Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Weakly Lefschetz symplectic manifolds


Authors: M. Fernández, V. Muñoz and L. Ugarte
Journal: Trans. Amer. Math. Soc. 359 (2007), 1851-1873
MSC (2000): Primary 53D05, 57R17, 53D35, 53C15
DOI: https://doi.org/10.1090/S0002-9947-06-04114-6
Published electronically: October 17, 2006
MathSciNet review: 2272152
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For a symplectic manifold, the harmonic cohomology of symplectic divisors (introduced by Donaldson, 1996) and of the more general symplectic zero loci (introduced by Auroux, 1997) are compared with that of its ambient space. We also study symplectic manifolds satisfying a weakly Lefschetz property, that is, the $ s$-Lefschetz property. In particular, we consider the symplectic blow-ups $ \widetilde{CP}{}^m$ of the complex projective space $ {CP}^m$ along weakly Lefschetz symplectic submanifolds $ M\subset{CP}^m$. As an application we construct, for each even integer $ s\geq 2$, compact symplectic manifolds which are $ s$-Lefschetz but not $ (s+1)$-Lefschetz.


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

  • 1. D. Auroux, Asymptotically holomorphic families of symplectic submanifolds, Geom. Funct. Anal. 7 (1997), 971-995.MR 1487750 (99b:57069)
  • 2. C. Benson and C.S. Gordon, Kähler and symplectic structures on nilmanifolds, Topology 27 (1988), 513-518. MR 0976592 (90b:53042)
  • 3. J.L. Brylinski, A differential complex for Poisson manifolds, J. Diff. Geom. 28 (1988), 93-114. MR 0950556 (89m:58006)
  • 4. G.R. Cavalcanti, The Lefschetz property, formality and blowing up in symplectic geometry, to appear in Trans. Amer. Math. Soc.
  • 5. S. K. Donaldson, Symplectic submanifolds and almost-complex geometry, J. Diff. Geom. 44 (1996), 666-705. MR 1438190 (98h:53045)
  • 6. M. Fernández, M. Gotay and A. Gray, Compact parallelizable four dimensional symplectic and complex manifolds, Proc. Amer. Math. Soc. 103 (1988), 1209-1212. MR 0955011 (90a:53039)
  • 7. M. Fernández, M. de León and M. Saralegui, A six dimensional compact symplectic solvmanifold without Kähler structures, Osaka J. Math. 33 (1996), 19-35. MR 1381616 (97d:53037)
  • 8. M. Fernández, V. Muñoz and J. Santisteban, Cohomologically Kähler manifolds with no Kähler metric, Internat. J. Math. Math. Sc. 52 (2003), 3315-3325. MR 2015766 (2004i:53122)
  • 9. M. Fernández, V. Muñoz, Formality of Donaldson submanifolds, Math. Zeit. 250 (2005), 149-175. MR 2136647 (2006a:53098)
  • 10. R. Gompf, A new construction of symplectic manifolds, Ann. Math. 142 (1995), 527-597. MR 1356781 (96j:57025)
  • 11. M. Gromov, A topological technique for the construction of solutions of differential equations and inequalities, Actes Congrés Intern. Math. (Nice 1970), Gauthier-Villars, Paris, No. 2, 221-225, 1971.MR 0420697 (54:8709)
  • 12. M. Gromov, Partial differential relations, Springer-Verlag, Berlin, 1986.MR 0864505 (90a:58201)
  • 13. A. Hattori, Spectral sequence in the de Rham cohomology of fibre bundles, J. Fac. Sci. Univ. Tokyo 8 (1960), 298-331. MR 0124918 (23:A2226)
  • 14. R. Ibáñez, Y. Rudyak, A. Tralle, and L. Ugarte, On symplectically harmonic forms on $ 6$-dimensional nilmanifolds, Comment. Math. Helv. 76 (2001), 89-109.MR 1819662 (2002b:53120)
  • 15. K. Kodaira, On the structure of compact complex analytic surfaces, I, Amer. J. Math. 86 (1964), 751-798. MR 0187255 (32:4708)
  • 16. J.L. Koszul, Crochet de Schouten-Nijenhuis et cohomologie, in Elie Cartan et les Math. d´Aujour d´Hui, Astérisque hors-série (1985), 251-271.MR 0837203 (88m:17013)
  • 17. A. Lichnerowicz, Les variétés de Poisson et les algébres de Lie associées, J. Diff. Geom. 12 (1977), 253-300. MR 0501133 (58:18565)
  • 18. A.I. Mal'cev, A class of homogeneous spaces, Izvestia Akademii Nauk S.S.S.R. Seriya Matematiceskaya 13 (1949), 9-32. English translation: Amer. Math. Soc. Transl. no. $ 39$, 1951.MR 0028842 (10:507d)
  • 19. O. Mathieu, Harmonic cohomology classes of symplectic manifolds, Comment. Math. Helv. 70 (1995), 1-9.MR 1314938 (96e:58004)
  • 20. D. McDuff, Examples of symplectic simply connected manifolds with no Kähler structure, J. Diff. Geom. 20 (1984), 267-277.MR 0772133 (86c:57036)
  • 21. V. Muñoz, F. Presas and I. Sols, Almost holomorphic embeddings in grassmanians with applications to singular symplectic submanifolds, J. Reine Angew. Math. 547 (2002), 149-189.MR 1900140 (2003f:53159)
  • 22. K. Nomizu, On the cohomology of compact homogeneous spaces of nilpotent Lie groups, Ann. of Math. 59 (1954), 531-538.MR 0064057 (16:219c)
  • 23. R. Paoletti, Symplectic subvarieties of projective fibrations over symplectic manifolds, Ann. Inst. Fourier, Grenoble 49 (1999), 1661-1672.MR 1723830 (2000h:53110)
  • 24. B. Shiffman, S. Zelditch, Asymptotics of almost holomorphic sections of ample line bundles on symplectic manifolds, J. Reine Angew. Math. 544 (2002), 181-222.MR 1887895 (2002m:58043)
  • 25. W.P. Thurston, Some simple examples of symplectic manifolds, Proc. Amer. Math. Soc. 55 (1976), 467-468. MR 0402764 (53:6578)
  • 26. D. Tischler, Closed $ 2$-forms and an embedding theorem for symplectic manifolds, J. Diff. Geom. 12 (1977), 229-235. MR 0488108 (58:7677)
  • 27. D. Yan, Hodge structure on symplectic manifolds, Adv. Math. 120 (1996), 143-154. MR 1392276 (97e:58004)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 53D05, 57R17, 53D35, 53C15

Retrieve articles in all journals with MSC (2000): 53D05, 57R17, 53D35, 53C15


Additional Information

M. Fernández
Affiliation: Departamento de Matemáticas, Facultad de Ciencia y Tecnología, Universidad del País Vasco, Apartado 644, 48080 Bilbao, Spain
Email: marisa.fernandez@ehu.es

V. Muñoz
Affiliation: Departamento de Matemáticas, Consejo Superior de Investigaciones Científicas, C/ Serrano 113bis, 28006 Madrid, Spain
Email: vicente.munoz@imaff.cfmac.csic.es

L. Ugarte
Affiliation: Departamento de Matemáticas, Facultad de Ciencias, Universidad de Zaragoza, Campus Plaza San Francisco, 50009 Zaragoza, Spain
Email: ugarte@unizar.es

DOI: https://doi.org/10.1090/S0002-9947-06-04114-6
Received by editor(s): February 9, 2005
Published electronically: October 17, 2006
Article copyright: © Copyright 2006 American Mathematical Society

American Mathematical Society