In the introduction to the Asterisque 82-83 volume on `Caractérisque d'Euler-Poincaré, Verdier writes:
Enfin signalons que la situation en caractéristique positive est loin d'être aussi satisfaisante que dans le cas transcendant. On aimerait pouvoir disposer d'un groupe de Grothendieck `sauvage' des faisceaux constructibles tels que deux faisceaux ayant même rang et même comportement sauvage en chaque point aient même classe dans ce groupe de Grothendieck et tel que ce groupe soit foncteur covariant. Des resultats récents de Laumon donnent des indications dans ce sens. On aimerait aussi disposer d'une théorie des classes d'homologie de Chern qui sereait une transformation naturelle de ce groupe de Grothendieck sauvage dans les groupes d'homologie étale.
My question is:
Has the situation improved in the intervening 32 years?
I recall in detail below some background and ask some more precise questions.
For a map of complex algebraic varieties $f: X \to Y$, one can define the pushforward of a constructible function $\xi$:
$$(f_* \xi)(p) = \sum_n n \chi(\xi^{-1}(n) \cap f^{-1}(p) )$$
This is a relative notion of Euler characteristic: $f_* (1_X)$ is the function on $Y$ whose value at $y$ is the Euler characteristic $\chi(X_y)$.
This notion is functorial: $(gf)_* = g_* f_*$.
The (naively) analogous notion in characteristic $p$ is not functorial. The standard counterexample is the Artin-Schreier map
$g:\mathbb{A}^1 \to \mathbb{A}^1$
$z \mapsto z^p - z$
Note $g_* 1 = p$. On the other hand, taking $f: \mathbb{A}^1 \to \mathrm{Spec}\,k$ the structure map, one has $f_* (g_* 1)) = p \ne 1 = (fg)_* 1$. The problem seems to be that the map is wildly ramified at infinity. The beginning of the above paragraph of Verdier is a request that this situation be remedied by introducing some intermediary between constructible sheaves and their stalkwise Euler characteristics which would record the possibility of wild ramification and thus be functorial under pushforward.
Is there now some such object?
The second part of Verdier's paragraph presumes that the answer to the question above is yes, and requests an analogue of Macpherson's Chern class transformation. Over $\mathbb{C}$, this is a natural transformation $c_{SM}: Con(\cdot) \to H_*(\cdot)$ between the functors (covariant with respect to proper pushforward) which respectively assign to a variety its constructible functions and homology; it has the property that if $X$ is proper and smooth then $c_{SM}(X) = c(TX) \cap [X]$.
Is there an analogue of the Macpherson Chern class transformation in characteristic $p$?
The Macpherson Chern class is defined in terms of something called the 'local Euler obstruction'. This is a constructible function $Eu_V$ attached to a variety $V$ which, roughly speaking, at a point $v\in V$ records the virtual number of zeroes of any extension of the radial vector field from the boundary of ball around $v$ to its interior. (One passes to the Nash blowup in order to make sense of this vector field.)
The Euler obstruction enjoys the following hyperplane formula: locally embedding $(V, v)$ in a smooth variety $Y$, taking some sufficiently small $\epsilon$ ball $B_\epsilon(v)$, and a generic projection $l: B_\epsilon(v) \to \mathbb{C}$, one finds that $l_* Eu_V$ is locally constant near $l(v)$. In other words, the Euler obstruction is annihilated by taking vanishing cycles with respect to a general linear function.
Is there an analogue of the local Euler obstruction in characteristic $p$, and is it annihilated by taking vanishing cycles with respect to a general linear function?
