On exotic Lagrangian tori in ℂℙ<sup>2</sup>

Vianna, Renato

doi:10.2140/gt.2014.18.2419

Cited by 45 publications

(71 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In [36], a similar result is proven for the monotone CP 2 #3CP 2 , and for some family of tori in the monotone (CP 1 ) 2m .…”

Section: To Mutually Different Hamiltonian Isotopy Classessupporting

confidence: 56%

“…The forthcoming work of Pascaleff-Tonkonog will present a proof for the wall-crossing formula ( [1,2,35], see also [15]). With that in hand, one can prove Theorem 1.1 for CP 2 #2CP 2 .…”

Section: Remark 12mentioning

confidence: 99%

“…Consider two monotone Lagrangian fibres of ATFs whose ATBDs are related via one mutation. The algebraic count of Maslov index 2 pseudo-holomorphic disks for these tori is expected to vary according to wall-crossing formulas [1,2,15,35]. In view of that we conjecture: Call (X, ω) the corresponding symplectic manifold.…”

Section: Remark 12mentioning

confidence: 99%

“…In particular, the blowup depends on the radius one takes. In a toric symplectic manifold, one can perform a blowup near an elliptic rank zero singularity ([31, Definition 4.2], [35,Definition 2.7]) and remain toric, provided one chooses a small enough ball compatible with the toric fibration, see Fig. 12.…”

Section: Monotonicity and Blowupmentioning

confidence: 99%

See 3 more Smart Citations

Infinitely many monotone Lagrangian tori in del Pezzo surfaces

Vianna

2017

Sel. Math. New Ser.

Self Cite

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

“…In [36], a similar result is proven for the monotone CP 2 #3CP 2 , and for some family of tori in the monotone (CP 1 ) 2m .…”

Section: To Mutually Different Hamiltonian Isotopy Classessupporting

confidence: 56%

“…The forthcoming work of Pascaleff-Tonkonog will present a proof for the wall-crossing formula ( [1,2,35], see also [15]). With that in hand, one can prove Theorem 1.1 for CP 2 #2CP 2 .…”

Section: Remark 12mentioning

confidence: 99%

Section: Remark 12mentioning

confidence: 99%

Section: Monotonicity and Blowupmentioning

confidence: 99%

See 2 more Smart Citations

Infinitely many monotone Lagrangian tori in del Pezzo surfaces

Vianna

2017

Sel. Math. New Ser.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Recent results by R. Vianna [48], [49] provide a major breakthrough in the understanding of monotone Lagrangian tori. He establishes the existence of an infinite number of Hamiltonian isotopy classes of monotone Lagrangian tori inside (CP 2 , ω FS ).…”

Section: 23mentioning

confidence: 99%

Lagrangian isotopy of tori in $${S^2\times S^2}$$ S 2 × S 2 and $${{\mathbb{C}}P^2}$$ C P 2

2016

View full text Add to dashboard Cite

Abstract. We show that, up to Lagrangian isotopy, there is a unique Lagrangian torus inside each of the following uniruled symplectic fourmanifolds: the symplectic vector space R 4 , the projective plane CP 2 , and the monotone S 2 × S 2 . The result is proven by studying pseudoholomorphic foliations while performing the splitting construction from symplectic field theory along the Lagrangian torus. A number of other related results are also shown. Notably, the nearby Lagrangian conjecture is established for T * T 2 , i.e. it is shown that every closed exact Lagrangian submanifold in this cotangent bundle is Hamiltonian isotopic to the zero-section.

show abstract

Tropical Lagrangians in toric del-Pezzo surfaces

Hicks

2021

Sel. Math. New Ser.

View full text Add to dashboard Cite

We look at how one can construct from the data of a dimer model a Lagrangian submanifold in $$(\mathbb {C}^*)^n$$ ( C ∗ ) n whose valuation projection approximates a tropical hypersurface. Each face of the dimer corresponds to a Lagrangian disk with boundary on our tropical Lagrangian submanifold, forming a Lagrangian mutation seed. Using this we find tropical Lagrangian tori $$L_{T^2}$$ L T 2 in the complement of a smooth anticanonical divisor of a toric del-Pezzo whose wall-crossing transformations match those of monotone SYZ fibers. An example is worked out for the mirror pair $$(\mathbb {CP}^2{\setminus } E, W), {\check{X}}_{9111}$$ ( CP 2 \ E , W ) , X ˇ 9111 . We find a symplectomorphism of $$\mathbb {CP}^2{\setminus } E$$ CP 2 \ E interchanging $$L_{T^2}$$ L T 2 and a SYZ fiber. Evidence is provided that this symplectomorphism is mirror to fiberwise Fourier–Mukai transform on $${\check{X}}_{9111}$$ X ˇ 9111 .

show abstract

On exotic Lagrangian tori in ℂℙ²

Cited by 45 publications

References 17 publications

Infinitely many monotone Lagrangian tori in del Pezzo surfaces

Infinitely many monotone Lagrangian tori in del Pezzo surfaces

Lagrangian isotopy of tori in $${S^2\times S^2}$$ S 2 × S 2 and $${{\mathbb{C}}P^2}$$ C P 2

Tropical Lagrangians in toric del-Pezzo surfaces

Contact Info

Product

Resources

About

On exotic Lagrangian tori in ℂℙ2

Cited by 45 publications

References 17 publications

Infinitely many monotone Lagrangian tori in del Pezzo surfaces

Infinitely many monotone Lagrangian tori in del Pezzo surfaces

Lagrangian isotopy of tori in $${S^2\times S^2}$$ S 2 × S 2 and $${{\mathbb{C}}P^2}$$ C P 2

Tropical Lagrangians in toric del-Pezzo surfaces

Contact Info

Product

Resources

About

On exotic Lagrangian tori in ℂℙ²