Moduli schemes associated to -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.

**[B]**F. A. Bogomolov,*Hamiltonian Kählerian manifolds*, Dokl. Akad. Nauk SSSR**243**(1978), no. 5, 1101–1104 (Russian). MR**514769****[C]**Clemens, H. ``Geometry of formal Kuranishi theory.'' Preprint, math.AG/9901084, Advances in Math., to appear.**[DM]**Ron Donagi and Eyal Markman,*Cubics, integrable systems, and Calabi-Yau threefolds*, Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry (Ramat Gan, 1993) Israel Math. Conf. Proc., vol. 9, Bar-Ilan Univ., Ramat Gan, 1996, pp. 199–221. MR**1360503****[DT]**S. K. Donaldson and R. P. Thomas,*Gauge theory in higher dimensions*, The geometric universe (Oxford, 1996) Oxford Univ. Press, Oxford, 1998, pp. 31–47. MR**1634503****[KS]**Katz, S., Sharpe, E. ``Notes on Ext groups and D-branes.'' Preprint University of Illinois at Urbana-Champaign (2002).**[Ko]**János Kollár,*Rational curves on algebraic varieties*, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 32, Springer-Verlag, Berlin, 1996. MR**1440180****[Ti]**Gang Tian,*Smoothness of the universal deformation space of compact Calabi-Yau manifolds and its Petersson-Weil metric*, Mathematical aspects of string theory (San Diego, Calif., 1986) Adv. Ser. Math. Phys., vol. 1, World Sci. Publishing, Singapore, 1987, pp. 629–646. MR**915841****[Tj]**Tjurin, A.K. ``Non-abelian analogues of Abel's theorem.'' Izv. Ross. Akad. Nauk Ser. Mat.**65**, no. 1 (2001), 133-196.**[To]**Andrey N. Todorov,*The Weil-Petersson geometry of the moduli space of 𝑆𝑈(𝑛≥3) (Calabi-Yau) manifolds. I*, Comm. Math. Phys.**126**(1989), no. 2, 325–346. MR**1027500****[V]**Claire Voisin,*Variations of Hodge structure of Calabi-Yau threefolds*, Lezioni Lagrange [Lagrange Lectures], vol. 1, Scuola Normale Superiore, Classe di Scienze, Pisa, 1996. MR**1658398****[W]**E. Witten,*Chern-Simons gauge theory as a string theory*, The Floer memorial volume, Progr. Math., vol. 133, Birkhäuser, Basel, 1995, pp. 637–678. MR**1362846**

Additional Information

**Herb Clemens**

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

Email:
clemens@math.ohio-state.edu

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

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