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# Section 1: Def of Algebraic varieties & singularities

Notation We denote $\mathbb R$ or $\mathbb C$ by $\Phi$.

Definition A subset $V\subseteq \Phi^m$ is called an algebraic set in $\Phi^m$ if $V$ is the common zero locus of some polynomials $f_i\in \Phi[x_1,\cdots ,x_m],i\in I$.

Definition A topological space $X$ is called irreducible if $X$ cannot be expressed as the union of two proper closed subsets.

Definition algebraic set + irreducible = algebraic variety.

Definition $I(V)\subseteq \Phi[x_1,\cdots ,x_m]$ is the ideal of all polynomials vanishing on $V$.

Fact $V$ is irreducible $\iff$ $I(V)$ is a prime.

Definition $\mathrm{dim}V=\mathrm{Trdeg}_\Phi \mathrm{Frac}(\Phi[x_1,\cdots ,x_m]/I(V))$.

Example

1. Hypersurfaces(here) can always be defined by one single polynomial.
2. $\mathbb A^1=\mathbb R$, then consider $f=x$, $V(f)=\{0\}$, $\mathrm{dim}V=\mathrm{Trdeg}_\mathbb R \mathbb R[x]/(x)=0$.

Definition A point $x\in V$ is called a nonsingular point / regular / simple if $$\mathrm{rank}(\partial f_i/\partial x_j(x))=\max_{y\in V}\mathrm{rank}(\partial f_i/\partial x_j(y))$$
otherwise it is called a singular point.

Fact The definition of singular point does not depend on the choice of generators of $I(V)$.

# Section 2: Why & How to study the singularities

Quick overview

1. Regular points are easy to study
2. We only study local properties around $x$ (analytically or algebraically)

Function $f$ defining equation of the hypersurface $V(f)$, then for any $x\in V\subseteq \mathbb C^m$, $K:=f^{-1}(0)\cap S_\varepsilon^{2m-1}(x)$ is a smooth manifold for small $\varepsilon>0$.

Example

1. If $x$ regular, then $K\simeq \mathbb R^{2m-2}\cap \mathbb S^{2m-1}\simeq \mathbb S^{2m-3}$.
2. $f=z_1^2+z_2^2,z=(0,0)$, then $K=\mathbb S^1\sqcup \mathbb S^1$.

Theorem (Brauner) Given $f=z_1^p+z_2^q$ where $p,q\in\mathbb N$ coprime, then $K$ is a $(p,q)$-torus knot.

proof: $z_1^p+z_2^q=0\iff |z_1|^p=|z_2|^q$ and $p\arg z_1=q\arg z_2+\pi(2k+1),k\in \mathbb Z$. So $K$ is the union of $p$ circles.

We will consider high dimensional cases in the future:

$$f=z_1^{a_1}+z_2^{a_2}+\cdots +z_n^{a_n},\quad a_i\in \mathbb N$$

Question: When $K$ is a sphere, Homeomorphic? Diffeomorphic?

Example $f(z)=z_1+z_2+\cdots +z_5$ homeomorphic to $\mathbb S^7$, but not diffeomorphic to $\mathbb S^7$ (Milnor).

Some observations in Regular case:

Corollary $V$ alg var on $\mathbb R$ or $\mathbb C$, then point $x^0\in V$

1. Regular
2. Isolated singular point(denoted by $\Sigma(V)$)

Every sufficiently small sphere $S_\varepsilon$ centered at $x^0$ intersects $V$ in a smooth manifold, maybe $\emptyset$.

Theorem $V\cap D_\varepsilon$ homeomorphic to cone over $V\cap S_\varepsilon$. Moreover $(D_\varepsilon,V\cap D_\varepsilon)$ is homeomorphic to $\mathrm{Cone}(\mathbb S_\varepsilon,V\cap \mathbb S_\varepsilon)$.

$\Phi:S_\varepsilon\to \mathbb S^1:z\mapsto f(z)/|f(z)|$ is a smooth map, then $\Phi$ is a smooth fiber bundle, called Milnor fibration.

Lemma Center $z^0$ of $S_\varepsilon$ regular, then $F_\theta$ diffeomorphic to open ball $\mathbb R^{2n}$.

# Section 3: The first main theorem - Fibration theorem

Theorem $\varepsilon$ small enough, then $\Phi:S_\varepsilon\setminus K\to \mathbb S^1$ is a smooth fiber bundle.

Definition Locally a product space, but globally may be not a product space is called a fiber bundle.

Data: $E,B,F$ topological spaces, $\pi:E\to B$ continuous surjective map, such that for any $b\in B$, there exists an open neighborhood $U$ of $b$ and a homeomorphism $h:\pi^{-1}(U)\to U\times F$ such that the following diagram commutes:

\begin{CD} \pi^{-1}(U) @>{h}>> U\times F \\ @V{\pi}VV @VV{\mathrm{proj}_1}V \\ U @>>{1}> U \end{CD}

Fibration: $\pi :E\to B$ satisfies the homotopy lifting property.

Analogous to Ehresmann's lemma:

Lemma $E,B$ smooth manifolds, $\pi:E\to B$ a proper submersion, then $\pi$ is a smooth fiber bundle.

Definition proper: the preimage of any compact set is compact.

Definition submersion: $\mathrm{rank}d\pi_x=\dim B$ for any $x\in E$.