  Two K\"ahler metrics on a complex manifold are called c-projectively
equivalent if their $J$-planar curves coincide. These curves are defined by the
property that the acceleration is complex proportional to the velocity. We give
an explicit local description of all pairs of c-projectively equivalent
K\"ahler metrics of arbitrary signature and use this description to prove the
classical Yano-Obata conjecture: we show that on a closed connected K\"ahler
manifold of arbitrary signature, any c-projective vector field is an affine
vector field unless the manifold is $CP^n$ with (a multiple of) the
Fubini-Study metric. As a by-product, we prove the projective Lichnerowicz
conjecture for metrics of Lorentzian signature: we show that on a closed
connected Lorentzian manifold, any projective vector field is an affine vector
field.
