Almost-Kähler smoothings of compact complex surfaces with A1 singularities

Caroline Vernier (1)

1 LMJL - Laboratoire de Mathématiques Jean Leray

This paper is concerned with the existence of metrics of constant Hermitian scalar curvature on almost-Kähler manifolds obtained as smoothings of a constant scalar curvature Kähler orbifold, with A1 singularities. More precisely, given such an orbifold that does not admit nontrivial holomorphic vector fields, we show that an almost-Kähler smoothing (Mε, ωε) admits an almost-Kähler structure (Jε, gε) of constant Hermitian curvature. Moreover, we show that for ε > 0 small enough, the (Mε, ωε) are all symplectically equivalent to a fixed symplectic manifold (M , ω) in which there is a surface S homologous to a 2-sphere, such that [S] is a vanishing cycle that admits a representant that is Hamiltonian stationary for gε.

Domaines: Géométrie différentielle [math.DG]

Caroline Vernier. Almost-Kähler smoothings of compact complex surfaces with A1 singularities. 2018. ⟨hal-01819616⟩