Locally conformal symplectic nilmanifolds with no locally conformal Kähler metrics

References

[1] D. Angella, G. Bazzoni and M. Parton, Structure of locally conformally symplectic Lie algebras and solvmanifolds, preprint, https://arxiv.org/abs/1704.01197.Search in Google Scholar

[2] G. Bande and D. Kotschick, Contact Pairs and Locally Conformal Symplectic Structures, Harmonic Maps and Differential Geometry, 85-98, Contemp. Math., 542, Amer. Math. Soc., Providence, RI, 2011.10.1090/conm/542/10700Search in Google Scholar

[3] A. Banyaga, Some properties of locally conformal symplectic structures, Comment. Math. Helv. 77 (2002) 383-398.10.1007/s00014-002-8345-zSearch in Google Scholar

[4] W. P. Barth, K. Hulek, C. A. M. Peters and A. Van de Ven, Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 4. Springer-Verlag, Berlin, 2004.10.1007/978-3-642-57739-0Search in Google Scholar

[5] G. Bazzoni, Vaisman nilmanifolds, Bull. Lond. Math. Soc. 49 (5) (2017), 824-830.10.1112/blms.12073Search in Google Scholar

[6] G. Bazzoni and J. C. Marrero, Locally conformal symplectic manifolds of the first kind, Bull. Sci. math. (2017), https://doi.org/10.1016/j.bulsci.2017.10.001.10.1016/j.bulsci.2017.10.001Search in Google Scholar

[7] G. Bazzoni and V. Muñoz, Classification of minimal algebras over any field up to dimension 6, Trans. Amer. Math. Soc. 364 (2), 1007-1028, 2012.10.1090/S0002-9947-2011-05471-1Search in Google Scholar

[8] G. Bazzoni and V. Muñoz, Manifolds which are complex and symplectic but not Kähler. In: Rassias T., Pardalos P. (eds) Essays in Mathematics and its Applications. Springer, ChamSearch in Google Scholar

[9] F. Belgun, On the metric structure of non-Kähler complex surfaces, Math. Ann. 317 (2000), no. 1, 1-40.Search in Google Scholar

[10] C. Benson and C. S. Gordon, Kähler and symplectic structures on nilmanifolds, Topology 27, 513-518, 1988.10.1016/0040-9383(88)90029-8Search in Google Scholar

[11] K. S. Brown, Cohomology of Groups, GTM 87, Springer-Verlag, 1982.10.1007/978-1-4684-9327-6Search in Google Scholar

[12] G. R. Cavalcanti, The Lefschetz property, formality and blowing up in symplectic geometry, Trans. Amer. Math. Soc. 359, no. 1, 333-348, 2007.10.1090/S0002-9947-06-04058-XSearch in Google Scholar

[13] S. Dragomir and L. Ornea, Locally conformal Kähler geometry, Progress in Mathematics 155, Birkhäuser Boston, 1998.10.1007/978-1-4612-2026-8Search in Google Scholar

[14] M. Fernández and V. Muñoz, An 8-dimensional non-formal simply connected symplectic manifold, Ann. of Math. (2) 167, no. 3, 1045-1054, 2008.10.4007/annals.2008.167.1045Search in Google Scholar

[15] M. Freedman, R. Hain and P. Teichner, Betti number estimates for nilpotent groups, Fields Medalists’ Lectures, 413-434, World Sci. Ser. 20th Century Math., 5, World. Sci. Publ., River Edge, NJ, 1997. 10.1142/9789812385215_0045Search in Google Scholar

[16] R. E. Gompf, A new construction of symplectic manifolds, Ann. of Math. (2), 142(3):527-595, 1995.10.2307/2118554Search in Google Scholar

[17] A. Gray and L. M. Hervella, The sixteen classes of almost Hermitian manifolds and their linear invariants, Ann. Mat. Pura Appl. (4), 123:35-58, 1980.10.1007/BF01796539Search in Google Scholar

[18] Z.-D. Guan, Toward a classification of compact nilmanifolds with symplectic structures, Int. Math. Res. Not. IMRN 2010, no. 22, 4377-4384.10.1093/imrn/rnq049Search in Google Scholar

