Gromov–Witten theory via roots and logarithms

Battistella, Luca¹ ; Nabijou, Navid² ; Ranganathan, Dhruv³

¹ Goethe Universität Frankfurt, Frankfurt, Germany, Università di Bologna, Bologna, Italy
² School of Mathematical Sciences, Queen Mary University of London, London, United Kingdom
³ Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge, United Kingdom

Geometry & topology, Tome 28 (2024) no. 7, pp. 3309-3355.

Résumé

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 multiroot 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. 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.

Détail

DOI : 10.2140/gt.2024.28.3309

Keywords: Gromov–Witten theory, logarithmic geometry, orbifolds, tropical curves, root stacks

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