Minimal and H-minimal submanifolds in toric geometry

Kotelskiy, Artem

doi:10.4310/jsg.2016.v14.n2.a3

Cited by 6 publications

(5 citation statements)

References 14 publications

(27 reference statements)

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“…The ideas of this section belong to Mironov, Panov, and Kotelskiy [30,31,24]. We give a different explanation of their results and prove some new lemmas.…”

Section: Intersection Of Quadrics and Lagrangian Submanifolds Ofmentioning

confidence: 97%

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Zoo of monotone Lagrangians in $\mathbb{C}P^n$

Oganesyan¹

2021

Preprint

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Let P ⊂ R m be a polytope of dimension m with n facets and a 1 , . . . , a n be the normal vectors to the facets of P . Assume that P is Delzant, Fano, and a 1 + . . . + a n = 0. We associate a monotone embedded Lagrangian L ⊂ CP n−1 to P . As an abstract manifold, the Lagrangian L fibers over (S 1 ) n−m−1 with fiber R P , where R P is defined by a system of quadrics in RP n−1 . The manifold R P is called the real toric space.We find an effective method for computing the Lagrangian quantum cohomology groups of the mentioned Lagrangians. Then we construct explicitly some set of wide and narrow Lagrangians.Our method yields many different monotone Lagrangians with rich topological properties, including non-trivial Massey products, complicated fundamental group and complicated singular cohomology ring.Interestingly, not only the methods of toric topology can be used to construct monotone Lagrangians, but the converse is also true: the symplectic topology of Lagrangians can be used to study the topology of R P . General formulas for the rings H * (R P , Z), H * (R P , Z 2 ) are not known. Since we have a method for constructing narrow Lagrangians, the spectral sequence of Oh can be used to study the singular cohomology ring of R P . This idea will be developed in a further paper.

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“…The ideas of this section belong to Mironov, Panov, and Kotelskiy [30,31,24]. We give a different explanation of their results and prove some new lemmas.…”

Section: Intersection Of Quadrics and Lagrangian Submanifolds Ofmentioning

confidence: 97%

“…Hamiltonian-minimal Lagrangian submanifolds in toric manifolds are studied in [30,31,24]. In particular, a Hamiltonian-minimal Lagrangian submanifold L ⊂ C n is associated to each Delzant polytope P .…”

Section: Introductionmentioning

confidence: 99%

Zoo of monotone Lagrangians in $\mathbb{C}P^n$

Oganesyan¹

2021

Preprint

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“…Mironov in [11] found a method for constructing Hamiltonian-minimal Lagrangian submanifolds of C n . See [10] for another point of view.…”

Section: Main Theorems: the Existence Partmentioning

confidence: 99%

Products and connected sums of spheres as monotone Lagrangian submanifolds

Oganesyan

Sun

2019

Preprint

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We obtain topological restrictions on Maslov classes of monotone Lagrangian submanifolds of C n . We also construct families of new examples of monotone Lagrangian submanifolds, which show that the restrictions on Maslov classes are sharp in certain cases. Contents 1. Introduction 1 Acknowledgements 4 2. Preliminaries 4 2.1. Local Floer cohomology 4 2.2. Spectral sequences of the Floer algebra 5 3. Main theorems: the existence part 6 4. Main theorems: the restriction part 11 5. Some explicit examples of monotone submanifolds of C n 15 6. Monotone Lagrangian submanifolds of CP n 16 References 18

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“…A construction explained in this section was discovered by Mironov in [15]. Later, the construction was studied by Panov and Kotelskiy in [16] and [17]. They applied their method to study minimal and Hamiltonian-minimal Lagrangians of toric manifolds.…”

Section: Lagrangian Submanifolds Of C Nmentioning

confidence: 99%

“…The following theorem was proved by Kotelskiy: Theorem 6.2. ( [17]) L is Lagrangian submanifold of C n with respect to the standard symplectic structure. Moreover, L is diffeomorphic to R × D Γ T Γ and L is Hamiltonianminimal (we use notations of section 3).…”

Section: Infinitely Many Non-isotopic Lagrangians In C Nmentioning

confidence: 99%

Non-isotopic monotone Lagrangian submanifolds of $\mathbb{C}^n$

Oganesyan

2019

Preprint

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Let P be a Delzant polytope in R k with n+k facets. We associate a closed Lagrangian submanifold L of C n to each Delzant polytope. We prove that L is monotone if and only if and only if the polytope P is Fano. Also, we pose the "Lagrangian version of Delzant theorem".Let n and p be even integers. Assume p is greater than 3, n > 2p. We construct p 2 monotone Lagrangian embeddings of S p−1 × S n−p−1 × T 2 into C n , no two of which are related by Hamiltonian isotopies. Some of these embeddings are smoothly isotopic and have equal minimal Maslov numbers, but they are not Hamiltonian isotopic. Also, we construct infinitely many non-monotone embeddings of S 2p−1 × S 2p−1 × T 2 into C 4p , no two of which are related by Hamiltonian isotopies.

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Minimal and H-minimal submanifolds in toric geometry

Cited by 6 publications

References 14 publications

Zoo of monotone Lagrangians in $\mathbb{C}P^n$

Zoo of monotone Lagrangians in $\mathbb{C}P^n$

Products and connected sums of spheres as monotone Lagrangian submanifolds

Non-isotopic monotone Lagrangian submanifolds of $\mathbb{C}^n$

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