Two-dimensional Lagrangian singularities and bifurcations of gradient lines I

Motivated by mirror symmetry, we consider a Lagrangian fibration X → B and Lagrangian maps f : L →֒ X → B, when L has dimension 2, exhibiting an unstable singularity, and study how their caustic changes, in a neighbourhood of the unstable singularity, when slightly perturbed. The integral curves of ∇fx, for x ∈ B, where fx(y) = f(y)−x ·y, called “gradient lines”, are then introduced, and a study of them, in order to analyse their bifurcation locus, is carried out. Supported by a JSPS Postdoctoral fellowship 2000 Mathematics Subject Classification: 37G25, 53D12, 70K60 E-Mail address: [email protected]