The Journal of Geometric Analysis

, Volume 28, Issue 4, pp 2984–3047 | Cite as

Several Special Complex Structures and Their Deformation Properties

  • Sheng Rao
  • Quanting ZhaoEmail author


We introduce a natural map from the space of pure-type complex differential forms on a complex manifold to the corresponding one on the infinitesimal deformations of this complex manifold. By use of this map, we generalize an extension formula in a recent work of K. Liu, X. Yang, and the first author. As direct corollaries, we prove several deformation invariance theorems for Hodge numbers. Moreover, we also study the Gauduchon cone and its relation with the balanced cone in the Kähler case, and show that the limit of the Gauduchon cone in the sense of D. Popovici for a generic fiber in a Kählerian family is contained in the closure of the Gauduchon cone for this fiber.


Deformations of complex structures Deformations and infinitesimal methods Formal methods Deformations Hermitian and Kählerian manifolds 

Mathematics Subject Classification

Primary 32G05 Secondary 13D10 14D15 53C55 



We would like to express our gratitude to Professors Daniele Angella, Kwokwai Chan, Huitao Feng, Jixiang Fu, Lei Fu, Conan Leung, Kefeng Liu, Dan Popovici, Fangyang Zheng, and Dr. Jie Tu, Yat-hin Suen, Xueyuan Wan, Jian Xiao, Xiaokui Yang, Wanke Yin, Shengmao Zhu for their useful advice or interest on this work. This work started when the first author was invited by Professor J.A. Chen to Taiwan University in May–July of 2013 with the support of the National Center for Theoretical Sciences, and was completed during his visit in June of 2015 to the Mathematics Department of UCLA. He takes this opportunity to thank them for their hospitality. Last but not least, the anonymous referee’s careful reading and valuable comments improve the statement significantly. Rao is partially supported by the National Natural Science Foundations of China No. 11301477, 11671305, and China Scholarship Council/University of California, Los Angeles Joint Scholarship Program. The corresponding author Zhao is partially supported by the Fundamental Research Funds for the Central Universities No. CCNU16A05013 and China Postdoctoral Science Foundation No. 2016M592356


