  There is an obstruction to the existence of K\"ahler -Einstein metrics which
is used to define the GIT weight for K-stability, and it has been extended to
various geometric problems. This survey paper considers such extended
obstructions to the existence problem of K\"ahler -Ricci solitons,
Sasaki-Einstein metrics and (conformally) Einstein-Maxwell K\"ahler metrics.
These three cases have a common feature that the obstructions are parametrized
by a space of vector fields. We see, in these three cases, the obstructions are
obtained as the derivative of suitable volume functionals. This tells us for
which vector fields we should try to solve the existence problems.
