We explicitly construct a Lie groupoid integrating the elliptic tangent bundle associated to a (possibly normal crossing) elliptic divisor, providing a necessary and sufficient topological condition for the existence of a Hausdorff integration. We also produce an explicit local model for the symplectic integration of an elliptic Poisson structure.
We construct a stable homotopy type invariant for any Legendrian submanifold in a jet bundle equipped with a linear-at-infinity generating family. We show that this spectrum lifts the generating family homology groups. When the generating family extends to a generating family for an embedded Lagrangian filling, we lift the Seidel isomorphism to the spectrum level. As applications, we establish topological constraints on Lagrangian fillings arising from generating families, algebraic constraints on whether generating families admit fillings, and lower bounds on how many fiber dimensions are needed to construct a generating family for a Legendrian.
We construct an infinite family of non-positive open books with structures. Combined with a result of Wendl, this allows us to give a complete answer to a long-standing question about the mapping class group of a compact surface with boundary: namely, we conclude that the monoid of monodromies supporting Stein-fillable contact structures is equal to the monoid of positive monodromies if and only if the surface is planar.
We use the symplectic rational blow-up to study some Lagrangian pinwheels in symplectic rational manifolds. In particular, we determine which symplectic forms in the threefold blow-up of C CP CP2 carry Lagrangian projective planes that can be made disjoint by a Hamiltonian isotopy. In addition, we show that such a disjunction is not possible in del Pezzo surfaces with Euler characteristic between 4 and 7. Finally, we determine which symplectic forms on S2 & times; S2 carry a Lagrangian L L3 L3, L3,1 pinwheel, answering a question of J. Evans.
In this article, we extend the methods from arXiv:2011.06568, where the five dimensional analogue of the three dimensional finite energy foliations introduced by Hofer--Wysocki--Zehnder was identified, to the case where there the underlying (IP) contact $5$-fold admits a $6$-dimensional (IP) symplectic filling. We show that the filling induces a moduli space of pseudo-holomorphic curves which is itself a symplectic filling of the standard $3$-sphere, and hence symplectomorphic to the $4$-dimensional ball. We further show that whenever the contact form on the $5$-fold is strictly convex, then this moduli space is a strictly convex domain, so that the induced Reeb dynamics at the boundary is in particular dynamically convex. For the circular restricted three-body problem, this implies that whenever the spatial dynamics (near one of the heavy masses) is strictly convex, the holomorphic shadow of arXiv:2011.06568 is dynamically convex; this is shown to hold for near-integrable cases close to the Kepler problem, and with mass ratio either zero or sufficiently close to 1.
