On the existence of (H,A)-stable sheaves on K3 or abelian surfaces

We give an existence result on (H,A)-stable sheaves on a K3 or abelian surface X with primitive triple of invariants (rank,first Chern class,Euler characteristics) in the integral cohomology lattice. Such a result yields the existence of singular projective Q-factorial symplectic terminalisations of certain moduli spaces of sheaves on X that are Gieseker semistable with respect to a nongeneral ample divisor.