Infinitely many exotic monotone Lagrangian tori in ℂℙ<sup>2</sup>

Vianna, Renato

doi:10.1112/jtopol/jtw002

Cited by 31 publications

(12 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This contrast with what we understand of a number of families of tori in two-dimensional examples, e.g. in the case of [ 5 , 14 , 36 ], for [ 5 , 53 ] or del Pezzo surfaces [ 3 , 54 ], and arguably most remarkably for log CY surfaces [ 26 , 27 , 46 ]; cluster structures associated to Grasmannians have also recenty been used to construct families of exotic Lagrangian tori in them [ 9 ].…”

Section: Floer-theoretic Propertiesmentioning

confidence: 67%

Families of monotone Lagrangians in Brieskorn–Pham hypersurfaces

Keating

2021

Math. Ann.

View full text Add to dashboard Cite

We present techniques, inspired by monodromy considerations, for constructing compact monotone Lagrangians in certain affine hypersurfaces, chiefly of Brieskorn–Pham type. We focus on dimensions 2 and 3, though the constructions generalise to higher ones. The techniques give significant latitude in controlling the homology class, Maslov class and monotonicity constant of the Lagrangian, and a range of possible diffeomorphism types; they are also explicit enough to be amenable to calculations of pseudo-holomorphic curve invariants. Applications include infinite families of monotone Lagrangian $$S^1 \times \Sigma _g$$ S 1 × Σ g in $${\mathbb {C}}^3$$ C 3 , distinguished by soft invariants for any genus $$g \ge 2$$ g ≥ 2 ; and, for fixed soft invariants, a range of infinite families of Lagrangians in Brieskorn–Pham hypersurfaces. These are generally distinct up to Hamiltonian isotopy. In specific cases, we also set up well-defined counts of Maslov zero holomorphic annuli, which distinguish the Lagrangians up to compactly supported symplectomorphisms. Inter alia, these give families of exact monotone Lagrangian tori which are related neither by geometric mutation nor by compactly supported symplectomorphisms.

show abstract

Section: Floer-theoretic Propertiesmentioning

confidence: 67%

Families of monotone Lagrangians in Brieskorn–Pham hypersurfaces

Keating

2021

Math. Ann.

View full text Add to dashboard Cite

show abstract

“…Recently, the use of almost toric fibrations has become an important tool in constructing new examples of Lagrangian tori. For example, Vianna has constructed infinitely many exotic tori in P 2 [Via16] and in del Pezzo surfaces [Via17]. For more details on almost toric fibrations, see [Sym01,LS10,Eva22].…”

Section: Introductionmentioning

confidence: 99%

“…

In [Via16], Vianna constructed infinitely many exotic Lagrangian tori in P 2 . We lift these tori to higher-dimensional projective spaces and show that they remain non-symplectomorphic.

…”

mentioning

confidence: 99%

Infinitely many monotone Lagrangian tori in del Pezzo surfaces

Vianna

2017

Sel. Math. New Ser.

View full text Add to dashboard Cite

We construct almost toric fibrations (ATFs) on all del Pezzo surfaces, endowed with a monotone symplectic form. Except for CP 2 #CP 2 , CP 2 #2CP 2 , we are able to get almost toric base diagrams (ATBDs) of triangular shape and prove the existence of infinitely many symplectomorphism (in particular Hamiltonian isotopy) classes of monotone Lagrangian tori in CP 2 #kCP 2 for k = 0, 3, 4, 5, 6, 7, 8. We name these tori. Using the work of Karpov-Nogin, we are able to classify all ATBDs of triangular shape. We are able to prove that CP 2 #CP 2 also has infinitely many monotone Lagrangian tori up to symplectomorphism and we conjecture that the same holds for CP 2 #2CP 2 . Finally, the Lagrangian tori n 1 ,n 2 ,n 3 p,q,r

show abstract

“…Leung and Symington [28]), and recently, almost-toric systems have proved to be of independent interest in symplectic topology (cf. Vianna [49,50,51]). While the classification problem of almost-toric systems has not been settled, even in the compact case, an important subclass of almost-toric systems has been completely understood: Pelayo and Vũ Ngo .…”

mentioning

confidence: 99%

“…Leung and Symington [28]), and recently, almost-toric systems have proved to be of independent interest in symplectic topology (cf. Vianna [49,50,51]).…”

mentioning

confidence: 99%

Faithful Semitoric Systems

Hohloch¹,

Sabatini²,

Sepe³

et al. 2018

SIGMA

View full text Add to dashboard Cite

Abstract. This paper consists of two parts. The first provides a review of the basic properties of integrable and almost-toric systems, with a particular emphasis on the integral affine structure associated to an integrable system. The second part introduces faithful semitoric systems, a generalization of semitoric systems (introduced by Vũ Ngo . c and classified by Pelayo and Vũ Ngo . c) that provides the language to develop surgeries on almost-toric systems in dimension 4. We prove that faithful semitoric systems are natural building blocks of almost-toric systems. Moreover, we show that they enjoy many of the properties that their (proper) semitoric counterparts do.

show abstract

Infinitely many exotic monotone Lagrangian tori in ℂℙ²

Cited by 31 publications

References 12 publications

Families of monotone Lagrangians in Brieskorn–Pham hypersurfaces

Families of monotone Lagrangians in Brieskorn–Pham hypersurfaces

Infinitely many monotone Lagrangian tori in del Pezzo surfaces

Faithful Semitoric Systems

Contact Info

Product

Resources

About

Infinitely many exotic monotone Lagrangian tori in ℂℙ2

Cited by 31 publications

References 12 publications

Families of monotone Lagrangians in Brieskorn–Pham hypersurfaces

Families of monotone Lagrangians in Brieskorn–Pham hypersurfaces

Infinitely many monotone Lagrangian tori in del Pezzo surfaces

Faithful Semitoric Systems

Contact Info

Product

Resources

About

Infinitely many exotic monotone Lagrangian tori in ℂℙ²