Symplectic surgeries from singularities

Given an isolated analytic hypersurface singularity 0 ∈ X0 := {f(z) = 0} ⊂ C one can form the smoothing Xt = {f(z) = t} and the resolution X̂ → X0, obtained by (repeatedly) blowing up the origin. These two associated spaces have the same link – that is, the intersection S ∩ f(0) – and there is a smooth surgery which replaces the smoothing by the resolution, or vice-versa. Thinking of the smoothing as Kähler (and so symplectic) by restriction of the standard Kähler form on C, the smoothing also looks somewhat like a resolution by symplectic parallel transport, described below. The result is that there is a canonical map or ‘Lagrangian blow up’ X0 ← Xt whose ‘exceptional locus’ (the inverse image of the singular point 0 ∈ X0) is a Lagrangian cycle, in fact a collection of Lagrangian spheres [21]. So the surgery replaces configurations of Lagrangian spheres by complex (symplectic) subvarieties. Symplectic parallel transport also shows that the Xts are all isomorphic as symplectic manifolds, so denoting any such symplectic manifold by X , we can denote this surgery by the diagram (motivated by smoothings and resolutions in algebraic geometry)