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Quelques propriétés symplectiques des variétés Kählériennes

Abstract : Symplectic geometry and complex geometry are closely related, in particular by Donaldson and Auroux’s asymptotically holomorphic techniques and by Eliashberg and Cieliebak’s work on pseudoconvexity. The work presented in this thesis is motivated by these two connections. We first give the following symplectic characterisation of Seshadri constants. In a complex manifold, the Seshadri constant of an integral Kähler class at a point is the upper bound on the capacities of standard balls admitting, for some Kähler form in this class, a codimension 0 holomorphic and iso-Kähler embedding centered at this point. This criterion was known by Eckl in 2014; we give a different proof of it. The second part is motivated by Donaldon’s following question: ‘Is every Lagrangian sphere of a complex projective manifold a vanishing cycle of a complex deformation to a variety with a conical singularity?’ On the one hand, we present every closed Lagrangian submanifold of a closed symplectic/Kähler manifold whose relative periods are integers as the lowest level set of a ‘convex’ exhaustion defined on the complement of a symplectic/complex hyperplane section. In the Kähler setting ‘complex’ means strictly plurisubharmonic while in the symplectic setting it refers to the existence of a Liouville pseudogradient. On the other hand, we prove that any Lagrangian sphere of a Stein domain which is the lowest level-set of a ‘convex’ function is a vanishing cycle of some complex deformation over the disc to a variety with a conical singularity.
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Submitted on : Monday, November 19, 2018 - 4:02:07 PM
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Alexandre Vérine. Quelques propriétés symplectiques des variétés Kählériennes. Géométrie différentielle [math.DG]. Université de Lyon, 2018. Français. ⟨NNT : 2018LYSEN038⟩. ⟨tel-01927097⟩

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