Moduli of PT-Semistable Objects I

We show boundedness for PT-semistable objects of any Chern classes on a smooth projective three-fold $X$. Then we show that the stack of objects in the heart $⟨{\text{Coh}}_{\le 1}(X),{\text{Coh}}_{\ge 2}(X)[1]⟩$ satisfies a version of the valuative criterion for completeness. In the remainder of the paper, we give a series of results on how to compute cohomology with respect to this heart.