Moduli of PT-Semistable Objects I

TL;DR

This paper establishes boundedness and completeness properties for PT-semistable objects on smooth projective three-folds, providing tools for computing cohomology in a specific heart of a derived category.

Contribution

It proves boundedness of PT-semistable objects with arbitrary Chern classes and shows the stack satisfies a valuative criterion for completeness, advancing moduli theory.

Findings

Boundedness of PT-semistable objects established

Stack of objects satisfies valuative criterion for completeness

Methods for computing cohomology in the heart developed

Abstract

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.