A note on Lagrangian barrier theorem by

We use the Gromov-Witten invariants and a nonsqueezing theorem by the author to generalize a theorem of Lagrangian barriers by P.Biran. A Kähler manifold is a triple consisting of a symplectic manifold (M,ω) and an integrable complex structure J compatible with ω on M . If [ω] ∈ H2(M2n,Z) it follows from Kodaira’s embedding theorem that there exists a smooth and reduced complex hypersurface Σ ⊂M such that its homology class [Σ] ∈ H2n−2(M) represents the Poincaré dual k[ω] ∈ H2(M) for some k ∈ N. Following [Bi] P = (M,ω, J ; Σ) is called a smoothly polarized Kähler manifold. Under the conditions that either dimRM ≤ 6 or ω|π2(M) = 0 the following two theorems were proved in Theorem 1.D and Theorem 4.A of [Bi] respectively. Theorem 1. If (M,ω) is a Kähler manifold with [ω] ∈ H2(M,Q), then for every ǫ > 0 there exists a Lagrangian CW-complex △ǫ ⊂ (M,ω) such that every symplectic embedding φ : B(ǫ) → (M,ω) must satisfy φ(B(ǫ)) ∩△ǫ 6= ∅. Theorem 2. If P = (M,ω, J ; Σ) is a n-dimensional polarized Kähler manifold of degree k then every symplectic embedding φ : B2n(λ) := {x ∈ R2n | |x| ≤ λ2} → ∗Partially supported by the NNSF 19971045 of China and Ministry of Education of China.