Wall-crossing and orientations for invariants counting coherent sheaves on CY fourfolds
<p>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-proje...
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| Other Authors: | |
| Format: | Thesis |
| Language: | English |
| Published: |
2021
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| Subjects: |
| _version_ | 1797086224350969856 |
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| author | Bojko, A |
| author2 | Joyce, D |
| author_facet | Joyce, D Bojko, A |
| author_sort | Bojko, A |
| collection | OXFORD |
| description | <p>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<sup>cs</sup><sub>X</sub> 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<sup>[n]</sup>
on the Hilbert scheme of points Hilb<sup>n</sup>
(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<sup>
4</sup>
. 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.</p> |
| first_indexed | 2024-03-07T02:19:03Z |
| format | Thesis |
| id | oxford-uuid:a345a408-b12f-4bd4-b5dc-db69cca85e48 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T02:19:03Z |
| publishDate | 2021 |
| record_format | dspace |
| spelling | oxford-uuid:a345a408-b12f-4bd4-b5dc-db69cca85e482022-03-27T02:25:47ZWall-crossing and orientations for invariants counting coherent sheaves on CY fourfoldsThesishttp://purl.org/coar/resource_type/c_db06uuid:a345a408-b12f-4bd4-b5dc-db69cca85e48Donaldson-Thomas invariantsEnglishHyrax Deposit2021Bojko, AJoyce, D<p>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<sup>cs</sup><sub>X</sub> 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<sup>[n]</sup> on the Hilbert scheme of points Hilb<sup>n</sup> (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<sup> 4</sup> . 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.</p> |
| spellingShingle | Donaldson-Thomas invariants Bojko, A Wall-crossing and orientations for invariants counting coherent sheaves on CY fourfolds |
| title | Wall-crossing and orientations for invariants counting coherent sheaves on CY fourfolds |
| title_full | Wall-crossing and orientations for invariants counting coherent sheaves on CY fourfolds |
| title_fullStr | Wall-crossing and orientations for invariants counting coherent sheaves on CY fourfolds |
| title_full_unstemmed | Wall-crossing and orientations for invariants counting coherent sheaves on CY fourfolds |
| title_short | Wall-crossing and orientations for invariants counting coherent sheaves on CY fourfolds |
| title_sort | wall crossing and orientations for invariants counting coherent sheaves on cy fourfolds |
| topic | Donaldson-Thomas invariants |
| work_keys_str_mv | AT bojkoa wallcrossingandorientationsforinvariantscountingcoherentsheavesoncyfourfolds |