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Dolbeault cohomology of nilmanifolds with left-invariant complex structure

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Complex and Differential Geometry

Part of the book series: Springer Proceedings in Mathematics ((PROM,volume 8))

Abstract

We discuss the known evidence for the conjecture that the Dolbeault cohomology of nilmanifolds with left-invariant complex structure can be computed as Lie-algebra cohomology and also mention some applications.

Mathematics Subject Classification (2010) Primary 53C56. Secondary 22E25, 32G05, 17B56.

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References

  1. Abbena, E., Garbiero, S., Salamon, S.: Almost Hermitian geometry on six dimensional nilmanifolds. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 30(1), 147–170 (2001).

    Google Scholar 

  2. Andrada, A., Salamon, S.: Complex product structures on Lie algebras. Forum Math. 17(2), 261–295 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  3. Barberis, M.L.: Affine connections on homogeneous hypercomplex manifolds. J. Geom. Phys. 32(1), 1–13 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  4. Barberis, M.L., Dotti, I.G., Verbitsky, M.: Canonical bundles of complex nilmanifolds, with applications to hypercomplex geometry. Math. Res. Lett. 16(2), 331–347 (2009)

    MATH  MathSciNet  Google Scholar 

  5. Benson, C., Gordon, C.S.: K¨ahler and symplectic structures on nilmanifolds. Topology 27(4), 513–518 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  6. Buser, P., Karcher, H.: Gromov’s almost flat manifolds, Ast´erisque, vol. 81. Soci´et´e Math´ematique de France, Paris (1981)

    Google Scholar 

  7. Catanese, F.: Deformation types of real and complex manifolds. In: Contemporary trends in algebraic geometry and algebraic topology (Tianjin, 2000), Nankai Tracts Math., vol. 5, pp.195–238. World Sci. Publ., River Edge, NJ (2002)

    Google Scholar 

  8. Catanese, F.: Deformation in the large of some complex manifolds. I. Ann. Mat. Pura Appl.(4) 183(3), 261–289 (2004)

    Google Scholar 

  9. Catanese, F., Frediani, P.: Deformation in the large of some complex manifolds. II. In: Recent progress on some problems in several complex variables and partial differential equations, Contemp. Math., vol. 400, pp. 21–41. Amer. Math. Soc., Providence, RI (2006)

    Google Scholar 

  10. Chevalley, C., Eilenberg, S.: Cohomology theory of Lie groups and Lie algebras. Trans. Amer.Math. Soc. 63, 85–124 (1948)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  12. Console, S., Fino, A.: On the de Rham cohomology of solvmanifolds, arXiv:0912.2006, (2009).

    Google Scholar 

  13. Console, S., Fino, A., Poon, Y.S.: Stability of abelian complex structures. Internat. J. Math. 17(4), 401–416 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  14. Cordero, L.A., Fern´andez, M., Gray, A., Ugarte, L.: Compact nilmanifolds with nilpotent complex structures: Dolbeault cohomology. Trans. Amer. Math. Soc. 352(12), 5405–5433 (2000)

    Google Scholar 

  15. Cordero, L.A., Fernandez, M., Ugarte, L., Gray, A.: Fr¨olicher spectral sequence of compact nilmanifolds with nilpotent complex structure. In: New developments in differential geometry, Budapest 1996, pp. 77–102. Kluwer Acad. Publ., Dordrecht (1999)

    Google Scholar 

  16. Corwin, L.J., Greenleaf, F.P.: Representations of nilpotent Lie groups and their applications.Part I, Cambridge Studies in Advanced Mathematics, vol. 18. Cambridge University Press, Cambridge (1990).

    Google Scholar 

  17. Dotti, I.G., Fino, A.: Hypercomplex nilpotent Lie groups. In: Global differential geometry: the mathematical legacy of Alfred Gray (Bilbao, 2000), Contemp. Math., vol. 288, pp. 310–314. Amer. Math. Soc., Providence, RI (2001)

    Google Scholar 

  18. Dyer, J.L.: A nilpotent Lie algebra with nilpotent automorphism group. Bull. Amer. Math. Soc. 76, 52–56 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  19. Fern´andez, M., Ivanov, S., Ugarte, L., Villacampa, R.: Compact supersymmetric solutions of the heterotic equations of motion in dimension 5. Nuclear Phys. B 820(1-2), 483–502 (2009).

    Google Scholar 

  20. Gra˜na, M., Minasian, R., Petrini, M., Tomasiello, A.: A scan for new N = 1 vacua on twisted tori, JHEP 05 (2007) 031, hep-th/0609124.

    Google Scholar 

  21. Green, B., Tao, T.: Linear equations in primes (2006), arXiv:math/0606088v2.

    Google Scholar 

  22. Gromov, M.: Almost flat manifolds. J. Differential Geom. 13(2), 231–241 (1978).

    MATH  MathSciNet  Google Scholar 

  23. Guan, D.: Modification and the cohomology groups of compact solvmanifolds. Electron. Res. Announc. Amer. Math. Soc. 13, 74–81 (electronic) (2007).

