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Stability conditions via spherical objects

An object in the bounded derived category D^b(Coh(X)) of coherent sheaves on a complex projective K3 surface X is spherical if it is rigid and simple. Although spherical objects form only a discrete set in the moduli stack of complexes, they determine much of the structure of X and D^b(Coh(X)). Here we show that a stability condition on D^b(Coh(X)) is determined by the stability of spherical objects.