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数学代写|复几何代写Complex Geometry代考|MATH3033 Manifolds and Varieties via Sheaves

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数学代写|复几何代写Complex Geometry代考|Sheaves of Functions

As we said above, we need to define sheaves in order eventually to define manifolds and varieties. We start with a more primitive notion. In many parts of mathematics, we encounter topological spaces with distinguished classes of functions on them: continuous functions on topological spaces, $C^{\infty}$-functions on $\mathbb{R}^n$, holomorphic functions on $\mathbb{C}^n$, and so on. These functions may have singularities, so they may be defined only over subsets of the space; we will be interested primarily in the case that these subsets are open. We say that such a collection of functions is a presheaf if it is closed under restriction. Given sets $X$ and $T$, let $\operatorname{Map}_T(X)$ denote the set of maps from $X$ to $T$. Here is the precise definition of a presheaf, or rather of the kind of presheaf we need at the moment.

Definition 2.1.1. Suppose that $X$ is a topological space and $T$ a nonempty set. A presheaf of $T$-valued functions on $X$ is a collection of subsets $\mathscr{P}(U) \subseteq \operatorname{Map}_T(U)$, for each open $U \subseteq X$, such that the restriction $\left.f\right|_V$ belongs to $\mathscr{P}(V)$ whenever $f \in \mathscr{P}(U)$ and $V \subset U$

The collection of all functions $\operatorname{Map}T(U)$ is of course a presheaf. Less trivially: Example 2.1.2. Let $T$ be a topological space. Then the set of continuous functions Cont ${X, T}(U)$ from $U \subseteq X$ to $T$ is a presheaf.

Example 2.1.3. Let $X$ be a topological space and let $T$ be a set. The set $T^P(U)$ of constant functions from $U$ to $T$ is a presheaf called the constant presheaf.

Example 2.1.4. Let $X=\mathbb{R}^n$. The sets $C^{\infty}(U)$ of $C^{\infty}$ real-valued functions form a presheaf.

Example 2.1.5. Let $X=\mathbb{C}^n$. The sets $\mathscr{O}(U)$ of holomorphic functions on $U$ form a presheaf. (A function of several variables is holomorphic if it is $C^{\infty}$ and holomorphic in each variable.)

Example 2.1.6. Let $L$ be a linear differential operator on $\mathbb{R}^n$ with $C^{\infty}$ coefficients (e.g., $\Sigma \partial^2 / \partial x_i^2$ ). Let $S(U)$ denote the space of $C^{\infty}$ solutions to $L f=0$ in $U$. This is a presheaf with values in $\mathbb{R}$.

Example 2.1.7. Let $X=\mathbb{R}^n$. The sets $L^p(U)$ of measurable functions $f: U \rightarrow \mathbb{R}$ satisfying $\int_U|f|^p<\infty$ form a presheaf.

数学代写|复几何代写Complex Geometry代考|Manifolds

As explained in the introduction, a manifold consists of a topological space with a distinguished class of functions that looks locally like $\mathbb{R}^n$. We now set up the language necessary to give a precise definition. Let $k$ be a field. Then $\operatorname{Map}_k(X)$ is a commutative $k$-algebra with pointwise addition and multplication.

Definition 2.2.1. Let $\mathscr{R}$ be a sheaf of $k$-valued functions on $X$. We say that $\mathscr{R}$ is a sheaf of algebras if each $R(U) \subseteq \operatorname{Map}_k(U)$ is a subalgebra when $U$ is nonempty. We call the pair $(X, \mathscr{R})$ a concrete ringed space over $k$ or simply a concrete $k$-space. We will sometimes refer to elements of $\mathscr{R}(U)$ as distinguished functions.

The sheaf $\mathscr{R}$ is called the structure sheaf of $X$. In this chapter, we usually omit the modifier “concrete,” but we will use it later on after we introduce a more general notion. Basic examples of $\mathbb{R}$-spaces are $\left(\mathbb{R}^n\right.$, Cont $\left._{\mathbb{R}^n, \mathbb{R}}\right)$ and $\left(\mathbb{R}^n, C^{\infty}\right)$, while $\left(\mathbb{C}^n, \mathscr{O}\right)$ is an example of a $\mathbb{C}$-space.

We also need to consider maps $F: X \rightarrow Y$ between such spaces. We will certainly insist on continuity, but in addition we require that when a distinguished function is precomposed with $F$, or “pulled back” along $F$, it remain distinguished.
Definition 2.2.2. A morphism of $k$-spaces $(X, \mathscr{R}) \rightarrow(Y, \mathscr{S})$ is a continuous map $F: X \rightarrow Y$ such that if $f \in \mathscr{S}(U)$, then $F^* f \in \mathscr{R}\left(F^{-1} U\right)$, where $F^* f=\left.f \circ F\right|_{f^{-1} U}$.
It is worthwhile noting that this completely captures the notion of a $C^{\infty}$, or holomorphic, map between Euclidean spaces.
Example 2.2.3. A $C^{\infty}$ map $F: \mathbb{R}^n \rightarrow \mathbb{R}^m$ induces a morphism $\left(\mathbb{R}^n, C^{\infty}\right) \rightarrow\left(\mathbb{R}^m, C^{\infty}\right)$ of $\mathbb{R}$-spaces, since $C^{\infty}$ functions are closed under composition. Conversely, if $F$ is a morphism, then the coordinate functions on $\mathbb{R}^m$ are expressible as $C^{\infty}$ functions of the coordinates of $\mathbb{R}^n$, which implies that $F$ is $C^{\infty}$.

Example 2.2.4. Similarly, a continuous map $F: \mathbb{C}^n \rightarrow \mathbb{C}^m$ induces a morphism of $\mathbb{C}$-spaces if and only if it is holomorphic.

This is a good place to introduce, or perhaps remind the reader of, the notion of a category [82]. A category $\mathscr{C}$ consists of a set (or class) of objects $\mathrm{Obj} \mathscr{C}$ and for each pair $A, B \in \mathscr{C}$, a set $\operatorname{Hom}{\mathscr{C}}(A, B)$ of morphisms from $A$ to $B$. There is a composition law $$\text { o: } \operatorname{Hom}{\mathscr{C}}(B, C) \times \operatorname{Hom}{\mathscr{C}}(A, B) \rightarrow \operatorname{Hom}{\mathscr{C}}(A, C),$$

and distinguished elements $i d_A \in \operatorname{Hom}{\mathscr{C}}(A, A)$ that satisfy (C1) associativity: $f \circ(g \circ h)=(f \circ g) \circ h$, (C2) identity: $f \circ \mathrm{id}_A=f$ and $\mathrm{id}_A \circ g=g$, whenever these are defined. Categories abound in mathematics. A basic example is the category of Sets. The objects are sets, $\operatorname{Hom}{\text {Sets }}(A, B)$ is just the set of maps from $A$ to $B$, and composition and $\mathrm{id}_A$ have the usual meanings. Similarly, we can form the category of groups and group homomorphisms, the category of rings and ring homomorphisms, and the category of topological spaces and continuous maps. We have essentially constructed another example. We can take the class of 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 this out for $k$-spaces.

数学代写|复几何代写Complex Geometry代考|Manifolds

o: $\operatorname{Hom} \mathscr{C}(B, C) \times \operatorname{Hom} \mathscr{C}(A, B) \rightarrow \operatorname{Hom} \mathscr{C}(A, C)$,

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