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 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 (Ĵε,ĝε) 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 forĝε.

Extremal metrics on fibrations

“…A novel feature in our approach is therefore that we will be applying Moser's trick to pull back our approximate solutions to a fixed formthe form constructed on the central fibre. We remark that working on the level of almost complex structures has been considered by Vernier [40], but her method is rather different, in that it does not involve the use of Moser's trick. Ultimately, she is interested in applying the methods in a non-Kähler setting, but even if the starting manifold is Kähler, her gluing method would break the integrability of the almost complex structure.…”