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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 $\bigoplus H^i_{{\acute{\mathrm{e}}\text {t}}}(X_{s'}, \mathbb{Q}_l)$?

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    $\begingroup$ Geometric points? And preserves the Chern classes of what? What if the fibers have different Picard ranks? $\endgroup$
    – Will Sawin
    Jul 25, 2020 at 19:07
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    $\begingroup$ @WillSawin Chern classes of the tangent bundle $\endgroup$
    – user145520
    Jul 25, 2020 at 19:14

1 Answer 1

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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.

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