Gromov-Witten theory via roots and logarithms
Orbifold and logarithmic structures provide independent routes to the virtual enumeration of curves with tangency orders for a simple normal crossings pair $(X|D)$. The theories do not coincide and their relationship has remained mysterious. We prove that the genus zero orbifold theories of multi-ro...
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| Main Authors: | , , |
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| Format: | Article (Journal) Chapter/Article |
| Language: | English |
| Published: |
23 Jun 2022
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| Edition: | Version v2 |
| In: |
Arxiv
Year: 2022, Pages: 1-35 |
| DOI: | 10.48550/arXiv.2203.17224 |
| Online Access: | Verlag, lizenzpflichtig, Volltext: https://doi.org/10.48550/arXiv.2203.17224 Verlag, lizenzpflichtig, Volltext: http://arxiv.org/abs/2203.17224 
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| Author Notes: | Luca Battistella, Navid Nabijou and Dhruv Ranganathan |
| Summary: | Orbifold and logarithmic structures provide independent routes to the virtual enumeration of curves with tangency orders for a simple normal crossings pair $(X|D)$. The theories do not coincide and their relationship has remained mysterious. We prove that the genus zero orbifold theories of multi-root stacks of strata blowups of $(X|D)$ converge to the corresponding logarithmic theory of $(X|D)$. With fixed numerical data, there is an explicit combinatorial criterion that guarantees when a blowup is sufficiently refined for the theories to coincide. There are two key ideas in the proof. The first is the construction of a naive Gromov-Witten theory, which serves as an intermediary between roots and logarithms. The second is a smoothing theorem for tropical stable maps; the geometric theorem then follows via virtual intersection theory relative to the universal target. The results import new computational tools into logarithmic Gromov-Witten theory. As an application, we show that the genus zero logarithmic Gromov-Witten theory of a pair is determined by the absolute Gromov-Witten theories of its strata. |
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| Item Description: | Version 1 vom 31 März 2022, Version 2 vom 23 Juni 2022 Gesehen am 14.10.2022 |
| Physical Description: | Online Resource |
| DOI: | 10.48550/arXiv.2203.17224 |