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Cohomologies of locally conformally symplectic manifolds and solvmanifolds

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Abstract

We study the Morse–Novikov cohomology and its almost-symplectic counterpart on manifolds admitting locally conformally symplectic structures. More precisely, we introduce lcs cohomologies and we study elliptic Hodge theory, dualities, Hard Lefschetz condition. We consider solvmanifolds and Oeljeklaus–Toma manifolds. In particular, we prove that Oeljeklaus–Toma manifolds with precisely one complex place, and under an additional arithmetic condition, satisfy the Mostow property. This holds in particular for the Inoue surface of type \(S^0\).

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References

  1. Angella, D.: The cohomologies of the Iwasawa manifold and of its small deformations. J. Geom. Anal. 23(3), 1355–1378 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  2. Angella, D., Bazzoni, G., Parton, M.: Four dimensional locally conformal symplectic Lie algebras. arXiv:1704.01197

  3. Angella, D., Kasuya, H.: Symplectic Bott–Chern cohomology of solvmanifolds. J. Symplectic Geom. arXiv:1308.4258

  4. Angella, D., Kasuya, H.: Hodge theory for twisted differentials. Complex Manifolds 1, 64–85 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  5. Angella, D., Tardini, N.: Quantitative and qualitative cohomological properties for non-Kähler manifolds. Proc. Amer. Math. Soc. 145(1), 273–285 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  6. Angella, D., Tomassini, A.: Inequalities à la Frölicher and cohomological decompositions. J. Noncommut. Geom. 9(2), 505–542 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Angella, D., Ugarte, L.: Locally conformal Hermitian metrics on complex non-Kähler manifolds. Mediterr. J. Math. 13(4), 2105–2145 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bande, G., Kotschick, D.: Contact pairs and locally conformally symplectic structures. Contemp. Math. 542, 85–95 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Banyaga, A.: Some properties of locally conformal symplectic structures. Comment. Math. Helv. 77(2), 383–398 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  10. Borel, A.: Linear Algebraic Groups. Graduate Texts in Mathematics, vol. 126, 2nd edn. Springer, New York (1991)

    Book  MATH  Google Scholar 

  11. Brylinski, J.-L.: A differential complex for Poisson manifolds. J. Differ. Geom. 28(1), 93–114 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  12. Cavalcanti, G.R.: New Aspects of the \(dd^c\)-textitlemma. Oxford University D. Phil thesis. arXiv:math/0501406

  13. Console, S., Fino, A.: Dolbeault cohomology of compact nilmanifolds. Transform. Groups 6(2), 111–124 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  14. Deligne, P., Griffiths, Ph, Morgan, J., Sullivan, D.: Real homotopy theory of Kähler manifolds. Invent. Math. 29(3), 245–274 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  15. Eliashberg, Y., Murphy, E.: Making cobordisms symplectic. arXiv:1504.06312

  16. Gini, R., Ornea, L., Parton, M., Piccinni, P.: Reduction of Vaisman structures in complex and quaternionic geometry. J. Geom. Phys. 56, 2501–2522 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gorbatsevich, V.V.: Symplectic structures and cohomology on some solvmanifolds. Sib. Math. J. 44(2), 260–274 (2003). (translated from Sibirsk. Mat. Zh. 44 (2003), no. 2, 322–342)

  18. Guedira, F., Lichnerowicz, A.: Géométrie des algèbres de Lie locales de Kirillov. J. Math. Pures Appl. 63, 407–484 (1984)

    MathSciNet  MATH  Google Scholar 

  19. Guillemin, V.: Symplectic Hodge theory and the \(d\delta \)-Lemma, preprint. Massachusetts Institute of Technology (2001)

  20. Haller, S., Rybicki, T.: On the group of diffeomorphisms preserving a locally conformal symplectic structure. Ann. Glob. Anal. Geom. 17(5), 475–502 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  21. Hasegawa, K.: Complex and Kähler structures on compact solvmanifolds. Conf. Symplectic Topol. J. Symplectic Geom. 3(4), 749–767 (2005)

    Article  MATH  Google Scholar 

  22. Hattori, A.: Spectral sequence in the de Rham cohomology of fibre bundles. J. Fac. Sci. Univ. Tokyo Sect. I(8), 289–331 (1960)

    MathSciNet  MATH  Google Scholar 

  23. Inoue, M.: On surfaces of class \(VII_{0}\). Invent. Math. 24, 269–310 (1974)

    Article  MathSciNet  Google Scholar 

  24. Kasuya, H.: Vaisman metrics on solvmanifolds and Oeljeklaus–Toma manifolds. Bull. Lond. Math. Soc. 45(1), 15–26 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. Kasuya, H.: Flat bundles and hyper-Hodge decomposition on solvmanifolds. Int. Math. Res. Not. 2015(19), 9638–9659 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  26. Kodaira, K.: On the structure of compact complex analytic surfaces. I. Am. J. Math. 86(4), 751–798 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  27. Kodaira, K., Spencer, D .C.: On deformations of complex analytic structures. III. Stability theorems for complex structures. Ann. Math. (2) 71, 43–76 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  28. Le, H.V., Vanzura, J.: Cohomology theories on locally conformal symplectic manifolds. Asian J. Math. 19(1), 45–82 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  29. Lyčagin, V.V.: Contact geometry and second-order nonlinear differential equations. Uspekhi Mat. Nauk 34, no. 1(205), 137–165 (1979)

