University of Cambridge
Birational Invariance of Punctured Log Gromov-Witten Theory and Intrinsic Mirror Constructions
Abstract
dc:description.abstractIn this thesis, we investigate and resolve various problems related to log Gromov-Witten theory and their application to mirror symmetry. We first prove for log Calabi-Yau varieties satisfying a semi-positivity assumption that the Gross-Siebert logarithmic mirror construction encodes solutions to enumerative problems considered in the non-archimedean construction of Keel and Yu, and use this to show the two approaches agree in most cases when both can be constructed. We also prove a classical-quantum period correspondence for smooth Fano pairs, with the classical periods encoded in the Gross-Siebert mirror construction, and in particular give enumerative meaning to generating series of regularized quantum periods. The second main result of this thesis is a study of the behavior of punctured log Gromov-Witten theory under log étale modifications X ̃ → X, generalizing an investigation first carried out by Abramovich and Wise. We show that the moduli space of stable log maps to X ̃ can be described explicitly in terms of the moduli space of stable log maps to X, together with understanding of the change in tropical moduli spaces. We use this result to resolve various foundational questions in punctured log Gromov-Witten theory, as well as to show a certain form of log étale invariance of the intrinsic mirror algebra.
Degree
thesis:*- Name dc:type.qualificationname
- Doctor of Philosophy (PhD)
- Level dc:type.qualificationlevel
- Doctoral
- Grantor dc:publisher.institution
- University of Cambridge
- Year dc:date.issued
- 2023
Author and committee
dc:creator, dc:contributor.*- Author dc:creator
-
- Johnston, Samuel
- Advisor dc:contributor.advisor
-
- Gross, Mark
Subjects
dc:subject × 3Rights
dc:rightsIdentifiers
dc:identifier.*- DOI dc:identifier.doi
- https://doi.org/10.17863/CAM.104123
- OAI identifier oai:identifier
- oai:www.repository.cam.ac.uk:1810/360923