Moduli Spaces of Stable Pairs in Donaldson-Thomas Theory

TL;DR

This paper constructs a moduli space for stable pairs on polarized smooth projective varieties using GIT, revealing a chamber structure and linking to Le Potier's coherent systems.

Contribution

It introduces a GIT-based construction of moduli spaces for $ ext{delta}$-semistable pairs and establishes their chamber structure and relation to existing moduli.

Findings

Constructed coarse moduli space for $ ext{delta}$-semistable pairs.

Proved a chamber structure for the moduli space.

Connected the moduli space to Le Potier's coherent systems.

Abstract

Let $(X,\mathcal{O}_X(1))$ be a polarized smooth projective variety over the complex numbers. Fix $\mathcal{D}\in \mathrm{coh}(X)$ and a nonnegative rational polynomial $\delta$. Using GIT we contruct a coarse moduli space for $\delta$-semistable pairs $(\mathcal{E},\phi)$ consisting of a coherent sheaf $\mathcal{E}$ and a homomorphism $\phi\colon \mathcal{D}\rightarrow \mathcal{E}$. We prove a chamber structure result and establish a connection to the moduli space of coherent systems constructed by Le Potier in \cite{LeP} and \cite{LeP2}.