Projective moduli spaces of complexes of sheaves

Abstract

Since the introduction of Bridgeland stability conditions, constructing moduli spaces of complexes has become an increasingly important task in algebraic geometry. However, the picture is complicated by the fact that the toolkit of Geometric Invariant Theory is frequently unavailable. In this work we showcase the use of determinantal line bundle techniques in constructing projective moduli space of complexes in two contexts. In Chapter 3, we establish projectivity of certain moduli spaces of Bridgeland semistable objects on a smooth, projective surface, and relate these moduli spaces to the Uhlenbeck compactification of the moduli of stable vector bundles. This is achieved by studying a determinantal line bundle constructed on Bridgeland moduli spaces by Bayer and Macrì. As an application, we give an argument showing that, under a coprime assumption, the moduli of Gieseker-stable sheaves is projective. In Chapter 4, we consider higher rank PT-stable pairs on a smooth, projective threefold. Using a determinantal line bundle, we construct a morphism from the moduli of PT-stable pairs to a projective scheme and show that the set-theoretic behavior of this map is closely analogous to that of the Uhlenbeck compactification.