  In this short note we prove that in many cases the failure of a variety to be
separably rationally connected is caused by the instability of the tangent
sheaf (if there are no other obvious reasons). A simple application of the
results proves that a smooth Fano complete intersection is separably rationally
connected if and only if it is separably uniruled. In particular, a general
such Fano complete intersection is separably rationally connected.
