  For complex Wigner-type matrices, i.e. Hermitian random matrices with
independent, not necessarily identically distributed entries above the
diagonal, we show that at any cusp singularity of the limiting eigenvalue
distribution the local eigenvalue statistics are universal and form a Pearcey
process. Since the density of states typically exhibits only square root or
cubic root cusp singularities, our work complements previous results on the
bulk and edge universality and it thus completes the resolution of the
Wigner-Dyson-Mehta universality conjecture for the last remaining universality
type in the complex Hermitian class. Our analysis holds not only for exact
cusps, but approximate cusps as well, where an extended Pearcey process
emerges. As a main technical ingredient we prove an optimal local law at the
cusp for both symmetry classes. This result is also used in the companion paper
[arXiv:1811.04055] where the cusp universality for real symmetric Wigner-type
matrices is proven.
