|
|||||
|
|
|
|
|
|
|
|
|
|
|
|
|
Gromov–Witten theory via
roots and logarithms
Luca Battistella, Navid Nabijou and Dhruv Ranganathan |
|
Geometry & Topology 28 (2024) 3309–3355
|
 |
Abstract |
|
Orbifold and logarithmic structures provide independent routes to the virtual enumeration of curves with tangency orders for a simple normal crossings pair . The theories do not coincide and their relationship has remained mysterious. We prove that the genus-zero orbifold theories of multiroot stacks of strata blowups of converge to the corresponding logarithmic theory of . With fixed numerical data, there is an explicit combinatorial criterion that guarantees when a blowup is sufficiently refined for the theories to coincide. The result identifies birational invariance as the crucial property distinguishing the logarithmic and orbifold theories. 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 a new set of 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. |
Keywords
Gromov–Witten theory, logarithmic geometry, orbifolds,
tropical curves, root stacks
|
Mathematical Subject Classification
Primary: 14A21, 14N35
|
References |
Publication
Received: 27 June 2022
Revised: 26 April 2023
Accepted: 27 May 2023
Published: 25 November 2024
Proposed: Jim Bryan
Seconded: Dan Abramovich, Dmitri Burago
|
Authors |
| Luca Battistella | |
| Goethe Universität Frankfurt Frankfurt Germany |
Università di Bologna Bologna Italy |
| Navid Nabijou | |
| School of Mathematical
Sciences Queen Mary University of London London United Kingdom |
|
| Dhruv Ranganathan | |
| Department of Pure Mathematics and
Mathematical Statistics University of Cambridge Cambridge United Kingdom |
|
| © 2024 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
Open Access made possible by participating institutions via Subscribe to Open.
