Posted on Categories:Complex Geometry, 复几何, 数学代写

# 数学代写|复几何代写Complex Geometry代考|COMP5045 Direct and Inverse Images

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## 数学代写|复几何代写Complex Geometry代考|Direct and Inverse Images

It is useful to transfer a sheaf from one topological space to another via a continuous map $f: X \rightarrow Y$. In fact, we have already done this in special cases. We start by explaining how to push a sheaf on $X$ down to $Y$.

Definition 3.7.1. Given a presheaf $\mathscr{F}$ on $X$, the direct image $f_* \mathscr{F}$ is a presheaf on $Y$ given by $f_* \mathscr{F}(U)=\mathscr{F}\left(f^{-1} U\right)$ with restrictions given by
$$\rho_{f^{-1} U f^{-1} V}: \mathscr{F}\left(f^{-1} U\right) \rightarrow \mathscr{F}\left(f^{-1} V\right) .$$
Lemma 3.7.2. Direct images of sheaves are sheaves.
Proof. Suppose that $f: X \rightarrow Y$ is a continuous map and $\mathscr{F}$ is a sheaf on $X$. Let $\left{U_i\right}$ be an open cover of $U \subseteq Y$, and $s_i \in f_* \mathscr{F}\left(U_i\right)$ a collection of sections that agree on the intersections. Then $\left{f^{-1} U_i\right}$ is an open cover of $f^{-1} U$, and we can regard $s_i \in \mathscr{F}\left(f^{-1} U_i\right)$ as a compatible collection of sections for it. Thus we can patch $s_i$ to get a uniquely defined $s \in f_* \mathscr{F}(U)=\mathscr{F}\left(f^{-1} U\right)$ such that $\left.s\right|{U_i}=s_i$. This proves that $f* \mathscr{F}$ is a sheaf.

Now we want to consider the opposite direction. Suppose that $\mathscr{G}$ is a sheaf on $Y$. We would like to pull it back to $X$. We will denote this by $f^{-1} \mathscr{G}$, since $f^*$ is reserved for something related to be defined later on. Naively, we can simply try to define
$$f^{-1} \mathscr{G}(U)=\mathscr{G}(f(U)) .$$
However, this does not yet make sense unless $f(U)$ is open. So as a first step, given any subset $S \subset Y$ of a topological space and a presheaf $\mathscr{G}$, define
$$\mathscr{G}(S)=\lim _{\rightarrow} \mathscr{G}(V)$$
as $V$ ranges over all open neighborhoods of $S$. When $S$ is a point, $\mathscr{G}(S)$ is just the stalk. An element of $\mathscr{G}(S)$ can be viewed as germ of a section defined in a neighborhood of $S$, where two sections define the same germ if their restrictions agree in a common neighborhood. If $S^{\prime} \subset S$, there is a natural restriction map $\mathscr{G}(S) \rightarrow \mathscr{G}\left(S^{\prime}\right)$ given by restriction of germs. So our naive attempt can now be made precise.

## 数学代写|复几何代写Complex Geometry代考|Differentials

With basic sheaf theory in hand, we can now construct sheaves of differential forms on manifolds and varieties in a unified way. In order to motivate things, let us start with a calculation. Suppose that $X=\mathbb{R}^n$ with coordinates $x_1, \ldots, x_n$. Given a $C^{\infty}$ function $f$ on $X$, we can develop a Taylor expansion about $\left(y_1, \ldots, y_n\right)$ :
$$f\left(x_1, \ldots, x_n\right)=f\left(y_1, \ldots, y_n\right)+\sum \frac{\partial f}{\partial x_i}\left(y_1, \ldots, y_n\right)\left(x_i-y_i\right)+O\left(\left(x_i-y_i\right)^2\right)$$
Thus the differential is given by
$$d f=f\left(x_1, \ldots, x_n\right)-f\left(y_1, \ldots, y_n\right) \bmod \left(x_i-y_i\right)^2 .$$
We can view $x_1, \ldots, x_n, y_1, \ldots, y_n$ as coordinates on $X \times X=\mathbb{R}^{2 n}$, so that $x_i-y_i=0$ defines the diagonal $\Delta$. Then $d f$ lies in the ideal of $\Delta$ modulo its square.

Let $X$ be a $C^{\infty}$ or complex manifold or an algebraic variety over a field $k$. We take $k=\mathbb{R}$ or $\mathbb{C}$ in the first two cases. We have a diagonal map $\delta: X \rightarrow X \times X$ given by $x \mapsto(x, x)$, and projections $p_i: X \times X \rightarrow X$. Let $\mathscr{I}{\Delta}$ be the ideal sheaf of the image of $\delta$, and let $\mathscr{I}{\Delta}^2 \subseteq \mathscr{I}{\Delta}$ be the sub-ideal sheaf locally generated by products of pairs of sections of $\mathscr{I}{\Delta}$. Then we define the sheaf of 1 -forms by
$$\Omega_X^1=\left.\left(\mathscr{I}{\Delta} / \mathscr{I}{\Delta}^2\right)\right|_{\Delta} .$$

## 数学代写|复几何代写Complex Geometry代考|Direct and Inverse Images

$$\rho_f^{-1} U f^{-1} V: \mathscr{F}\left(f^{-1} U\right) \rightarrow \mathscr{F}\left(f^{-1} V\right)$$

$U \subseteq Y$ ，和 $s_i \in f_* \mathscr{F}\left(U_i\right)$ 在交叉点上达成一致的部分的集合。然后 left 的分隔符缺失或无法识别



## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。