Exact Lagrangian immersions with a single double point

Abstract

We show that if a closed orientable 2 k 2k -manifold K K , k > 2 k>2 , with Euler characteristic χ ( K ) ≠ − 2 \chi (K)\ne -2 admits an exact Lagrangian immersion into C 2 k \mathbb {C}^{2k} with one transverse double point and no other self-intersections, then K K is diffeomorphic to the sphere. The proof combines Floer homological arguments with a detailed study of moduli spaces of holomorphic disks with boundary in a monotone Lagrangian submanifold obtained by Lagrange surgery on K K .