Moduli schemes associated to $K$-trivial threefolds as gradient schemes

Author:
Herb Clemens

Journal:
J. Algebraic Geom. **14** (2005), 705-739

DOI:
https://doi.org/10.1090/S1056-3911-05-00413-3

Published electronically:
May 12, 2005

MathSciNet review:
2147351

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Abstract | References | Additional Information

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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**65**, no. 1 (2001), 133–196.

Additional Information

**Herb Clemens**

Affiliation:
Mathematics Department, Ohio State University, 231 W. 18th Ave., Columbus, Ohio 43210

Email:
clemens@math.ohio-state.edu

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