Moduli schemes associated to -trivial threefolds as gradient schemes
Abstract: 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.
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Affiliation: Mathematics Department, Ohio State University, 231 W. 18th Ave., Columbus, Ohio 43210
Received by editor(s): August 26, 2004
Received by editor(s) in revised form: February 28, 2005, and March 17, 2005
Published electronically: May 12, 2005
Additional Notes: Partially supported by NSF grant DMS-0200895