# Zariski-homeomorphisms

This question is motivated by two questions at MO and at MSE.

I am interested in homeomorphism types of (irreducible) complex-projective varieties with respect to the Zariski topology. Any two Zariski-homeomorphic varieties have to have the same dimension, but there are more subtle Zariski-topological invariants as well. Examples of Zariski-homeomorphisms are provided by biregular maps between varieties and the action of the Galois group $Gal({\mathbb C}/{\mathbb Q})$. Furthermore, clearly, any bijection between (irreducible) projective curves is a Zariski-homeomorphism. On the other hand, it appears that the extreme flexibility of Zariski topology in dimension 1 is just an accident. For instance, it is not hard to prove (and it was essentially known in the 19th century) that the only Zariski-homeomorphisms ${\mathbb C}P^n\to {\mathbb C}P^n$ ($n\ge 2$) are compositions of projective transformations and elements of $Gal({\mathbb C}/{\mathbb Q})$.

Question 1. Suppose that $X, Y$ are (irreducible) complex-projective varieties of dimension $\ge 2$. Is it true that the set of Zariski-homeomorphisms $X\to Y$ is obtained by composing biregular maps (not necessarily between $X$ and $Y$) with actions of some elements of $Gal({\mathbb C}/{\mathbb Q})$?

Question 2. (Somewhat weaker.) Suppose that $X, Y$ are Zariski-homeomorphic (irreducible) complex-projective varieties of dimension $\ge 2$. Is it true that there exists a Zariski-homeomorphism $X\to Y$ obtained by composing a biregular map $X'\to Y'$ with actions of some elements of $Gal({\mathbb C}/{\mathbb Q})$?

• The following papers may be relevant to your question: Hrushovski, Zilber, Zariski geometries, Bullettins of the AMS (1993); Hrushovski, Zilber, Zariski geometries, Journal of the AMS (1996); Zilber, Zariski geometries: geometry from the logician's point of view (2009). – Qfwfq Feb 23 '14 at 15:10
• @Qfwfq: Thank you, I will take a look! – Moishe Kohan Feb 23 '14 at 23:04