  We study the eigenvalue behaviour of large complex correlated Wishart
matrices near an interior point of the limiting spectrum where the density
vanishes (cusp point), and refine the existing results at the hard edge as
well. More precisely, under mild assumptions for the population covariance
matrix, we show that the limiting density vanishes at generic cusp points like
a cube root, and that the local eigenvalue behaviour is described by means of
the Pearcey kernel if an extra decay assumption is satisfied. As for the hard
edge, we show that the density blows up like an inverse square root at the
origin. Moreover, we provide an explicit formula for the $1/N$ correction term
for the fluctuation of the smallest random eigenvalue.
