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

Full-text PDF

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]**Bogomolov, F.A., ``Hamiltonian Kählerian manifolds.'' Dokl. Akad. Nauk SSSR**243**, no. 5 (1978), 1101-1104. MR**0514769 (80c:32024)****[C]**Clemens, H. ``Geometry of formal Kuranishi theory.'' Preprint, math.AG/9901084, Advances in Math., to appear.**[DM]**Donagi, R., Markman, E. ``Cubics, integrable systems, and Calabi-Yau threefolds.'' Proc. Hirzebruch 65-birthday Conference in Algebraic Geometry. Edited by M. Teicher, Israel Math. Conf. Proc.**9**(1996). MR**1360503 (97f:14039)****[DT]**Donaldson, S.K., Thomas, R. ``Gauge theory in higher dimensions.''*The Geometric Universe: Science, Geometry and the work of Roger Penrose*, Edited by S.A. Huggett et al., Oxford University Press: 1998. MR**1634503 (2000a:57085)****[KS]**Katz, S., Sharpe, E. ``Notes on Ext groups and D-branes.'' Preprint University of Illinois at Urbana-Champaign (2002).**[Ko]**Kollár, J.*Rational curves on algebraic varieties.*Ergebnisse der Mathematik und Grenzgebiete 3. Folge, vol. 32, Springer-Verlag: 1996. MR**1440180 (98c:14001)****[Ti]**Tian, G. ``Smoothness of the universal deformation space of compact Calabi-Yau manifolds and its Petersson-Weil metric.''*Math. Aspects of String Theory*. Yau, S.-T.(ed.). Singapore: World Scientific (1987). MR**0915841****[Tj]**Tjurin, A.K. ``Non-abelian analogues of Abel's theorem.'' Izv. Ross. Akad. Nauk Ser. Mat.**65**, no. 1 (2001), 133-196.**[To]**Todorov, A. ``The Weil-Petersson Geometry of the Moduli Space of (Calabi-Yau) Manifolds I.'' Commun. Math. Phys.**126**(1989), 325-346. MR**1027500 (91f:32022)****[V]**Voisin, C.*Variations of Hodge Structure of Calabi-Yau Threefolds.*Leziono LaGrange, Pub.Cl.Sc. della Scuola Normale Sup., Pisa, (1996), 7-14. MR**1658398 (99m:14023)****[W]**Witten, E. ``Chern-Simons gauge theory as a string theory.'' Floer Memorial Volume: Progr. Math.**133**, Birkhäuser (1995), 637-678. MR**1362846 (97j:57052)**

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