4
$\begingroup$

Let $X$ be a scheme of finite type over $\mathbb{C}$ and let $Z \hookrightarrow X$ be a closed subscheme. Consider the associated closed inclusion $Z_{an} \hookrightarrow X_{an}$ between their analytifications (regarded as topological spaces). Is this a strong neighborhood deformation retract? By this I mean, can I find for every neighborhood $U$ of a point $z$ in $Z_{an}$ another neighborhood $V \subseteq U$ such that $V$ deformation retracts onto $V \cap Z_{an}$? If not in general, is this true for some additional assumptions on $X$ and $Z$?

Thanks!

Improve this question
$\endgroup$
4
  • 1
    $\begingroup$ I think this is true and follows from the existence of triangulations. I am not sure of the original reference for this, but it is probably contained in works of Hironaka and/or Lojasiewicz. There are probably several more recent references as well (see, e.g, papers of Bierstone-- Milman). $\endgroup$
    – naf
    Commented Sep 16, 2015 at 5:32
  • $\begingroup$ If $X$ is an algebraic variety and $Z$ is a subvariety, then indeed the classical triangulation results (due originally to Lojasiewicz) show that one can triangulate $X$ with $Z$ as a subcomplex. Hironaka has a paper discussing the proof. $\endgroup$
    – Dan Ramras
    Commented Sep 16, 2015 at 17:09
  • $\begingroup$ Thanks for the comments. I'm having trouble tracking down exactly which paper this occurs in. Any idea the title? Thanks again! $\endgroup$
    – David Carchedi
    Commented Sep 17, 2015 at 0:54
  • $\begingroup$ Hironaka, Triangulations of Algebraic Sets, Poc. Symp. in Pure Math, Vol. 29, 1975. The book of Coste, Bochnak, and Roy discusses this as well, but I think they only handle the compact case. $\endgroup$
    – Dan Ramras
    Commented Sep 19, 2015 at 4:07

0

Reset to default

You must log in to answer this question.