Toward a topological characterization of symplectic manifolds

Abstract

A fibration-like structure called a hyperpencil is defined on a smooth, closed 2n-manifold X, generalizing a linear system of curves on an algebraic variety. A deformation class of hyperpencils is shown to determine an isotopy class of symplectic structures on X. This provides an inverse to Donaldson's program for constructing linear systems on symplectic manifolds. In dimensions ≤ 6, work of Donaldson and Auroux provides hyperpencils on any symplectic manifold, and the author conjectures that this extends to arbitrary dimensions. In dimensions where this holds, the set of deformation classes of hyperpencils canonically maps onto the set of isotopy classes of rational symplectic forms up to positive scale, topologically determining a dense subset of all symplectic forms up to an equivalence relation on hyperpencils. In particular, the existence of a hyperpencil topologically characterizes those manifolds in dimensions ≤ 6 (and perhaps in general) that admit symplectic structures.