Autour du programme de Calabi, méthodes de recollement

Abstract : We study 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 $(J_\varepsilon, g_\varepsilon)$ of constant Hermitian curvature. Moreover, we show that for $ \varepsilon > 0$ small enough, the $(M_\varepsilon, \omega_\varepsilon)$ are all symplectically equivalent to a fixed symplectic manifold $(M , \omega)$ 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 $g_\varepsilon$.
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Caroline Vernier. Autour du programme de Calabi, méthodes de recollement. Géométrie différentielle [math.DG]. Université Bretagne Loire, 2018. Français. ⟨tel-01912801v1⟩

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