on Brieskorn Manifolds

Question

Brieskorn showed that $X_{k}=\{(z_0, \dots, z_k)\in \mathbb{C}^{k+1}| -z_{0}^{3}+\sum_{i=1}^{k} z_{i}^{2}=0\}$(k odd, k>2) is a topological manifold. Is it a smooth manifold?

In general, let $a_1, \dots, a_n \in \mathbb{N}$, and $M(a_1, \dots, a_n)=\{(z_1, \dots, z_n)\in \mathbb{C}^{n}| \sum_{i = 1}^{n} z_{i}^{a_i}=0\}$.

When is $M(a_1, \dots, a_n)$ a topological manifold?

When is $M(a_1,\dots, a_n)$ a smooth manifold?

When is $M(a_1, \dots, a_n)$ a topological manifold but it has no smooth structure?

When does $M(a_1, \dots, a_n)$ admit a smooth structure but the smooth structure is not unique?

Answer 1

Let me consider the question about smoothness.

The answer is that the complex variety $M(a_1, \ldots, a_n)$ is never a smooth manifold if $a_i \geq 2$. In fact, in this case $M(a_1, \ldots, a_n)$ has an isolated singular point at the origin, hence we can apply the following general result:

Proposition. A complex variety can never be a smooth manifold throughout a neighborhood of a singular point.

For a proof, see Milnor's book Singular Points of Complex Hypersurfaces, page 13.

Regarding the question "When is $M(a_1, \ldots, a_n)$ a topological manifold?", I do not know the complete answer, but only some partial results about the corresponding Milnor links. For instance, denoting by $2, \ldots, 2$ a string of $2k-1$ elements equal to $2$, Brieskorn proved that the Milnor links at the origin of $$M(3,\, 6r+1, \, 2, \ldots, 2) \quad \textrm{and} \quad M(3, \, 2, \ldots, 2)$$ are homotophy spheres of dimension $4k$ and $4k-2$, respectively. See here for more details.

Answer 2

Here are some comments to Francesco Polizzi's answer:

Question 1 has been answered completely by Brieskorn in "Beispiele zur Differentialtopologie von Singularitäten", Invent. Math. 2, 1966, 1-14, a link to the paper can be found in Ranicki's link collection for exotic spheres. Assuming $n>3$, $M(a_1,\dots,a_n)$ is a topological manifolds if and only if the link $M(a_1,\dots,a_n)\cap S^{2n-1}$ is a homotopy sphere. The latter is (by Satz 1 in Brieskorn's paper) equivalent to either of the following:

• $\Delta_a(1)=1$, where $a=(a_1,\dots,a_n)$, $$\Delta_a(t)=\prod_{0<i_k<a_k} (t-\zeta_{a_1}^{i_1}\cdots\zeta_{a_n}^{i_n}), \zeta_{a_k}=e^{2\pi i/a_k} $$ is the characteristic polynomial of the monodromy automorphism on $H_{n-1}(\Xi_a,\mathbb{Z})$ with $\Xi_a=\{(z_1,\dots,z_n)\in \mathbb{C}^n\mid \sum_{i=1}^nz_i^{a_i}=1\}$.

• there is a graph $G_a$ associated to $a=(a_1,\dots,a_n)$ with vertex set $\{a_1,\dots,a_n\}$, where $a_i$ and $a_j$ are connected by an edge if and only if $gcd(a_i,a_j)>1$. This graph either a) has two isolated points or b) has a single isolated point and one component with an odd number of vertices such that for all vertices $a_i,a_j$ in this component we have $gcd(a_i,a_j)=2$.

As an alternative to Francesco Polizzi's argument, Question 2 about smoothness can also be answered using the results of Brieskorn's paper. In the cases where $M(a_1,\dots,a_n)$ is a topological manifold, Brieskorn actually determines the smooth structure of the link sphere. If $M(a_1,\dots,a_n)$ is a smooth manifold, then the link sphere must have the standard smooth structure. However, there are plenty examples (such as the $M(2,\dots,2,3)$) where the smooth structure is exotic, hence the topological manifold can not be smoothed. For the exact conditions (using signature computations), see Brieskorn's paper or Section 3 of Hirzebruch's "Singularities and exotic spheres" (also found in the abovementioned link collection).

Question 3 is then a combination of the above two.