Brieskorn manifold

Example

Let $n \ge 1$ and let $a_0, \dots, a_n \ge 2$ be integers. The Brieskorn manifold is $$\Sigma(a_0, \dots, a_n) = \set{z \in \C^{n+1} : z_0^{a_0} + z_1^{a_1} + \cdots + z_n^{a_n} = 0} \cap S^{2n+1},$$ where $S^{2n+1}$ is the unit sphere in $\C^{n+1}$. It is the link at $0$ of the hypersurface $f^{-1}(0)$, where $f = z_0^{a_0} + \cdots + z_n^{a_n}$.

Check: the partial derivatives of $f$ are $a_j z_j^{a_j - 1}$, which vanish together only at $z = 0$. So $f$ has an isolated singularity at $0$ and $f^{-1}(0)$ is smooth away from $0$. Since $f$ is weighted homogeneous, the sphere of radius $1$ gives the same link as any small sphere. So $\Sigma(a_0, \dots, a_n)$ is a smooth closed manifold of dimension $2n - 1$. The Milnor number of $f$ is $(a_0 - 1)(a_1 - 1) \cdots (a_n - 1)$, because the monomials $z_0^{b_0} \cdots z_n^{b_n}$ with $0 \le b_j \le a_j - 2$ form a basis of the local algebra.