Smoothing singular constant scalar curvature Kähler surfaces and minimal Lagrangians

Given a complex surface X with singularities of class T and no nontrivial holomorphic vector field, endowed with a Kähler class Ω0, we consider smoothings (Mt,Ωt), where Ωt is a Kähler class on Mt degenerating to Ω0. Under an hypothesis of nondegeneracy of the smoothing at each singular point, we prove that if X admits a constant scalar curvature Kähler metric in Ω0, then Mt admits a constant scalar curvature Kähler metric in Ωt for small t.In addition, we construct small Lagrangian stationary spheres which represent Lagrangian vanishing cycles when t is small.