×

zbMATH — the first resource for mathematics

Hodge theory and deformations of maps. (English) Zbl 0845.14007
The paper under review is an abstraction of previous results of the authors into the most general (non)-sensical context. The general idea, formulated in vague terms, is that the image of a Hodge-type group under a functorial homomorphism into an infinitesimal deformation group lands in the unobstructed deformations. Dually given a functorial homomorphism from an obstruction group into a Hodge type group, the effective obstructions lie in the kernel.
As applications the author considers deformations of maps, especially immersions (generalizing a result by C. Voisin to general \(q\)-symplectic complex manifolds, to the effect that the deformation spaces of Lagrangian immersions are smooth) and fiber spaces. The author concludes by considering conditions for curves to move, getting results for symplectic \(n\)-folds, and considering the structure of Kähler manifolds, whose bundles of holomorphic two-forms are spanned outside a finite set of points.

MSC:
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
PDF BibTeX XML Cite
Full Text: Numdam EuDML
References:
[1] Bloch, S. Semi-regularity and de Rham cohomology , Invent. math. 17 (1972), 51-66. · Zbl 0254.14011 · doi:10.1007/BF01390023 · eudml:142158
[2] Deligne, P. Théorème de Lefschetz et critères de dégénérescence de suits spectrales , Publ. Math. IHES, 35 (1968), 197-226. · Zbl 0159.22501 · doi:10.1007/BF02698925 · numdam:PMIHES_1968__35__107_0 · eudml:103884
[3] Fujiki, A. On automorphism groups of compact Kähler manifolds , Invent. Math. 44 (1978), 225-258. · Zbl 0367.32004 · doi:10.1007/BF01403162 · eudml:142529
[4] Lichnérowicz, A. Sur les transformations analytiques des variétés Kähleriennes . C.R. Acad. Sci. Paris 244 (1957), 3011-3014. · Zbl 0080.37501
[5] Narasimhan, M.S. , Seshadri, C.S. Stable and unitary vector bundles on a compact Riemann surface . Ann. of Math. 82 (1965), 540-567. · Zbl 0171.04803 · doi:10.2307/1970710
[6] Ran, Z. Deformations of manifolds with torsion or negative canonical bundle . J. Algebraic Geometry, 1 (1992), 279-292. · Zbl 0818.14003
[7] Ran, Z. Lifting of cohomology and deformations of certain holomorphic maps . Bull. AMS 26 (1992), 113-117. · Zbl 0749.32010 · doi:10.1090/S0273-0979-1992-00244-6 · arxiv:math/9201267
[8] Ran, Z. Hodge theory and the Hilbert scheme . J. Differential Geometry 37 (1993), 191-198; Erratum & Addendum (to appear). · Zbl 0804.14004 · doi:10.4310/jdg/1214453428
[9] Ran, Z. The structure of Gauss-like maps . Compositio Math. 32 (1984), 171-177 · Zbl 0547.14004 · numdam:CM_1984__52_2_171_0 · eudml:89657
[10] Ran, Z. Deformations of maps . In: Algebraic curves and projective geometry , E. Ballico, C. Cilberto, eds., Lecture notes in Math 1389. Berlin: Springer 1989. · Zbl 0708.14006
[11] Ran, Z. Deformations of Calabi-Yau Kleinfolds . In: Essays on mirror manifolds , S. T. Yau ed. pp. 451-457. Hong Kong: International Press 1992. · Zbl 0827.32021
[12] Ran, Z. Canonical infinitesimal deformations . Preprint. · Zbl 1060.14016 · arxiv:math/9810041
[13] Schlessinger, M. Functors of Artin rings , Trans. AMS 130 (1968), 208-222. · Zbl 0167.49503 · doi:10.2307/1994967
[14] Ueno, K. Classification theory of algebraic varieties and compact complex spaces. Lecture notes in Math. 439 . Berlin: Springer 1975. · Zbl 0299.14007
[15] Voisin, C. Sur la stabilité des sous-variétés lagrangiennes des variétés symplectiques holomorphes . Preprint. · Zbl 0765.32012
[16] Zariski, O. Theorem of Riemann-Roch for high multiples of an effective divisor on an algebraic surface . Ann. Math. 76 (1962), 560-615. · Zbl 0124.37001 · doi:10.2307/1970376
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.