Geometry of symplectic flux and Lagrangian torus fibrations

Abstract

Symplectic flux measures the areas of cylinders swept in the process of a Lagrangian isotopy. We study flux via a numerical invariant of a Lagrangian submanifold that we define using its Fukaya algebra. The main geometric feature of the invariant is its concavity over isotopies with linear flux.We derive constraints on flux, Weinstein neighbourhood embeddings and holomorphic disk potentials for Gelfand-Cetlin fibres of Fano varieties in terms of their polytopes. We show that Calabi-Yau SYZ fibres have unobstructed Floer theory under a general assumption. We also describe the space of fibres of almost toric fibrations on the complex projective plane up to Hamiltonian isotopy, and provide other applications.

Viterbo conjecture for Zoll symmetric spaces

“…Hence for v being J-holomorphic, where J ∈ J(M, ω) satisfies the above property, and perhaps decreasing ǫ ′′ to 0 < ǫ ′ ≤ ǫ ′′ to make the intersection with {r = (r 0 + ǫ ′ )/k} transverse, we obtain Remark 26. The approach of the adapted choice of the almost complex structure was developed, for a different purpose, in discussions with D. Tonkonog and R. Vianna in the course of preparation of [90]. The proof of Lemma 31 is similar to that of Lemma 27, and the proof of Lemma 32 is similar to that of Lemma 29, with the only modification being running the argument of [33,Proposition 4.2] for relative Seidel invariants [56,57,58,60].…”

Algebraic and symplectic viewpoint on compactifications of two-dimensional cluster varieties of finite type

“…The flux f(γ) measures the symplectic area of a cylinder swept by a cycle γ ∈ H 1 (T 2 , Z) as we move in a path of Lagrangian fibres connecting (0, 0) to (p 1 , p 2 ). [See, for instance, [30] for a more complete understanding of flux in ATFs.] So, in practice, we visualise the base minus a set of cuts (one for each nodal fibre) affinely embedded into R 2 endowed with the standard affine structure.…”

“…This ensures the existence of a monotone fibre, that can be detected by the intersection point of the lines in the diagrams that go through the nodes and are in the direction of the cuts. A symplectomorphism class invariant of these monotone fibres (the star-shape) is shown [30] to be given by the interior of the polytope seen in H 1 (T 2 , R) ∼ = R 2 , as we forget the nodes and cuts. So there is a symplectomorphism identifying two monotone fibres of an ATF, if and only if, the associated polytopes are related under SL(2; Z).…”

“…In this case, the mutation cycle induced by the nodal singularities in X does provides us with a 5-cycle of distinct monotone Lagrangian tori. Recall that monotone fibres of non-SL(2, Z) related diagrams are distinct [30].…”

An-type surface singularity and nondisplaceable Lagrangian tori

“…The nondisplaceable fibers here are analogues of the β i -type fibers in [31] where they discussed all possible Lagrangian tori of CP 2 arising as fibers of almost toric fibrations. Moreover, with Kähler parameters carefully chosen we have different degenerations to the same monotone symplectic manifold.…”