Computing motivic Donaldson-Thomas invariants

Abstract

This thesis develops a method (dimensional reduction) to compute motivic Donaldson–Thomas invariants. The method is then employed to compute these invariants in several different cases. Given a moduli scheme with a symmetric obstruction theory a Donaldson– Thomas type invariant can be defined by integrating Behrend’s function over the scheme. Motivic Donaldson–Thomas theory aims to provide a more refined invariant associated to each such moduli space a virtual motive. From the modern point of view motivic Donaldson–Thomas invariants should be defined for a three dimensional Calabi–Yau category. These categories often arise in a geometric context as the derived category of representations of a quiver with potential. Provided the potential has a linear factor we are able to reduce the problem of computing the corresponding virtual motives to a much simpler one. This includes geometric examples coming from local curves which we compute explicitly.