# 数学代写|黎曼曲面代写Riemann surface代考|Local structure of singularities

#### Doug I. Jones

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## 数学代写|黎曼曲面代写Riemann surface代考|Local structure of singularities

There is a basic result about the local structure of singular Riemann surfaces (they are more often called analytic spaces in the literature), which justifies the moral definition we gave in the first lecture. It also says that all local information near singular points can be captured algebraically.
2.14 Theorem (Weierstrass Preparation Theorem): Let $F(z, w)$ be holomorphic near $(0,0)$ and assume that
$$F(0,0)=0, \quad \frac{\partial F}{\partial w}(0,0)=0, \quad \ldots, \quad \frac{\partial^{n-1} F}{\partial w^{n-1}}(0,0)=0, \quad \text { but } \quad \frac{\partial^n F}{\partial w^n}(0,0) \neq 0 .$$
Then

• (weak form) there exists a function $\Phi$ of the form
$$\Phi(z, w)=w^n+f_{n-1}(z) w^{n-1}+\cdots+f_1(z) w+f_0(z)$$
with $f_0, \ldots, f_{n-1}$ analytic near $z=0$, whose zero-set agrees with that of $F$ near 0 .
• (strong form) there exists, additionally, a holomorphic function $u(z, w)$, non-zero near $(z, w)=(0,0)$, such that
$$F(z, w)=\Phi(z, w) u(z, w)$$
Moreover, this factorization of $F$ is unique.

## 数学代写|黎曼曲面代写Riemann surface代考|Removal of singularities

If we are not interested in the structure of a singularity, there arises the natural question how this singularity can be ‘removed’, or resolved (the official term). In algebraic geometry, this is done by a procedure called normalization. There is an analytic way to describe that; I shall do so informally, without attempting to define all terms or prove the statements.

There is no way to resolve the singularity of a surface $S$ while keeping it in $\mathbb{C}^2$, so let us first clarify the question. First, it can be shown that the singularities of analytic sets are isolated. (This is plausible enough, as they are the zeroes of the gradient of $F$ ). The correct question is: can we find an abstract Riemann surface $R$ (necessarily non-singular, in view of our definition), mapping holomorphically to $S$, so that the map is bi-holomorphic at the regular points of $S$ ? If so,we say that $R$ resolves the singularities of $S$; in effect, we have replaced the singular points of $S$ with smooth points (in $R$ ). The answer is, it can always be done, and $R$ is unique up to isomorphism.

Note, first, that there are two kinds of singularities: topological ones and purely analytic ones. An example of a topological singularity is the solution set of $w^2-z^2=0$ near the origin, whose neighbourhood is homeomorphic to the union of two disks meeting at their centre. Such singularities arise when the defining power series splits into distinct factors – in this case, $(z-w)(z+w)$. The first step in the resolution is then clear: we separate the disks by replacing their union with a disjoint union. There will now be two points in $R$ maping to the singular point of $S$.

In general, if the power series $F$ defining our surface near $s$ can be factored into terms which are not units in the ring of holomorphic functions near $s$, we replace its zero-set by the disjoint union of the zero-sets of the factors.

The simplest example of a purely analytic singularity is $z^3-w^2=0$, near the origin. As we shall see, the zero-set $S$ is locally homeomorphic to the disc; however, neither coordinate can be used to define the structure of a non-singular surface on $S$. Instead, this can be done by the analytic map $u \mapsto(z, w)=\left(u^2, u^3\right)$, which defines a homeomorphism from $\mathbb{C}$ to $S$. Away from zero, the map is bi-holomorphic, because $u$ can be recovered as $w / z$.

The following theorem shows that our example is no accident. Say that a power series $F(z, w)$ is irreducible near $(0,0)$ if, for any factorization $F=F_1(z, w) \cdot F_2(z, w)$ into power series, near $(0,0)$, one of the factors does not vanish at $(0,0)$ (and hence is multiplicatively invertible there). This excludes the possibility of decomposing the zero-set, as in the previous example.

# 黎曼曲面代写

## 数学代写|黎曼曲面代写Riemann surface代考|Local structure of singularities

2.14定理(Weierstrass准备定理):设$F(z, w)$在$(0,0)$附近全纯，并设
$$F(0,0)=0, \quad \frac{\partial F}{\partial w}(0,0)=0, \quad \ldots, \quad \frac{\partial^{n-1} F}{\partial w^{n-1}}(0,0)=0, \quad \text { but } \quad \frac{\partial^n F}{\partial w^n}(0,0) \neq 0 .$$

(弱形式)存在一个形式的函数$\Phi$
$$\Phi(z, w)=w^n+f_{n-1}(z) w^{n-1}+\cdots+f_1(z) w+f_0(z)$$

(强形式)另外存在一个全纯函数$u(z, w)$，在$(z, w)=(0,0)$附近非零，使得
$$F(z, w)=\Phi(z, w) u(z, w)$$

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