  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.) An analogous algebro-geometric construction for
the holomorphic Chern-Simons functional is presented giving the local analytic
moduli scheme of a vector bundle. An analogous gradient scheme construction for
Brill-Noether 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.
