Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)



Lifting of cohomology and unobstructedness of certain holomorphic maps

Author: Ziv Ran
Journal: Bull. Amer. Math. Soc. 26 (1992), 113-117
MSC (2000): Primary 32G05; Secondary 14D15, 32G13
MathSciNet review: 1102754
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let f be a holomorphic mapping between compact complex manifolds. We give a criterion for f to have unobstructed deformations, i.e. for the local moduli space of f to be smooth: this says, roughly speaking, that the group of infinitesimal deformations of f, when viewed as a functor, itself satisfies a natural lifting property with respect to infinitesimal deformations. This lifting property is satisfied e.g. whenever the group in question admits a 'topological' or Hodge-theoretic interpretation, and we give a number of examples, mainly involving Calabi-Yau manifolds, where that is the case.

References [Enhancements On Off] (What's this?)

  • [B] F. A. Bogomolov, Hamiltonian Kählerian manifolds, Dokl. Akad. Nauk SSSR 243 (1978), no. 5, 1101–1104 (Russian). MR 514769
  • [H] Eiji Horikawa, On deformations of holomorphic maps. III, Math. Ann. 222 (1976), no. 3., 275–282. MR 0417458,
  • [R1] Ziv Ran, Deformations of maps, Algebraic curves and projective geometry (Trento, 1988) Lecture Notes in Math., vol. 1389, Springer, Berlin, 1989, pp. 246–253. MR 1023402,
  • [R2] Ziv Ran, Stability of certain holomorphic maps, J. Differential Geom. 34 (1991), no. 1, 37–47. MR 1114451
  • [R3] Ziv Ran, Deformations of manifolds with torsion or negative canonical bundle, J. Algebraic Geom. 1 (1992), no. 2, 279–291. MR 1144440
  • [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
  • [To] A. N. Todorov, The Weil-Petersson geometry of the moduli space of $ SU(n \geq 3)$ (Calabi-Yau) manifolds, preprint IHES, November, 1988.
  • [V] Claire Voisin, Sur la stabilité des sous-variétés lagrangiennes des variétés symplectiques holomorphes, Complex projective geometry (Trieste, 1989/Bergen, 1989) London Math. Soc. Lecture Note Ser., vol. 179, Cambridge Univ. Press, Cambridge, 1992, pp. 294–303 (French). MR 1201391,

Similar Articles

Retrieve articles in Bulletin of the American Mathematical Society with MSC (2000): 32G05, 14D15, 32G13

Retrieve articles in all journals with MSC (2000): 32G05, 14D15, 32G13

Additional Information

Article copyright: © Copyright 1992 American Mathematical Society

American Mathematical Society