Lagrangian non-intersections

Lagrangian intersections in contact geometry

Lagrangian intersections in contact geometry ( First draft )

A contactomorphism is a diffeomorphism ψ : M → M which preserves the contact structure ξ. This means that ψα = eα for some smooth function h : M → . A contact isotopy is a smooth family of contactomorphisms ψs : M → M such that ψ sα = e sα for all s. Any such isotopy with ψ0 = id and h0 = 0 is generated by Hamiltonian functions Hs : M → as follows. Each Hamiltonian function Hs determines a Hami...

Lagrangian Intersections and the Serre Spectral Sequence

For a transversal pair of closed Lagrangian submanifolds L, L of a symplectic manifold M so that π1(L) = π1(L′) = 0 = c1|π2(M) = ω|π2(M) and a generic almost complex structure J we construct an invariant with a high homotopical content which consists in the pages of order ≥ 2 of a spectral sequence whose differentials provide an algebraic measure of the highdimensional moduli spaces of pseudo-h...

Remarks on Lagrangian Intersections in Toric Manifolds

We consider two natural Lagrangian intersection problems in the context of symplectic toric manifolds: displaceability of torus orbits and of a torus orbit with the real part of the toric manifold. Our remarks address the fact that one can use simple cartesian product and symplectic reduction considerations to go from basic examples to much more sophisticated ones. We show in particular how rig...

Lifting Non-proper Tropical Intersections

We prove that if X,X are closed subschemes of a torus T over a non-Archimedean field K, of complementary codimension and with finite intersection, then the stable tropical intersection along a (possibly positive-dimensional, possibly unbounded) connected component C of Trop(X) ∩ Trop(X) lifts to algebraic intersection points, with multiplicities. This theorem requires potentially passing to a s...