  Let C be the complex field and K=C((x,y)) or K=C((x))(y). Let G be a
connected linear algebraic group over K. Under the assumption that the
K-variety G is K-rational, i.e. that the function field is purely transcendant,
it was proved that a principal homogeneous space of G has a rational point over
K as soon as it has one over each completion of K with respect to a discrete
valuation. In this paper we show that one cannot in general do without the
K-rationality assumption. To produce our examples, we introduce a new type of
obstruction. It is based on higher reciprocity laws on a 2-dimensional scheme.
We also produce a family of principal homogeneous spaces for which the refined
obstruction controls exactly the existence of rational points.
