The Spectral Density Function for the Laplacian on High Tensor Powers of a Line Bundle

ω(Ju, Jv) = ω(u, v) and ω(·, J ·) ≫ 0. The combination thus defines an associated Riemannian metric β(·, ·) = ω(·, J ·). Any symplectic manifold possesses such a structure. We will assume further that ω is ‘integral’ in the cohomological sense. This means we can find a complex hermitian line bundle L → X with hermitian connection ∇ whose curvature is −iω. Recently, beginning with Donaldson’s seminal paper, [5], the notion of “nearly holomorphic” or “asymptotically holomorphic” sections of L⊗k has attracted a fair amount of attention. Let us recall that one natural way to define spaces of such sections is by means of an analogue of the ∂-Laplacian [2, 3].