Cohomology and Obstructions II: Curves on K-trivial threefolds

TL;DR

This paper develops algebraic geometric constructions for moduli spaces of curves and vector bundles on K-trivial threefolds, linking them with gradient schemes and the Abel-Jacobi map.

Contribution

It introduces new algebraic geometric methods to describe moduli spaces of curves and bundles on K-trivial threefolds using gradient schemes and extends the Abel-Jacobi map framework.

Findings

Construction of local Hilbert schemes as gradient schemes

Algebro-geometric description of the moduli of vector bundles

New formulation of the Abel-Jacobi map for threefolds

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.) 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.