Vanishing cycles of symplectic foliations
Fabio Gironella, Klaus Niederkrüger, Lauran Toussaint
TL;DR
The paper develops a high-dimensional analogue of Novikov vanishing cycles for strong symplectic foliations and proves that any Lagrangian vanishing cycle in such a foliation on a closed manifold must be trivial, thereby extending rigidity phenomena from dimension three to higher dimensions. The authors introduce a precise notion of a Lagrangian vanishing cycle modeled on a closed manifold $S$, and they construct a robust analytic framework for leafwise pseudo-holomorphic discs using domain-dependent almost complex structures to establish transversality and compactness. Central to the argument is a leafwise Bishop family near the vanishing core, whose moduli space yields uniform energy bounds and topological constraints that forbid nontrivial cycles. In addition to the main obstruction theorem, the work provides explicit construction methods to modify foliations (opening leaves, preserving vanishing cycles) and yields infinite families of taut, codimension-one foliations containing nontrivial Lagrangian vanishing cycles, highlighting the nuanced balance between topology and symplectic geometry in higher-dimensional foliations.
Abstract
Several results in recent years have shown that the usual generalizations of taut foliations to higher dimensions, based only on topological concepts, lead to a theory that lacks the complexity of its 3-dimensional counterpart. Instead, we propose strong symplectic foliations as natural candidates for such a generalization and we prove in this article that they do yield some interesting rigidity results, such as potentially topological obstructions on the underlying ambient manifold. We introduce a high-dimensional generalization of 3-dimensional vanishing cycles for symplectic foliations, which we call Lagrangian vanishing cycles, and prove that they prevent a symplectic foliation from being strong, just as vanishing cycles prevent tautness in dimension 3 due to the classical result of Novikov from 1964. We then describe, in every codimension, examples of symplectically foliated manifolds which admit Lagrangian vanishing cycles, but for which more classical arguments fail to obstruct strongness. In codimension 1, this is achieved by a rather explicit modification of the symplectic foliation, which allows us to open up closed leaves having non-trivial holonomy on both sides, and is thus of independent interest. Since there is no comprehensive source on holomorphic curves with boundary in symplectic foliations, we also give a detailed introduction to much of the analytic theory, in the hope that it might serve as a reference for future work in this direction.