Moduli schemes associated to $K$-trivial threefolds as gradient schemes

On a threefold with trivial canonical bundle, Kuranishi theory gives an algebro-geometry construction of the (local analytic) Hilbert scheme of curves at a smooth holomorphic curve as a gradient scheme, that is, the zero-scheme of the exterior derivative of a holomorphic function on a (finite-dimensional) polydisk. (The corresponding fact in an infinite-dimensional setting was long ago discovered by physicists.) This parallels the way the holomorphic Chern-Simons functional gives the local analytic moduli scheme for vector bundles on a Calabi-Yau threefold. An analogous gradient scheme construction for Noether-Lefschetz loci on ample divisors is also given. Finally, using a structure theorem of Donagi-Markman, we present a new formulation of the Abel-Jacobi mapping into the intermediate Jacobian of a threefold with trivial canonical bundle.