Almost-Kähler smoothings of compact complex surfaces with $A_1$ singularities

Abstract

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 $A_1$ singularities. More precisely, given such an orbifold that does not admit nontrivial holomorphic vector fields, we show that an almost-Kähler smoothing $(M_{\epsilon },{\omega }_{\epsilon })$ admits an almost-Kähler structure $({\hat{J}}_{\epsilon },{\hat{g}}_{\epsilon })$ of constant Hermitian curvature. Moreover, we show that for $\epsilon >0$ small enough, the $(M_{\epsilon },{\omega }_{\epsilon })$ are all symplectically equivalent to a fixed symplectic manifold $(\hat{M},\hat{\omega })$ 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 ${\hat{g}}_{\epsilon }$.