Semi-Local Exotic Lagrangian Tori in Dimension Four

We study exotic Lagrangian tori in dimension four. In certain Stein domains B d p q (which naturally appear in almost toric fibrations) we find d + 1 families of monotone Lagrangian tori which are mutually distinct, up to symplectomorphisms. We prove that these remain distinct under embeddings of B d p q into geometrically bounded symplectic four-manifolds. We show that there are infinitely many different such embeddings when X is compact and (almost) toric and hence conclude that X contains arbitrarily many Lagrangian tori which are distinct up to symplectomorphisms of X . In dimension four arbitrarily many different Lagrangian tori were previously known only in del Pezzo surfaces. Neither the embedded tori, nor the ambient space X needs to be monotone for our methods to work.