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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.

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