Toward a topological characterization of symplectic manifolds. (English) Zbl 1084.53072

The paper introduces a notion of a hyperpencil on a \(2n\)-dimensional manifold \(X\). It is a map \(f:X\to \mathbb{CP}^{n-1}\) which generalizes a linear system of curves on an algebraic variety. The main result of the paper states that if a \(2n\)-dimensional smooth, closed, oriented manifold admits a hyperpencil then it admits a symplectic structure. This is in the spirit of the author’s construction of a symplectic structure on a symplectic Lefschetz fibration (and both generalize the Thurston construction of a symplectic structure on certain surface bundles). More precisely, for a smooth, closed, oriented manifold with a hyperpencil there exists a contractible set of symplectic structures (in the cohomology class defined by the pencil). This defines a map from deformation classes of hyperpencils to isotopy classes of symplectic forms. On the other hand, by the results of Donaldson and Auroux, any symplectic manifold of dimension at most 6 admits a hyperpencil. The author conjectures that this extends to arbitrary dimension. In dimensions where the conjecture holds, the existence of a hyperpencil topologically characterizes those manifolds that admit symplectic structures. The paper is well written and contains nice examples.


53D35 Global theory of symplectic and contact manifolds
57R17 Symplectic and contact topology in high or arbitrary dimension
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