0

Can you prove the following

Let $X$ be a scheme of finite type over $\mathbb{C}$, then the weak homotopy type of the complex-analytic space $X(\mathbb{C})$ is the weak homotopy type of a finite CW complex

without using Hironaka's resolution of singularities (preferably no resolutions at all) and no theorems of J. Lurie? Here is an example of a proof that does not meet our criteria.

  • Isn't this a duplicate: https://mathoverflow.net/questions/26927/how-to-prove-that-a-projective-variety-is-a-finite-cw-complex ? – Denis Nardin Jun 07 '19 at 10:09
  • @DenisNardin it might be, I have to verify the article of Hironaka mentioned there. And your comment made me realize I wanted to ask a different question. –  Jun 07 '19 at 10:19
  • 1
    Let me also mention that it is fairly trivial to reduce to the affine case using the strong form of the Seifert-Van Kampen theorem (that $U\cup V$ is the homotopy pushout of $U←U\cap V→V$ where $U,V$ are open subsets). The proof I like most happens to have been written by Lurie, but it's a much older result than that, so maybe it'd qualify :). – Denis Nardin Jun 07 '19 at 10:31

0 Answers0