2002/06/20 by Herbert Clemens, Clemens, Herbert
Mathematics · #Algebraic Geometry (math.AG) #FOS: Mathematics #math.AG
paper · pdf · doi:10.48550/arxiv.math/0206219
30 pages, contributions by Richard Thomas and Claire Voisin, latex2e file
arxiv created 2002/10/08 · arxiv updated 2009/11/30
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.