Almost-Kähler 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-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 $(M_\varepsilon , \omega_\varepsilon)$ admits an almost-Kähler structure $(\hat{J}_\varepsilon, \hat{g}_\varepsilon)$ of constant Hermitian curvature. Moreover, we show that for $\varepsilon \gt 0$ small enough, the $(M_\varepsilon, \omega_\varepsilon)$ 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 $\hat{g}_\varepsilon$.

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Received 6 November 2018

Accepted 11 October 2019

Published 29 October 2020