Wall-crossing and orientations for invariants counting coherent sheaves on CY fourfolds

Thesis

Borisov–Joyce and Oh–Thomas defined virtual invariants counting sheaves on Calabi–Yau fourfolds. Similarly to Donaldson invariants, these depend on existence and choice of orientations on moduli spaces of coherent sheaves. The first part of the thesis addresses this question for quasi-projective Calabi–Yau fourfolds, generalizing the work of Cao– Gross–Joyce. The orientations on compactly supported perfect complexes are expressed in terms of a pull-back of gauge-theoretic ones which live on the classifying space C^cs_X of compactly supported K-theory. The proof relies on a choice of a compactification, which allows us to directly obtain orientability of moduli spaces of stable pairs. In the second part of the thesis, we study the conjectural wall-crossing formulae of Gross– Joyce–Tanaka. We begin, by addressing the conjecture of Cao–Kool , which expresses the virtual integrals of a tautological line bundle L^[n] on the Hilbert scheme of points Hilbⁿ (X) in terms of the MacMahon function. We also obtain a prediction for the K-theoretic refinement of this invariant proposed by Nekrasov, which coincides with the expectations from the result for C ⁴ . Studying the invariants further, we find a universal transformation relating them to integrals on Hilbert schemes of points for elliptic surfaces. To understand this, we recover the previously known results for Quot-schemes on elliptic surfaces using similar wallcrossing arguments. We will further study this in to recover and generalize the full result of Arbesfeld–Johnson–Lim–Oprea–Pandharipande for surfaces including divisor contributions.

Authors: Bojko, A

• DOI: 10.5287/ora-qr0gnmnkg
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: Invariants; Hilbert schemes; Orientations; Vertex algebras; Gauge theory; Wall-crossing; Moduli problems; Counting sheaves; Donaldson--Thomas theory
• Subjects: Donaldson-Thomas invariants