This paper is concerned with the existence of metrics of constant Hermitian scalar curvature on almost-Kahler manifolds obtained as smoothings of a constant scalar curvature Kahler orbifold, with A1 singularities. More precisely, given such an orbifold that does not admit nontrivial holomorphic vector fields, we show that an almost-Kahler smoothing (Me, ωe) admits an almost-Kahler structure (Je, ge) of constant Hermitian curvature. Moreover, we show that for e > 0 small enough, the (Me, ωe) 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 ge.

中文翻译：

---

具有 $A_1​​$ 奇点的紧凑复杂曲面的几乎 Kähler 平滑

本文关注的是在几乎凯勒流形上存在恒定厄米标量曲率的度量，该流形作为恒定标量曲率 Kahler orbifold 的平滑获得，具有 A1 奇点。更准确地说，给定这样一个不允许非平凡全纯向量场的轨道，我们证明了几乎卡勒平滑 (Me, ωe) 允许恒定厄米曲率的近似卡勒结构 (Je, ge)。此外，我们表明，对于足够小的 e > 0，(Me, ωe) 都辛等价于一个固定的辛流形 (M , ω)，其中有一个表面 S 与一个 2 球体同源，使得 [S ] 是一个消失环，它允许代表是 ge 的哈密顿平稳。