Moduli spaces of complexes of coherent sheaves

Thesis

In this thesis we consider problems related to Joyce’s vertex algebra construction and the topology of stabilized moduli spaces. We first compute the homology of the moduli stack of objects in the derived category of a smooth complex projective variety X in class D, showing that the rational cohomology ring is freely generated by tautological classes. This is used to identify Joyce’s construction with a generalized super-lattice vertex algebra on the rational K-theory of X^an. Then, we prove orientability of moduli spaces of (complexes of) coherent sheaves on projective Calabi–Yau 4-folds–this has applications to defining a C-linear enumerative invariant theory for Calabi–Yau 4-folds. This result is based on joint work with Yalong Cao and Dominic Joyce. Lastly, we consider a connective even complex-oriented homology theory E with associated formal group law F and show that, given a Künneth isomorphism, replacing ordinary homology with E in Joyce’s construction yields a vertex F -algebra–this result is based on joint work with Markus Upmeier.

Authors: Gross, J

• DOI: 10.5287/ora-zrz0ybemx
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Subjects: Moduli Spaces; Calabi-Yau Manifolds; Algebraic Geometry; Algebraic Topology