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 $\langle \Coh_{\leq 1}(X), \Coh_{\geq 2}(X)[1] \rangle$ 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.