On the log-local principle for the toric boundary

Pierrick Bousseau; Andrea Brini; Michel van Garrel

doi:10.1112/blms.12566

On the log-local principle for the toric boundary

Pierrick Bousseau, Andrea Brini, Michel van Garrel

Mathematics

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Abstract

Let (Formula presented.) be a smooth projective complex variety and let (Formula presented.) be a reduced normal crossing divisor on (Formula presented.) with each component (Formula presented.) smooth, irreducible and numerically effective. The log–local principle put forward in van Garrel et al. (Adv. Math. 350 (2019) 860–876) conjectures that the genus 0 log Gromov–Witten theory of maximal tangency of (Formula presented.) is equivalent to the genus 0 local Gromov–Witten theory of (Formula presented.) twisted by (Formula presented.). We prove that an extension of the log–local principle holds for (Formula presented.) a (not necessarily smooth) (Formula presented.) -factorial projective toric variety, (Formula presented.) the toric boundary, and descendant point insertions.

Original language	English
Pages (from-to)	161-181
Journal	Bulletin of the London Mathematical Society
Volume	54
Issue number	1
Early online date	7 Mar 2022
DOIs	https://doi.org/10.1112/blms.12566
Publication status	E-pub ahead of print - 7 Mar 2022

Bibliographical note

Funding Information:
We thank Helge Ruddat for getting this project started by asking us to compute some local Gromov–Witten invariants of and for helpful discussions all along. We thank Travis Mandel for detailed explanations on [ 23, 24 ]. We are grateful to Miles Reid and Andrea Petracci for some discussions on the classification of toric varieties with numerically effective divisors. We thank Makoto Miura for a discussion on Hibi varieties. Finally, it is a pleasure to thank Navid Nabijou and Dhruv Ranganathan for discussions on their parallel work [ 26 ]. This project has been supported by the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska‐Curie Grant Agreement Number: 746554 (M. van Garrel), the Engineering and Physical Sciences Research Council under Grant Agreement ref. EP/S003657/2 (A. Brini) and by Dr. Max Rössler, the Walter Haefner Foundation and the ETH Zürich Foundation (P. Bousseau and M. van Garrel).

Publisher Copyright:
© 2022 The Authors. Bulletin of the London Mathematical Society is copyright © London Mathematical Society.

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Cite this

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abstract = "Let (Formula presented.) be a smooth projective complex variety and let (Formula presented.) be a reduced normal crossing divisor on (Formula presented.) with each component (Formula presented.) smooth, irreducible and numerically effective. The log–local principle put forward in van Garrel et al. (Adv. Math. 350 (2019) 860–876) conjectures that the genus 0 log Gromov–Witten theory of maximal tangency of (Formula presented.) is equivalent to the genus 0 local Gromov–Witten theory of (Formula presented.) twisted by (Formula presented.). We prove that an extension of the log–local principle holds for (Formula presented.) a (not necessarily smooth) (Formula presented.) -factorial projective toric variety, (Formula presented.) the toric boundary, and descendant point insertions.",

author = "Pierrick Bousseau and Andrea Brini and {van Garrel}, Michel",

note = "Funding Information: We thank Helge Ruddat for getting this project started by asking us to compute some local Gromov–Witten invariants of and for helpful discussions all along. We thank Travis Mandel for detailed explanations on [ 23, 24 ]. We are grateful to Miles Reid and Andrea Petracci for some discussions on the classification of toric varieties with numerically effective divisors. We thank Makoto Miura for a discussion on Hibi varieties. Finally, it is a pleasure to thank Navid Nabijou and Dhruv Ranganathan for discussions on their parallel work [ 26 ]. This project has been supported by the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska‐Curie Grant Agreement Number: 746554 (M. van Garrel), the Engineering and Physical Sciences Research Council under Grant Agreement ref. EP/S003657/2 (A. Brini) and by Dr. Max R{\"o}ssler, the Walter Haefner Foundation and the ETH Z{\"u}rich Foundation (P. Bousseau and M. van Garrel). Publisher Copyright: {\textcopyright} 2022 The Authors. Bulletin of the London Mathematical Society is copyright {\textcopyright} London Mathematical Society.",

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On the log-local principle for the toric boundary

Abstract

Bibliographical note

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