  A variety is rationally connected if two general points can be joined by a
rational curve. A higher version of this notion is rational simple
connectedness, which requires suitable spaces of rational curves through two
points to be rationally connected themselves. We prove that smooth, complex,
weighted complete intersections of low enough degree are rationally simply
connected. This result has strong arithmetic implications for weighted complete
intersections defined over the function field of a smooth, complex curve.
Namely, it implies that these varieties satisfy weak approximation at all
places, that R-equivalence of rational points is trivial, and that the Chow
group of zero cycles of degree zero is zero.
