Ho Hai Phung

Let $X$ be a smooth connected variety over a field $k$ of characteristic
zero and $x$ be a $k$-point of $X$. The differential fundamental group
$\pi(X,x)$ is an affine $k$-group scheme, the representation category of
which is equivalent (by means of the fiber functor at $x$) to the category
of vector bundles with flat connections on $X$. There is a natural map
from the group cohomology of representations of $\pi(X,x)$ and de Rham
cohomology of the corresponding connections. When $X$ is a projective
curve of positive genus these maps are isomorphisms. In this talk we
discuss the similar question when $X$ is a smooth projective curve over a
field of positive characteristic. In these new settings, bundles with
flat connections are replaced by stratified bundles and de Rham
cohomology is replaced by Grothendieck's infinitesimal cohomology (also
called stratified cohomology). It turns out that the natural maps from
group cohomology to stratified cohomology are isomorphisms. These
isomorphisms seem to be a coincidence, since the two cohomology theories
(stratified and group cohomologies) are different from their nature. We
show that the two cohomologies do not agree when computed on stratified
ind-bundles by explicit examples.
This is joint work with Vo Quoc Bao and Dao Van Thinh.