  30. de León, M., López, B., Belén, J.C., Marrero, E.Padrón: On the computation of the Lichnerowicz–Jacobi cohomology. J. Geom. Phys. 44(4), 507–522 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  31. Mathieu, O.: Harmonic cohomology classes of symplectic manifolds. Comment. Math. Helv. 70(1), 1–9 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  32. Merkulov, S.A.: Formality of canonical symplectic complexes and Frobenius manifolds. Int. Math. Res. Not. 1998(14), 727–733 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  33. Millionshchikov, D.V.: Cohomology of solvable Lie algebras, and solvmanifolds. Mat. Zametki 77(1), 67–79 (2005). (Mat. Zametki 77 (2005), no. 1, 67–79; translation in Math. Notes 77 (2005), no. 1-2, 61–71)

  34. Mostow, G.D.: Cohomology of topological groups and solvmanifolds. Ann. Math. (2) 73(1), 20–48 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  35. Neukirch, J.: Algebraic Number Theory. Springer, New York (1999)

    Book  MATH  Google Scholar 

  36. Novikov, S.P.: Multi-valued functions and functionals. An analogue of Morse theory. Sov. Math. Dokl. 24, 222–226 (1981)

    MATH  Google Scholar 

  37. Novikov, S.P.: The Hamiltonian formalism and a multi-valued analogue of Morse theory. Russ. Math. Surv. 37, 1–56 (1982)

    Article  MATH  Google Scholar 

  38. Oeljeklaus, K., Toma, M.: Non-Kähler compact complex manifolds associated to number fields. Ann. Inst. Fourier (Grenoble) 55(1), 161–171 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  39. Ornea, L., Verbitsky, M.: in preparation

  40. Otiman, A.: Morse–Novikov cohomology of locally conformally Kähler surfaces. arXiv:1609.07675

  41. Pazhitnov, A.V.: An analytic proof of the real part of Novikov’s inequalities. Sov. Mat. Dokl. 35, 456–457 (1987)

    MathSciNet  MATH  Google Scholar 

  42. Raghunathan, M.S.: Discrete Subgroups of Lie Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 68. Springer, New York (1972)

    Book  Google Scholar 

  43. Roman, S.: Field Theory. Springer, New York (2006)

    MATH  Google Scholar 

  44. Sage Mathematics Software (Version 6.10). The Sage Developers (2015). http://www.sagemath.org

  45. Schweitzer, M.: Autour de la cohomologie de Bott–Chern, Prépublication de l’Institut Fourier no. 703 (2007). arXiv:0709.3528

  46. Thurston, W.P.: Some simple examples of symplectic manifolds. Proc. Amer. Math. Soc. 55(2), 467–468 (1976)

    MathSciNet  MATH  Google Scholar 

  47. Tseng, L.-S., Yau, S.-T.: Cohomology and Hodge theory on symplectic manifolds: I. J. Differ. Geom. 91(3), 383–416 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  48. Tseng, L.-S., Yau, S.-T.: Cohomology and Hodge theory on symplectic manifolds: II. J. Differ. Geom. 91(3), 417–443 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  49. Vuletescu, V.: LCK metrics on Oeljeklaus–Toma manifolds versus Kronecker’s theorem. Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 57(105), no. 2, 225–231 (2014)

  50. Weintraub, S.: Galois Theory. Springer, New York (2006)

    MATH  Google Scholar 

  51. Yan, D.: Hodge structure on symplectic manifolds. Adv. Math. 120, 143–154 (1996)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank Giovanni Bazzoni, Liviu Ornea, Luis Ugarte, Victor Vuletescu, for interesting discussions. The first-named and the third-named authors would like to thank also Adriano Tomassini for his constant support and encouragement and for useful discussions. The second-named author is also grateful for constructive discussions to Andrei Sipoş and Miron Stanciu and would like to thank Liviu Ornea for his constant guidance. Part of this work has been done during the stay of the first-named author at Universitatea din Bucureşti with the support of an ICUB Fellowship: he would like to thank Liviu Ornea and Victor Vuletescu for the invitation, and the whole Department for the warm hospitality.

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Authors and Affiliations

  1. Dipartimento di Matematica e Informatica “Ulisse Dini”, Università degli Studi di Firenze, viale Morgagni 67/a, 50134, Firenze, Italy

    Daniele Angella

  2. Institute of Mathematics “Simion Stoilow” of the Romanian Academy, 21, Calea Grivitei Street, 010702, Bucharest, Romania

    Alexandra Otiman

  3. Faculty of Mathematics and Computer Science, University of Bucharest, 14 Academiei Str., Bucharest, Romania

    Alexandra Otiman

  4. Dipartimento di Matematica, Università di Pisa, largo Bruno Pontecorvo 5, 56127, Pisa, Italy

    Nicoletta Tardini

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  1. Daniele Angella
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  2. Alexandra Otiman
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  3. Nicoletta Tardini
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Correspondence to Nicoletta Tardini.

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Dedicated to Professor Paolo Piccinni on the occasion of his 65th birthday. Buon compleanno!

D. Angella is supported by the SIR2014 Project RBSI14DYEB “Analytic aspects in complex and hypercomplex geometry”, by ICUB Fellowship for Visiting Professor, and by GNSAGA of INdAM. N. Tardini is supported by Project PRIN “Varietà reali e complesse: geometria, topologia e analisi armonica” and by GNSAGA of INdAM.

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Angella, D., Otiman, A. & Tardini, N. Cohomologies of locally conformally symplectic manifolds and solvmanifolds. Ann Glob Anal Geom 53, 67–96 (2018). https://doi.org/10.1007/s10455-017-9568-y

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