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Register a new userConformal symplectic geometry of cotangent bundles
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We prove a version of the Arnold conjecture for Lagrangian submanifolds of conformal symplectic manifolds: a Lagrangian $L$ which has non-zero Morse-Novikov homology for the restriction of the Lee form $beta$ cannot be disjoined from itself by a $C^0$-small Hamiltonian isotopy. Furthermore for generic such isotopies the number of intersection points equals at least the sum of the free Betti numbers of the Morse-Novikov homology of $beta$. We also give a short exposition of conformal symplectic geometry, aimed at readers who are familiar with (standard) symplectic or contact geometry.
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We study the geometry of manifolds carrying symplectic pairs consisting of two closed 2-forms of constant ranks, whose kernel foliations are complementary. Using a variation of the construction of Boothby and Wang we build contact-symplectic and contact pairs from symplectic pairs.
We describe how the result in [1] extends to prove the existence of a Serre type spectral sequence converging to the symplectic homology SH_*(M) of an exact Sub-Liouville domain M in a cotangent bundle T*N. We will define a notion of a fiber-wise symplectic homology SH_*(M,q) for each point q in N, which will define a graded local coefficient system on N. The spectral sequence will then have page two isomorphic to the homology of N with coefficients in this graded local system.
We classify symplectically foliated fillings of certain contact foliated manifolds. We show that up to symplectic deformation, the unique minimal symplectically foliated filling of the foliated sphere cotangent bundle of the Reeb foliation in the 3-sphere is the associated disk cotangent bundle. En route to the proof, we study another foliated manifold, namely the product of a circle and an annulus with almost horizontal foliation. In this case, the foliated unit cotangent bundle does not have a unique minimal symplectic filling. We classify the foliated fillings of this manifold up to symplectic deformation equivalence using combinatorial invariants of the filling.
The paper was withdrawn due to a gap in the proof of Lemma 3.
We analyze two different fibrations of a link complement M constructed by McMullen-Taubes, and studied further by Vidussi. These examples lead to inequivalent symplectic forms on a 4-manifold X = S x M, which can be distinguished by the dimension of the primitive cohomologies of differential forms. We provide a general algorithm for computing the monodromies of the fibrations explicitly, which are needed to determine the primitive cohomologies. We also investigate a similar phenomenon coming from fibrations of a class of graph links, whose primitive cohomology provides information about the fibration structure.
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