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

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