[19] K. Hasegawa, Minimal models of nilmanifolds, Proc. Amer. Math. Soc. 106, no. 1, 65-71, 1989.10.1090/S0002-9939-1989-0946638-XSearch in Google Scholar

[20] D. Huybrechts, Complex geometry. An introduction, Universitext. Springer-Verlag, Berlin, 2005.Search in Google Scholar

[21] H. Kasuya, Vaisman metrics on solvmanifolds and Oeljeklaus-Toma manifolds, Bull. Lond. Math. Soc. 45, no. 1, 15-26, 2013.10.1112/blms/bds057Search in Google Scholar

[22] T. Kashiwada and S. Sato, On harmonic forms on compact locally conformal Kähler manifolds with parallel Lee form, Ann. Fac. Sci. Kinshasa, Zaire. 6, 17-29, 1980.10.2996/kmj/1138036121Search in Google Scholar

[23] K. Kodaira, On the structure of compact complex analytic surfaces. I, Amer. J. Math. 86, 751-798, 1964.10.2307/2373157Search in Google Scholar

[24] D. McDuff, Examples of symplectic simply connected manifolds with no Kähler structure, J. Diff. Geom. 20, 267-277, 1984.10.4310/jdg/1214438999Search in Google Scholar

[25] D. McDuff, D. Salamon, Introduction to symplectic topology, Oxford Science, 1995.Search in Google Scholar

[26] A. Mal’cev, On a class of homogeneous spaces, Izv. Akad. Nauk. Armyan. SSSR Ser. Mat. 13 (1949), 201-212.Search in Google Scholar

[27] J. Martinet, Formes de contact sur les varieties de dimension 3, Lect. Notes in Math, 20 (9) (1971), 142-163.10.1007/BFb0068901Search in Google Scholar

[28] K. Nomizu, On the cohomology of compact homogeneous space of nilpotent Lie group, Ann. of Math. (2) 59 (1954), 531-538.10.2307/1969716Search in Google Scholar

[29] K. Oeljeklaus and M. Toma, Non-Kähler compact complex manifolds associated to number fields, Ann. Inst. Fourier (Grenoble) 55, no. 1, 161-171, 2005.10.5802/aif.2093Search in Google Scholar

[30] J. Oprea and A. Tralle, Symplectic manifolds with no Kähler structure, Lecture Notes in Mathematics, 1661, Springer-Verlag, Berlin, 1997.Search in Google Scholar

[31] L. Ornea and M. Pilca, Remarks on the product of harmonic forms, Pacific J. of Math. 250, no. 2, 353-363, 2011.10.2140/pjm.2011.250.353Search in Google Scholar

[32] L. Ornea and M. Verbitsky, A report on locally conformally Kähler manifolds, Harmonic Maps and Differential Geometry, 135-149, Contemp. Math., 542, Amer. Math. Soc., Providence, RI, 2011.10.1090/conm/542/10703Search in Google Scholar

[33] H. Sawai, Locally conformal Kähler structures on compact nilmanifolds with left-invariant complex structures, Geom. Dedicata 125 (2007), 93-101.10.1007/s10711-007-9140-1Search in Google Scholar

[34] W. Thurston, Some simple examples of symplectic manifolds, Proc. Amer. Math. Soc. 55 no. 2, 467-468, 1976.10.1090/S0002-9939-1976-0402764-6Search in Google Scholar

[35] L. Ugarte, Hermitian structures on six-dimensional nilmanifolds, Transform. Groups 12 (2007), no. 1, 175-202.Search in Google Scholar

[36] I. Vaisman, On locally conformal almost Kähler manifolds, Israel J. Math. 24 (1976), no. 3-4, 338-351.Search in Google Scholar

[37] I. Vaisman, Generalized Hopf manifolds, Geom. Dedicata, 13 (1982), 231-255.10.1007/BF00148231Search in Google Scholar

[38] I. Vaisman, Locally Conformal Symplectic Manifolds, Internat. J. Math. & Math. Sci. 8 (3) (1985), 521-536.10.1155/S0161171285000564Search in Google Scholar