Vanishing cycles of symplectic foliations

TL;DR

This paper introduces Lagrangian vanishing cycles in symplectic foliations, demonstrating their role in obstructing strong symplectic structures and providing new rigidity results in higher dimensions.

Contribution

It generalizes vanishing cycles to higher-dimensional symplectic foliations and shows their impact on the topology and rigidity of such structures.

Findings

Lagrangian vanishing cycles prevent symplectic foliations from being strong.

Explicit modifications of foliations can create examples with Lagrangian vanishing cycles.

Provides foundational analytic theory for holomorphic curves in symplectic 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.