Generalized Donaldson-Thomas Invariants Via Kirwan Blowups

Download or read book Generalized Donaldson-Thomas Invariants Via Kirwan Blowups written by Michail Savvas and published by . This book was released on 2018 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: In this thesis, we develop a virtual cycle approach towards generalized Donaldson-Thomas theory of Calabi-Yau threefolds. Starting with an Artin moduli stack parametrizing semistable sheaves or perfect complexes, we construct an associated Deligne-Mumford stack, called its Kirwan partial desingularization, with an induced semi-perfect obstruction theory of virtual dimension zero, and define the generalized Donaldson-Thomas invariant via Kirwan blowups as the degree of the corresponding virtual cycle. The key ingredients are a generalization of Kirwan's partial desingularization procedure and recent results from derived symplectic geometry regarding the local structure of stacks of sheaves and perfect complexes on Calabi-Yau threefolds. Examples of applications include Gieseker stability of coherent sheaves and Bridgeland and polynomial stability of perfect complexes. In the case of Gieseker semistable sheaves, this new Donaldson-Thomas invariant is invariant under deformations of the complex structure of the Calabi-Yau threefold. More generally, deformation invariance is true under appropriate assumptions which are expected to hold in all cases.