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We show that for a Zariski general hypersurface $V$ of degree $M+1$ in${\mathbb P}^{M+1}$ for $M\geqslant 5$ there are no Galois rational covers$X\dashrightarrow V$ of degree $d\geqslant 2$ with an abelian Galois group,where $X$ is a rationally connected variety. In particular, there are norational maps $X\dashrightarrow V$ of degree 2 with $X$ rationally connected.This fact is true for many other families of primitive Fano varieties as welland motivates a conjecture on absolute rigidity of primitive Fano varieties.

Rationally connected rational double covers of primitive Fano varieties