Almost-Kähler smoothings of compact complex surfaces with A₁ singularities

2018/06/20 by Caroline Vernier, Vernier, Caroline
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #math.DG

paper · pdf · doi:10.48550/arxiv.1806.07773

arxiv created 2018/06/20 · arxiv updated 2018/06/21

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 (M_ε, ω_ε) admits an almost-Kähler structure ( J_ε, g_ε) of constant Hermitian curvature. Moreover, we show that for ε>0 small enough, the (M_ε, ω_ε) 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 g_ε.

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