Compact Exact Lagrangian Intersections in Cotangent Bundles via Sheaf Quantization

Abstract

We show that the cardinality of the transverse intersection of two compact exact Lagrangian submanifolds in a cotangent bundle is bounded from below by the dimension of the Hom space of sheaf quantizations of the Lagrangians in Tamarkin's category. Our sheaf-theoretic method can also deal with clean and degenerate Lagrangian intersections. k∈Z dim H k (M ; L) (1.4) for any rank 1 locally constant sheaf L on M over any field k. In particular, #(L 1 ∩ L 2 ) ≥ k∈Z dim H k (M ; k).

Estimating Reeb chords using microlocal sheaf theory

“…It says that certain sheaf homomorphisms descend to Morse theory. A similar result in sheaf theory can also been found in [32,Section 4.3].…”

Sheaf quantization and intersection of rational Lagrangian immersions

“…(5.36) Proof. (i) The proof is essentially the same as that of [Ike19,§4.3]. The only and slight difference appears in checking that δ :…”

“…For the estimates for the number of the intersection points, similarly to [Ike19] we study the object Hom ⋆ (F (0,a) , F H [0,a] ), where F H [0,a] denotes the Hamiltonian deformation of F [0,a] associated with φ H 1 . We find that its microsupport is related to the intersection # (y, y ′ ) ∈ L × L ι(y) = φ where q : M × R × R/θZ → R/θZ is the projection, ℓ : R → R/θZ is the quotient map, and Ω + = {τ > 0} ⊂ T * (M × R × R/θZ) with (t; τ ) being the homogeneous coordinate on T * (R/θZ).…”

“…An important and hard part in the proof is the study of the right-hand side of (1.8). The difficulty comes from the singularity of the microsupports of F (0,a) and F [0,a] while previous works such as [Gui19,Ike19] dealt with quantizations with non-singular microsupports. We study an action of H * (L) = i∈Z H i (L; F 2 ) on the right-hand side of (1.8) and even compute it explicitly in the case where the intersection is transverse.…”

Sheaf quantization and intersection of rational Lagrangian immersions

“…For the bounds for the number of the intersection points, similarly to [15] we study the object Hom (F (0,a) , F H [0,a] ), where F H [0,a] denotes the Hamiltonian deformation of F [0,a] associated with φ H 1 . We find that its microsupport is related to the intersection # (y, y…”