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arXiv · 2106.02618

Examples of symplectic non-leaves

Fabio Gironella·Lauran Toussaint
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Abstract

This paper deals with the following question: which manifolds can be realized as leaves of codimension-1 symplectic foliations on closed manifolds? We first observe that leaves of symplectic foliations are necessarily strongly geometrically bounded. We show that a symplectic structure which admits an exhaustion by compacts with (convex) contact boundary can be deformed to a strongly geometrically bounded one. We then give examples of smooth manifolds which admit a strongly geometrically bounded symplectic form and can be realized as a smooth leaf, but not as a symplectic leaf for any choice of symplectic form on them. Lastly, we show that the (complex) blowup of 2n-dimensional Euclidean space at infinitely many points, both admits strongly geometrically bounded symplectic forms for which it can and cannot be realized as a symplectic leaf.

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Fabio Gironella, Lauran Toussaint. 2021-06-04. Examples of symplectic non-leaves. https://doi.org/10.4171/rmi%2F1520

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