Let $ f \in \mathbb{C}[z_1, \ldots, z_n]$ be a polynomial such that $f'(z) \neq 0$ if $z \neq 0$ ($f'$ means $\left( \frac{\partial f}{\partial z_1}, \ldots, \frac{\partial f}{\partial z_n}\right)$ ). Let $S = S^{2n-1} \subset \mathbb{C}^n$ be a sphere in $\mathbb{C}^n$.

Is the set $L = \{ z \in S: f(z) = 0 \}$ a smooth manifold?

If yes, is it a boundary of some smooth closed manifold?


This is a well-known fact, and I'm not sure if this question is suitable for MO. Anyway, I will give a short answer.

Let $X=V(f) \subset \mathbb{C}^{n}$ be an affine hypersurface with at most an isolated singularity at the origin and let $L= X \cap S_{\varepsilon}$ be the intersection of $X$ with a sphere $S_{\varepsilon}=S^{2n-1}_{\varepsilon}$ centered at the origin and having sufficiently small radius $\varepsilon$.

Then Milnor proved that $L$ is a smooth manifold of real dimension $2n-3$, which is called the Milnor link of the singularity. Such a link is unknotted if and only if $0$ is a regular point for $X$, i.e. if and only if $df(0) \neq 0$.

Milnor also proved that the map $$\theta \colon S_{\varepsilon} \setminus L \longrightarrow S^1, \quad x \mapsto \frac{f(x)}{|f(x)|}$$ is a smooth locally trivial fibration, called the Milnor fibration. Any fibre $F_a:=\theta^{-1}(a)$ is a smooth open manifold of dimension $2n-2$. Its closure $\overline{F}_a$ is a manifold with boundary, such that $\partial \overline{F}_a =L$.

The classical reference is Milnor's book Singular points of complex hypersurfaces. See also Dimca's book Singularities and Topology of Hypersurfaces, Chapter 3.


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