Almost-K\"ahler smoothings of compact complex surfaces with $A_1$ singularities

This paper is concerned with the existence of metrics of constant Hermitian scalar curvature on almost-K"ahler manifolds obtained as smoothings of a constant scalar curvature K"ahler orbifold, with $A_1$ singularities. More precisely, given such an orbifold that does not admit nontrivial holomorphic vector fields, we show that an almost-K"ahler smoothing $(M_\varepsilon, \omega_\varepsilon)$admitsanalmost-K\ddot{a}hlerstructure$(\hat J_\varepsilon, \hat g_\varepsilon)$ 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 $(\hat M, \hat \omega)$inwhichthereisasurface$S$homologoustoa2-sphere,suchthat$[S]$ is a vanishing cycle that admits a representant that is Hamiltonian stationary for ${\widehat{g}}_{\epsilon }$.