We show that for a local, discretely valued field $F$, with residue characteristic $p$, and a variety $𝒰$ over $F$, the map $\rho :\mathrm{Gal}(F^{\mathrm{sep}}/F)\to \mathrm{Out}\left({\pi }_{1,\mathrm{geom}}^{\left(p^{\text{'}}\right)}\right(𝒰\left)\right)$ to the outer automorphisms of the prime to $p$ geometric étale fundamental group of $𝒰$ maps the wild inertia onto a finite image. We show that under favourable conditions $\rho$ depends only on the reduction of $𝒰$ modulo a power of the maximal ideal of $F$. The proofs make use of the theory of logarithmic schemes.

Every compact Kähler surface is deformation equivalent to a projective surface. In particular, topologically Kähler surfaces and projective surfaces cannot be distinguished. Kodaira had asked whether this continues to hold in higher dimensions. We explain the construction of a series of counter-examples due to C. Voisin, which yields compact Kähler manifolds of dimension at least four whose rational homotopy type is not realized by any projective manifold.