂ Y and any g ∈ O(W ), the composite

− g 1 ◦φ:φ (W ) − → C is in O(φ 1 W ).

11.3 Exercise Give another equivalent definition of an abstract variety, regular functions and regular morphisms, in terms of overlapping affine open patches, in the spirit of the definition of a manifold in terms of overlapping coordinate patches.

11.4 Note In the above definition, one can replace the field C by any algebraically closed field k of arbitrary characteristic, to get the definition of the category of abstract varieties over k.

11.5 An abstract variety X is said to be non-singular at a point P if the local ring O X,P is regular. This generalises the earlier definition of non-singularity for quasi-projective varieties.

11.6 If X and Y are abstract varieties, then a product variety X × Y together with regular morphisms (projections) p 1 :X × Y → X and p 2 :X ×Y→Y exists in the category of abstract varieties. It is a simple exercise to manufacture (X × Y, p 1 ,p 2 ) by taking affine open covers (U i ) and (V j ) respectively of X and Y , taking the products U i ×V j of the affine open sets, and then gluing them together as in Exercise 11.3.

11.7 If the diagonal subvariety ∆ X ⊂ X × X is closed then X is called separated. Note that all quasi-projective varieties are automatically separated by Exercise 8.16.