# Homotopy Ehresmann and deformation invariance of $l$-adic Chern classes

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

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

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.