Cohomologies on Hypercomplex Manifolds

Chapter
Part of the Springer INdAM Series book series (SINDAMS, volume 21)

Abstract

We review some cohomological aspects of complex and hypercomplex manifolds and underline the differences between both realms. Furthermore, we try to highlight the similarities between compact complex surfaces on one hand and compact hypercomplex manifolds of real dimension 8 with holonomy of the Obata connection in \(\mathrm{SL}(2, \mathbb{H})\) on the other hand.

References

  1. 1.
    D. Angella, Cohomological aspects of non-Kähler manifolds. Ph.D. thesis, Università di Pisa, 2013. arXiv:1302.0524Google Scholar
  2. 2.
    D. Angella, A. Tomassini, On the \(\partial \overline{\partial }\)-Lemma and Bott–Chern cohomology. Invent. Math. 192, 71–81 (2013)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    D. Angella, G. Dloussky, A. Tomassini, On Bott–Chern cohomology of compact complex surfaces. Ann. Mat. Pura Appl. (4) 195, 199–217 (2016)Google Scholar
  4. 4.
    B. Banos, A. Swann, Potentials for hyper-Kähler metrics with torsion. Classical Quantum Gravity 21, 3127–3135 (2004)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    M.L. Barberis, I. Dotti Miatello, Hypercomplex structures on a class of solvable Lie groups. Q. J. Math. Oxford Ser. (2) 47, 389–404 (1996)Google Scholar
  6. 6.
    M.L. Barberis, I.G. Dotti, M. Verbitsky, Canonical bundles of complex nilmanifolds, with applications to hypercomplex geometry. Math. Res. Lett. 16, 331–347 (2009)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    F. Battaglia, A hypercomplex Stiefel manifold. Differ. Geom. Appl. 6, 121–128 (1996)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    C.P. Boyer, A note on hyper-Hermitian four-manifolds. Proc. Am. Math. Soc. 102, 157–164 (1988)MathSciNetMATHGoogle Scholar
  9. 9.
    C.P. Boyer, K. Galicki, B.M. Mann, Hypercomplex structures on Stiefel manifolds. Ann. Glob. Anal. Geom. 14, 81–105 (1996)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    N. Buchdahl, On compact Kähler surfaces. Ann. Inst. Fourier (Grenoble) 49, 287–302 (1999)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    M.M. Capria, S.M. Salamon, Yang–Mills fields on quaternionic spaces. Nonlinearity 1, 517–530 (1988)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    P. Deligne, P. Griffith, J. Morgan, D. Sullivan, Real homotopy theory of Kähler manifolds. Invent. Math. 29, 245–274 (1975)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    A. Fino, G. Grantcharov, Properties of manifolds with skew-symmetric torsion and special holonomy. Adv. Math. 189, 439–450 (2004)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    G. Grantcharov, Y.S. Poon, Geometry of hyper-Kähler connections with torsion. Commun. Math. Phys. 213, 19–37 (2000)CrossRefMATHGoogle Scholar
  15. 15.
    G. Grantcharov, M. Lejmi, M. Verbitsky, Existence of HKT metrics on hypercomplex manifolds of real dimension 8 (2014). arXiv:1409.3280Google Scholar
  16. 16.
    D. Joyce, Compact hypercomplex and quaternionic manifolds. J. Differ. Geom. 35, 743–761 (1992)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    D. Joyce, Compact Manifolds with Special Holonomy. Oxford Mathematical Monographs (Oxford University Press, Oxford, 2000)MATHGoogle Scholar
  18. 18.
    D. Kaledin, Integrability of the twistor space for a hypercomplex manifold. Sel. Math. (N.S.) 4, 271–278 (1998)Google Scholar
  19. 19.
    Ma. Kato, Compact differentiable 4-folds with quaternionic structures. Math. Ann. 248, 79–96 (1980)Google Scholar
  20. 20.
    K. Kodaira, On the structure of compact complex analytic surfaces I. Am. J. Math. 86, 751–798 (1964)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    A. Lamari, Courants kählériens et surfaces compactes. Ann. Inst. Fourier (Grenoble) 49, 263–285 (1999)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    M. Lejmi, P. Weber, Quaternionic Bott–Chern cohomology and existence of HKT metrics. Q. J. Math. 1–24. doi 10.1093/qmath/haw060Google Scholar
  23. 23.
    M. Lübke, A. Teleman, The Kobayashi–Hitchin Correspondence (World Scientific, River Edge, NJ, 1995)CrossRefMATHGoogle Scholar
  24. 24.
    S. Merkulov, L. Schwachhöfer, Classification of irreducible holonomies of torsion-free affine connections. Ann. Math. (2) 150, 77–149 (1999)Google Scholar
  25. 25.
    Y. Miyaoka, Kähler metrics on elliptic surfaces. Proc. Jpn. Acad. 50, 533–536 (1974)CrossRefMATHGoogle Scholar
  26. 26.
    M. Obata, Affine connections on manifolds with almost complex, quaternionic or Hermitian structure. Jpn. J. Math. 26, 43–77 (1956)CrossRefMATHGoogle Scholar
  27. 27.
    H. Pedersen, Y.S. Poon, Inhomogeneous hypercomplex structures on homogeneous manifolds. J. Reine Angew. Math. 516, 159–181 (1999)MathSciNetMATHGoogle Scholar
  28. 28.
    D. Popovici, Deformation limits of projective manifolds: Hodge numbers and strongly Gauduchon metrics. Invent. Math. 194, 515–534 (2013)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    S.M. Salamon, Differential geometry of quaternionic manifolds. Ann. Sci. Ec. Norm. Super 19, 31–55 (1986)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Y.T. Siu, Every K3 surface is Kähler. Invent. Math. 73, 139–150 (1983)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    A. Soldatenkov, Holonomy of the Obata connection in SU(3). Int. Math. Res. Not. IMRN, 15, 3483–3497 (2012)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Ph. Spindel, A. Sevrin, W. Troost, A. Van Proeyen, Extended supersymmetric σ-models on group manifolds. Nucl. Phys. B 308, 662–698 (1988)MathSciNetCrossRefGoogle Scholar
  33. 33.
    A. Teleman, The pseudo-effective cone of a non-Kählerian surface and applications. Math. Ann. 335, 965–989 (2006)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    M. Verbitsky, Hyperkähler manifolds with torsion, supersymmetry and Hodge theory. Asian J. Math. 6, 679–712 (2002)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    C. Voisin, Théorie de Hodge et géométrie algébrique complexe. Cours Spécialisés, vol. 10 (Société Mathématique de France, Paris, 2002), pp. viii+595Google Scholar
  36. 36.
    D. Widdows, A Dolbeault-type double complex on quaternionic manifolds. Asian J. Math. 6, 253–275 (2002)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    C.-C. Wu, On the geometry of superstrings with torsion. Ph.D. thesis, Harvard University, ProQuest LLC, Ann Arbor, MI, 2006Google Scholar

Copyright information

© Springer International Publishing AG, a part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of MathematicsBronx Community College of CUNYBronxUSA
  2. 2.Département de MathématiquesUniversité libre de BruxellesBrusselsBelgium

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