The Lefschetz property, formality and blowing up in symplectic geometry
HTML articles powered by AMS MathViewer
- by Gil Ramos Cavalcanti PDF
- Trans. Amer. Math. Soc. 359 (2007), 333-348 Request permission
Abstract:
In this paper we study the behaviour of the Lefschetz property under the blow-up construction. We show that it is possible to reduce the dimension of the kernel of the Lefschetz map if we blow up along a suitable submanifold satisfying the Lefschetz property. We use this, together with results about Massey products, to construct compact nonformal symplectic manifolds satisfying the Lefschetz property.References
- I. K. Babenko and I. A. Taĭmanov, Massey products in symplectic manifolds, Mat. Sb. 191 (2000), no. 8, 3–44 (Russian, with Russian summary); English transl., Sb. Math. 191 (2000), no. 7-8, 1107–1146. MR 1786415, DOI 10.1070/SM2000v191n08ABEH000497
- I. K. Babenko and I. A. Taĭmanov, On nonformal simply connected symplectic manifolds, Sibirsk. Mat. Zh. 41 (2000), no. 2, 253–269, i (Russian, with Russian summary); English transl., Siberian Math. J. 41 (2000), no. 2, 204–217. MR 1762178, DOI 10.1007/BF02674589
- Chal Benson and Carolyn S. Gordon, Kähler and symplectic structures on nilmanifolds, Topology 27 (1988), no. 4, 513–518. MR 976592, DOI 10.1016/0040-9383(88)90029-8
- Jean-Luc Brylinski, A differential complex for Poisson manifolds, J. Differential Geom. 28 (1988), no. 1, 93–114. MR 950556
- G. R. Cavalcanti, New aspects of the $dd^c$-lemma, Ph.D. thesis, Oxford University, 2004, math.DG/0501406.
- Luis A. Cordero, M. Fernández, and A. Gray, Symplectic manifolds with no Kähler structure, Topology 25 (1986), no. 3, 375–380. MR 842431, DOI 10.1016/0040-9383(86)90050-9
- Pierre Deligne, Phillip Griffiths, John Morgan, and Dennis Sullivan, Real homotopy theory of Kähler manifolds, Invent. Math. 29 (1975), no. 3, 245–274. MR 382702, DOI 10.1007/BF01389853
- S. K. Donaldson, Symplectic submanifolds and almost-complex geometry, J. Differential Geom. 44 (1996), no. 4, 666–705. MR 1438190
- Marisa Fernández and Vicente Muñoz, Formality of Donaldson submanifolds, Math. Z. 250 (2005), no. 1, 149–175. MR 2136647, DOI 10.1007/s00209-004-0747-8
- Robert E. Gompf, A new construction of symplectic manifolds, Ann. of Math. (2) 142 (1995), no. 3, 527–595. MR 1356781, DOI 10.2307/2118554
- M. L. Gromov, A topological technique for the construction of solutions of differential equations and inequalities, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars, Paris, 1971, pp. 221–225. MR 0420697
- M. Gualtieri, Generalized complex geometry, Ph.D. thesis, Oxford University, 2003, math.DG/0401221.
- Nigel Hitchin, Generalized Calabi-Yau manifolds, Q. J. Math. 54 (2003), no. 3, 281–308. MR 2013140, DOI 10.1093/qjmath/54.3.281
- Raúl Ibáñez, Yuli Rudyak, Aleksy Tralle, and Luis Ugarte, On certain geometric and homotopy properties of closed symplectic manifolds, Proceedings of the Pacific Institute for the Mathematical Sciences Workshop “Invariants of Three-Manifolds” (Calgary, AB, 1999), 2003, pp. 33–45. MR 1953319, DOI 10.1016/S0166-8641(02)00041-X
- Jean-Louis Koszul, Crochet de Schouten-Nijenhuis et cohomologie, Astérisque Numéro Hors Série (1985), 257–271 (French). The mathematical heritage of Élie Cartan (Lyon, 1984). MR 837203
- Gregory Lupton and John Oprea, Symplectic manifolds and formality, J. Pure Appl. Algebra 91 (1994), no. 1-3, 193–207. MR 1255930, DOI 10.1016/0022-4049(94)90142-2
- Olivier Mathieu, Harmonic cohomology classes of symplectic manifolds, Comment. Math. Helv. 70 (1995), no. 1, 1–9. MR 1314938, DOI 10.1007/BF02565997
- Dusa McDuff, Examples of simply-connected symplectic non-Kählerian manifolds, J. Differential Geom. 20 (1984), no. 1, 267–277. MR 772133
- Dusa McDuff and Dietmar Salamon, Introduction to symplectic topology, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. Oxford Science Publications. MR 1373431
- S. A. Merkulov, Formality of canonical symplectic complexes and Frobenius manifolds, Internat. Math. Res. Notices 14 (1998), 727–733. MR 1637093, DOI 10.1155/S1073792898000439
- Timothy James Miller, On the formality of $(k-1)$-connected compact manifolds of dimension less than or equal to $4k-2$, Illinois J. Math. 23 (1979), no. 2, 253–258. MR 528561
- Katsumi Nomizu, On the cohomology of compact homogeneous spaces of nilpotent Lie groups, Ann. of Math. (2) 59 (1954), 531–538. MR 64057, DOI 10.2307/1969716
- A. N. Paršin, On a certain generalization of Jacobian manifold, Izv. Akad. Nauk SSSR Ser. Mat. 30 (1966), 175–182 (Russian). MR 0196770
- Yuli Rudyak and Aleksy Tralle, On Thom spaces, Massey products, and nonformal symplectic manifolds, Internat. Math. Res. Notices 10 (2000), 495–513. MR 1759504, DOI 10.1155/S1073792800000271
- David Tischler, Closed $2$-forms and an embedding theorem for symplectic manifolds, J. Differential Geometry 12 (1977), no. 2, 229–235. MR 488108
- Dong Yan, Hodge structure on symplectic manifolds, Adv. Math. 120 (1996), no. 1, 143–154. MR 1392276, DOI 10.1006/aima.1996.0034
Additional Information
- Gil Ramos Cavalcanti
- Affiliation: Mathematical Institute, University of Oxford, St. Giles 24-29, Oxford, OX1 3BN, United Kingdom
- MR Author ID: 757552
- Email: gilrc@maths.ox.ac.uk
- Received by editor(s): November 14, 2004
- Published electronically: August 15, 2006
- Additional Notes: This research was supported by CAPES (Coordenação de Aperfeiçoamento de Pessoal de Nível Superior, Ministério da Educação e Cultura), Brazilian Government, Grant 1326/99-6
- © Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 359 (2007), 333-348
- MSC (2000): Primary 53D35; Secondary 57R19
- DOI: https://doi.org/10.1090/S0002-9947-06-04058-X
- MathSciNet review: 2247894