Let $S$ be a connected scheme of finite type over $\overline{\mathbb{F}\_p}$. Let $\pi:X\to S$ be a smooth proper morphism such that each fiber over a closed point has a trivial étale fundamental group. Let $s, s'\in S$ be two closed points.


For $l\neq p$ is there an equivalence between the localized-at-$l$ étale homotopy types of $X\_s$ and $X\_{s'}$?


If there is an equivalence can it be chosen to map the total Chern class of $X\_s$ in $\bigoplus H^i\_{{\acute{\mathrm{e}}\text {t}}}(X\_s, \mathbb{Q}\_l)$ to the total Chern class of $X\_{s'}$ in $\bigoplus H^i\_{{\acute{\mathrm{e}}\text {t}}}(X\_{s'}, \mathbb{Q}\_l)$?



Homotopy Ehresmann and deformation invariance of $l$-adic Chern classes
By finding a path between $s$ and $s'$, it suffices to consider the case $S$ local strictly henselian, $s$ the closed (geom.) point, $s'\to S$ some other geometric point. Cutting down with curves, we can even assume $S$ is the spectrum of a strictly henselian discrete valuation ring, and that $s'$ is the geometric generic point.


The inclusion $i\colon X\_s \hookrightarrow X$ induces an isomorphism on fundamental groups (SGA1 Exp. X Thm. 2.1) and cohomology of local systems (proper base change), and hence an equivalence of (profinite) etale homotopy types (combine Theorems 4.3 and 12.5 in Artin-Mazur).


Moreover, we have $i^\* T\_{X/S} = T\_{X\_s/s}$, and hence the total Chern class of $X$ (relative to $S$) corresponds to the total Chern class of $X\_s$ under the $\mathbf{Q}\_\ell$-cohomology isomorphism induced by $i^\*$.


We can therefore replace $X\_s$ with $X$ at this point (and $T\_{X\_s/s}$ with $T\_{X/S}$). The advantage of doing so is that there is a map $$\bar j\colon X\_{s'}\to X.$$


Local acyclicity for smooth morphisms (SGA 4.3 Exp. XV Thm 2.1) implies now that $\bar j$ induces an isomorphism on the prime-to-$p$ quotient of the fundamental group and on the cohomology of any prime-to-$p$ local system. This implies again that the map induced by $\bar j$ on prime-to-$p$ or pro-$\ell$ etale homotopy types is an equivalence. And again, since $\bar j^\* T\_{X/S}= T\_{X\_{s'}/s'}$, this is compatible with the total Chern class.


Composing the inverse equivalences to the one induced by $i$ with the one induced by $\bar j$ yields the desired equivalence.


