Singularities of integrable Hamiltonian systems: a criterion for non-degeneracy, with an application to the Manakov top

Let (M,\omega) be a symplectic 2n-manifold and h_1,...,h_n be functionally independent commuting functions on M. We present a geometric criterion for a singular point P\in M (i.e. such that {dh_i(P)}_{i=1}^n are linearly dependent) to be non-degenerate in the sence of Vey-Eliasson. Then we apply Fomenko's theory to study the neighborhood U of the singular Liouville fiber containing saddle-saddle singularities of the Manakov top. Namely, we describe the singular Liouville foliation on U and the `Bohr-Sommerfeld' lattices on the momentum map image of U. A relation with the quantum Manakov top studied by Sinitsyn and Zhilinskii (SIGMA 3 2007, arXiv:math-ph/0703045) is discussed.Comment: Application to the Manakov top (Sections 3,4) extende

Computations in monotone Floer theory

Floer theory is a rich collection of tools for studying symplectic manifolds and their Lagrangian submanifolds with the help of holomorphic curves. Its origins lie in estimating the numbers of equilibria in Hamiltonian dynamics, and more recently it has become a major component of the Homological Mirror Symmetry conjecture. This work presents several new computations in Floer theory which combine the use of geometric symmetries, naturally arising in various contexts, with advanced algebraic structures related to Floer theory, like the string maps and the Fukaya category. The three main directions of our study are: the Floer cohomology for a pair of commuting symplectomorphisms; the Fukaya algebra of a Lagrangian submanifold invariant under a circle action; and rigidity properties of non-monotone Lagrangian submanifolds based on the use of low-area versions of the string maps. In each of the three mentioned setups we provide concrete applications of our general results to the study of symplectic manifolds. For example, we prove that Dehn twists in most projective hypersurfaces have infinite order in the symplectic mapping class group; prove that the real projective space split-generates the Fukaya category of the complex projective space and therefore must intersect any other Lagrangian submanifold that is nontrivial in that Fukaya category; and we exhibit a continuous family of Lagrangian tori in the complex projective plane that cannot be made disjoint from the standard Clifford torus by a Hamiltonian isotopy

Geometry of symplectic flux and Lagrangian torus fibrations

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.Comment: 53 pages, 15 figures; various minor improvement