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 〈Coh⩽1(X),Coh⩾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.