  We introduce several families of filtrations on the space of vector bundles
over a smooth projective variety. These filtrations are defined using the large
k asymptotics of the kernel of the Dolbeault Dirac operator on a bundle twisted
by the kth power of an ample line bundle. The filtrations measure the failure
of the bundle to admit a holomorphic structure. We study compatibility under
the Chern isomorphism of these filtrations with the Hodge filtration on
cohomology.
