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Consider a hypersurface $V(f) \subseteq \mathbb{C}^{n+1}$ with an isolated singularity at the origin. If $L := V(f) \cap S^{2n+1}_\epsilon$ is the link of $V(f)$ (with $S^{2n+1}_\epsilon$ a sufficiently small sphere centered at the origin), then it is known that the intersection of $V(f)$ with the ball bounded by $S^{2n+1}_\epsilon$ is homeomorphic to the cone over $L$.

Is more true? In particular, does $L$, specified as a submanifold of $S^{2n+1}_\epsilon$, completely determine the analytic type of the singularity?

Perhaps the following weaker statement is true: If $L \subseteq S^{2n+1}_\epsilon$ is embedded as an unknotted sphere, is $V(f)$ non-singular at $0$?

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  • $\begingroup$ do you know Mumford's thesis? maths.ed.ac.uk/~aar/papers/mumfordsin.pdf $\endgroup$
    – roy smith
    Commented May 4, 2016 at 17:09
  • $\begingroup$ What do you mean when you write, "In particular, does $L$, specified as a submanifold of $S^{2n+1}_\epsilon$, completely determine the analytic type of the singularity?" Since these submanifolds of spheres vary with $\epsilon$, I assume that you mean "up to diffeomorphism". But now consider a nonisotrivial family of smooth hypersurfaces in $\mathbb{CP}^n$ and the cones over these. $\endgroup$
    – Jason Starr
    Commented Oct 2, 2020 at 8:45
  • $\begingroup$ I meant something more like "up to isotopy" than "up to diffeomorphism"; e.g. in the case $n = 2$, $L$ is an actual link in $S^3$, and I certainly want to remember more than the number of connected components. (I have a vague recollection of seeing a more canonical definition of the link, avoiding the dependence on $\epsilon$). Your counterexample then shows that the most we could hope for is that the isotopy class of the link determines the deformation-equivalence class of the singularity. $\endgroup$
    – dorebell
    Commented Oct 8, 2020 at 23:44
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    $\begingroup$ I think that the isotopy class (maybe even with some extra structure coming from the contact manifold structure on the sphere) does not determine the deformation class. Consider the blowings up of projective 3-space at both a twisted cubic (rational) curve and a plane cubic (elliptic) curve. Choose a projective embedding of each having the same degree on the strict transform of a general line and on a (rational curve) fiber of the exceptional divisor over the center of the blowing up. In the h-principle book by Eliashberg and Mishachev they prove a symplectic isotopy theorem . . . $\endgroup$
    – Jason Starr
    Commented Oct 15, 2020 at 9:41
  • $\begingroup$ The example in my previous comment is wrong, but Eliashberg-Mishachev does yield a counterexample. I will try to post it as an answer soon . . . $\endgroup$
    – Jason Starr
    Commented Oct 21, 2020 at 22:37

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