Symplectic Hodge theory, harmonicity, and Thom duality

Abstract

We study the notion of harmonicity in the sense of symplectic geometry, and investigate the geometric properties of harmonic Thom forms and distributional Thom currents, dual to different types of submanifolds. We show that the harmonic Thom form associated to a symplectic submanifold is nowhere vanishing. We also construct symplectic smoothing operators which preserve the harmonicity of distributional currents and using these operators, construct harmonic Thom forms for co-isotropic submanifolds, which unlike the harmonic forms associated with symplectic submanifolds, are supported in an arbitrary tubular neighborhood of the manifold.