Smoothing singular constant scalar curvature Kähler surfaces and minimal Lagrangians

Olivier Biquard (1, 2) , Yann Rollin (3)

1 UPMC - Université Pierre et Marie Curie - Paris 6
2 DMA - Département de Mathématiques et Applications - ENS Paris
3 Géométrie et analyse globale

Given a complex surface X with singularities of class T and no nontrivial holomorphic vector field, endowed with a Kähler class Ω 0 , we consider smoothings (M t , Ω t ), where Ω t is a Kähler class on M t degenerating to Ω 0 .Under an hypothesis of non degeneracy of the smoothing at each singular point, we prove that if X admits a constant scalar curvature Kähler metric in Ω 0 , then M t admits a constant scalar curvature Kähler metric in Ω t for small t.In addition, we construct small Lagrangian stationary spheres which represent Lagrangian vanishing cycles when t is small.

Domaines: Géométrie différentielle [math.DG]

Olivier Biquard, Yann Rollin. Smoothing singular constant scalar curvature Kähler surfaces and minimal Lagrangians. Advances in Mathematics, 2015, 285, pp.980-1024. ⟨10.1016/j.aim.2015.08.013⟩. ⟨hal-02928847⟩