Infinitely many monotone Lagrangian tori in del Pezzo surfaces
type: Item
instance of: scholarly article
title: Infinitely many monotone Lagrangian tori in del Pezzo surfaces
zbMATH Open document ID: 1377.53101
author: Renato Ferreira de Velloso Vianna
published in: Selecta Mathematica. New Series
publication date: 2017-07-14T00:00:00Z
full work available at URL: https://arxiv.org/abs/1602.03356
review text: The author constructs almost toric fibrations on del Pezzo surfaces endowed with a monotone symplectic form. He gets the existence of infinitely many symplectomorphism classes of monotone Lagrangian tori inside \(\mathbb{CP}^1\times\mathbb{CP}^1\) and \(\mathbb{CP}^2 \sharp_k\overline{\mathbb{CP}^2}\) for \(k\in\{0,1,3,4,5,6,7,8\}\), see Theorems 1.1 and 1.6. Conjecturally, the same holds true also for \(k=2\), thanks to the forthcoming work by Pascaleff and Tonkonog, see Conjecture 1.3. The techniques and tools includes: nodal trades, nodal slides, transferring the cut, mutations, and an invariant called boundary Maslov-2 convex hull. The authors also relates his work with other papers, among them [\textit{V. Shende} et al., ``On the combinatorics of exact Lagrangian surfaces'', Preprint, \url{arXiv:1603.07449}; \textit{A. Keating}, Geom. Funct. Anal. 25, No. 6, 1822--1901 (2015; Zbl 1338.32025); \textit{B. V. Karpov} and \textit{D. Yu. Nogin}, Izv. Math. 62, No. 3, 429--463 (1998; Zbl 0949.14026); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 62, No. 3, 3--38 (1998); \textit{P. Hacking} and \textit{Y. Prokhorov}, Compos. Math. 146, No. 1, 169--192 (2010; Zbl 1194.14054); \textit{K. Fukaya} et al., Int. Math. Res. Not. 2012, No. 13, 2942--2993 (2012; Zbl 1250.53077)].
reviewed by: Daniele Angella
Mathematics Subject Classification ID: 53D12
Mathematics Subject Classification ID: 53D05
zbMATH DE Number: 6745647
zbMATH Keywords: almost toric fibration
zbMATH Keywords: almost toric base diagram
zbMATH Keywords: del Pezzo surface
zbMATH Keywords: monotone Lagrangian torus
MaRDI profile type: Publication
OpenAlex ID: W2269544036
Wikidata QID: Q59614143
arXiv ID: 1602.03356
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DOI: 10.1007/S00029-017-0312-Z
description: scientific article