Cohomology and Obstructions II: Curves on K-trivial threefolds

Details Cohomology and Obstructions II: Curves on K-trivial threefolds Cohomology and Obstructions II: Curves on K-trivial threefolds

2002-06-20

arxiv.org/abs/math/0206219v2

Mathematics

Algebraic Geometry

30 pages, contributions by Richard Thomas and Claire Voisin, latex2e file

Scientific paper

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.

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