3
$\begingroup$

I am looking for an implicit function theorem which holds also on singular spaces, at least if the singularities are "mild".

For example, let $0 = z^2 - x y + z w + w^2 + \epsilon w$ define a hypersurface in $\mathbb{C}^4$ with coordinates $(x,y,z,w)$ and a parameter $\epsilon \in \mathbb{C}$. Let $h \colon \mathbb{C}^4 \to \mathbb{C}$ be the projection to the last coordinate $w$. The restriction of $h$ to this hypersurface is not submersive in $0$ if $\epsilon \neq 0$, hence the implicit function theorem cannot be applied in $0$ for finding a (local, holomorphic) section of $h$.

However, for $\epsilon = 0$ the hypersurface is singular in $0$, and a (holomorphic) section $f \colon \mathbb{C} \to \mathbb{C}^4$ of $h$ is given by $\zeta \mapsto (3\zeta,\zeta,\zeta,\zeta)$. At least when looking at the Zariski tangent spaces, the map $h$ is actually submersive in $0$, and my question is whether the existence of the section $f$ of $h$ can be guaranteed locally by some version of the implicit function theorem, ideally also for other types of singularities.

$\endgroup$
  • 1
    $\begingroup$ According to A'Campo, singularity theory is the branch of mathematics covering those rare cases where implicit function theorem does not work :) $\endgroup$ – Alex Degtyarev Feb 9 '15 at 21:39
  • 1
    $\begingroup$ If seriously, similar things (maybe not quite what you want) are related to Milnor fibers, so you may want to check Milnor's book on singularities of complex hypersurfaces. $\endgroup$ – Alex Degtyarev Feb 9 '15 at 21:42

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.