    Google Scholar 

  24. Hirzebruch, F.: Topological methods in algebraic geometry. Classics in Mathematics. Springer-Verlag, Berlin (1995).

    MATH  Google Scholar 

  25. Host, B., Kra, B.: Nonconventional ergodic averages and nilmanifolds. Ann. of Math. (2) 161(1), 397–488 (2005).

    Google Scholar 

  26. Ketsetzis, G., Salamon, S.: Complex structures on the Iwasawa manifold. Adv. Geom. 4(2), 165–179 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  27. Knapp, A.W.: Lie groups beyond an introduction, Progress in Mathematics, vol. 140, second edn. Birkh¨auser Boston Inc., Boston, MA (2002)

    Google Scholar 

  28. Kobayashi, S., Nomizu, K.: Foundations of differential geometry. Vol. II. Interscience Tracts in Pure and Applied Mathematics, No. 15 Vol. II. Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney (1969)

    Google Scholar 

  29. Kodaira, K.: On the structure of compact complex analytic surfaces. II. Amer. J. Math. 88, 682–721 (1966)

    Article  MATH  MathSciNet  Google Scholar 

  30. Kuranishi, M.: On the locally complete families of complex analytic structures. Ann. of Math. (2) 75, 536–577 (1962)

    Google Scholar 

  31. Lange, H., Birkenhake, C.: Complex abelian varieties, Grundlehren der Mathematischen Wissenschaften, vol. 302. Springer-Verlag, Berlin (1992)

    Google Scholar 

  32. Lauret, J.: A canonical compatible metric for geometric structures on nilmanifolds. Ann. Global Anal. Geom. 30(2), 107–138 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  33. Maclaughlin, C., Pedersen, H., Poon, Y.S., Salamon, S.: Deformation of 2-step nilmanifolds with abelian complex structures. J. London Math. Soc. (2) 73(1), 173–193 (2006)

    Google Scholar 

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

    Google Scholar 

  35. Nakamura, I.: Complex parallelisable manifolds and their small deformations. J. Differential Geometry 10, 85–112 (1975)

    MATH  MathSciNet  Google Scholar 

  36. Nomizu, K.: On the cohomology of compact homogeneous spaces of nilpotent Lie groups. Ann. of Math. (2) 59, 531–538 (1954)

    Google Scholar 

  37. Pittie, H.V.: The nondegeneration of the Hodge-de Rham spectral sequence. Bull. Amer. Math. Soc. (N.S.) 20(1), 19–22 (1989)

    Google Scholar 

  38. Rollenske, S.: The Fr¨olicher spectral sequence can be arbitrarily non-degenerate. Math. Ann. 341(3), 623–628 (2008).

    Google Scholar 

  39. Rollenske, S.: The Kuranishi space of complex parallelisable nilmanifolds (2008). arXiv:0803.2048, to appear in JEMS.

    Google Scholar 

  40. Rollenske, S.: Geometry of nilmanifolds with left-invariant complex structure and deformations in the large. Proc. Lond. Math. Soc. (3) 99(2), 425–460 (2009).

    Google Scholar 

  41. Rollenske, S.: Lie-algebra Dolbeault-cohomology and small deformations of nilmanifolds. J. Lond. Math. Soc. (2) 79(2), 346–362 (2009).

    Google Scholar 

  42. Salamon, S.M.: Complex structures on nilpotent Lie algebras. J. Pure Appl. Algebra 157(2-3), 311–333 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  43. Ugarte, L.: Hermitian structures on six-dimensional nilmanifolds. Transform. Groups 12(1), 175–202 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  44. Vinberg, E.B., Gorbatsevich, V.V., Shvartsman, O.V.: Discrete subgroups of Lie groups. In: Lie groups and Lie algebras, II, Encyclopaedia Math. Sci., vol. 21, pp. 1–123, 217–223. Springer, Berlin (2000)

    Google Scholar 

  45. Weibel, C.A.: An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38. Cambridge University Press, Cambridge (1994)

    Google Scholar 

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Authors and Affiliations

  1. Mathematisches Institut, Rheinische Friedrich-Wilhelms-Universität Bonn, Endenicher Allee 60, 53115, Bonn, Germany

    Sönke Rollenske

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  1. Sönke Rollenske
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Correspondence to Sönke Rollenske .

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Editors and Affiliations

  1. , Institut für Algebraische Geometrie, Leibniz Universität Hannover, Welfengarten 1, Hannover, 30167, Germany

    Wolfgang Ebeling

  2. FB Mathematik, Inst. Mathematik, Universität Hannover, Welfengarten 1, Hannover, 30167, Germany

    Klaus Hulek

  3. , Institut für Differentialgeometrie, Leibniz Universität Hannover, Welfengarten 1, Hannover, 30167, Germany

    Knut Smoczyk

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Rollenske, S. (2011). Dolbeault cohomology of nilmanifolds with left-invariant complex structure. In: Ebeling, W., Hulek, K., Smoczyk, K. (eds) Complex and Differential Geometry. Springer Proceedings in Mathematics, vol 8. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-20300-8_18

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