  1. 1.
    Alessandrini, L., Bassanelli, G.: Positive \(\partial {\overline{\partial }}\)-closed currents and non-Kähler geometry. J. Geom. Anal. 2, 291–361 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alessandrini, L., Bassanelli, G.: Modifications of compact balanced manifolds. C. R. Math. Acad. Sci. Paris 320, 1517–1522 (1995)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Angella, D.: The cohomologies of the Iwasawa manifold and its small deformations. J. Geom. Anal. 23(3), 1355–1378 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Barannikov, S., Kontsevich, M.: Frobenius manifolds and formality of Lie algebras of polyvector fields. Int. Math. Res. Notices 4, 201–215 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Barlet, D.: Espace analytique réduit des cycles analytiques complexes d’un espace analytique complexe de dimensin finie. In: Fonctions de Plusieurs Variables Complexes II(Sém. Franois Norguet, 1974–1975). Lecture Notes in Mathematics, vol. 482, pp. 1–158. Springer, New York (1975)Google Scholar
  6. 6.
    Barth, W., Hulek, K., Peters, C., Van de Ven, A.: Compact complex surfaces, Second edition. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 4. Springer, Berlin (2004)Google Scholar
  7. 7.
    Boucksom, S.: On the volume of a line bundle. Int. J. Math. 13(10), 1043–1063 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Boucksom, S.: Divisorial Zariski decomposition on compact complex manifolds. Ann. Sci. École Norm. Sup. 37(1), 45–76 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Boucksom, S., Demailly, J.-P., Paun, M., Peternell, T.: The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension. J. Algebraic Geom. 22, 201–248 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Boucksom, S., Eyssidieux, P., Guedj, V., Zeriahi, A.: Monge-Ampère equations in big cohomology classes. Acta Math. 205, 199–262 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Chan, K., Suen, Y.: A Chern-Weil approach to deformations of pairs and its applications. Complex Manifolds. arXiv:1406.6753v3 (2016)
  12. 12.
    Chiose, I., Rasdeaconu, R., Suvaina, I.: Balanced metrics on uniruled manifolds. arXiv:1408.4769v1
  13. 13.
    Clemens, H.: Geometry of formal Kuranishi theory. Adv. Math. 198, 311–365 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Console, S., Fino, A., Poon, Y.-S.: Stability of abelian complex structures. Int. J. Math. 17(4), 401–416 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Demailly, J.-P.: Regularization of closed positive currents and intersection theory. J. Algebraic Geom. 1, 361–409 (1992)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Demailly, J.-P., Peternell, T., Schneider, M.: Compact complex manifolds with numerically effective tangent bundles. J. Algebraic Geom. 3, 295–345 (1994)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Demailly, J.-P., Paun, M.: Numerical characterization of the Kähler cone of a compact Kähler manifold. Ann. Math. 159, 1247–1274 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Ehresmann, C.: Sur les espaces fibres differentiables. C. R. Acad. Sci. Paris 224, 1611–1612 (1947)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Friedman, R.: On threefolds with trivial canonical bundle: complex geometry and Lie theory (Sundance, UT, 1989). Proc. Sympos. Pure Math. 53, 103–134 (1991)CrossRefGoogle Scholar
  20. 20.
    Frölicher, A.: Zur Differentialgeometrie der komplexen Strukturen. Math. Ann. 129, 50–95 (1955)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Fu, J., Li, J., Yau, S.-T.: Balanced metrics on non-Kähler Calabi-Yau threefolds. J. Differ. Geom. 20, 81–129 (2012)CrossRefzbMATHGoogle Scholar
  22. 22.
    Fu, J., Xiao, J.: Relations between the Kähler cone and the balanced cone of a Kähler manifold. Adv. Math. 263, 230–252 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Gauduchon, P.: Le théorèm de l’excentricité nulle. C. R. Acad. Sc. Paris. Série A-B 285, 387–390 (1977)zbMATHGoogle Scholar
  24. 24.
    Grauert, H.: Ein Theorem der analytischen Garbentheorie und die Modulräume komplexer Strukturen (German). Inst. Hautestud. Sci. Publ. Math. 5, 233–292 (1960)zbMATHGoogle Scholar
  25. 25.
    Huang, L.: On joint moduli spaces. Math. Ann. 302(1), 61–79 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Kodaira, K.: On the structure of compact complex analytic surfaces. I. Am. J. Math. 86, 751–798 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Kodaira, K.: Complex Manifolds and Deformations of Complex Structures. Grundlehren der Mathematischen Wissenschaften, vol. 283. Springer, New York (1986)CrossRefzbMATHGoogle Scholar
  28. 28.
    Kodaira, K., Spencer, D.: On deformations of complex analytic structures, III. Stability theorems for complex structures. Ann. Math. 71(2), 43–76 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Kuranishi, M.: New proof for the existence of locally complete families of complex structures. In: Proc. Conf. Complex Analysis (Minneapolis, 1964), pp. 142–154. Springer, Berlin (1965)Google Scholar
  30. 30.
    Lamari, A.: Courants kählériens et surfaces compactes. Ann. Inst. Fourier 49(1), 263–285 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Li, Yi: On deformations of generalized complex structures the generalized Calabi-Yau case. arXiv:hep-th/0508030v2 (2005)
  32. 32.
    Liu, K., Rao, S.: Remarks on the Cartan formula and its applications. Asian J. Math. 16(1), 157–170 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Liu, K., Rao, S., Yang, X.: Quasi-isometry and deformations of Calabi-Yau manifolds. Invent. Math. 199(2), 423–453 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Liu, K., Sun, X., Yau, S.-T.: Recent development on the geometry of the Teichmüller and moduli spaces of Riemann surfaces. In: Surveys in Differential Geometry, Vol. XIV. Geometry of Riemann Surfaces and Their Moduli Spaces, pp. 221–259 (2009)Google Scholar
  35. 35.
    Latorre, A., Ugarte, L., Villacampa, R.: On the Bott-Chern cohomology and balanced Hermitian nilmanifold. Int. J. Math. 25(6), 1450057 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Michelsohn, M.L.: On the existence of special metrics in complex geometry. Acta Math. 49(3–4), 261–295 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Morrow, J., Kodaira, K.: Complex Manifolds. Holt, Rinehart and Winston Inc, New York (1971)zbMATHGoogle Scholar
  38. 38.
    Nakamura, I.: Complex parallelisable manifolds and their small deformations. J. Differ. Geom. 10, 85–112 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Newlander, A., Nirenberg, L.: Complex analytic coordinates in almost complex manifolds. Ann. Math. 65(2), 391–404 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Popovici, D.: Deformation limits of projective manifolds: Hodge numbers and strongly Gauduchon metrics. Invent. Math. 194(3), 515–534 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Popovici, D.: Aeppli Cohomology classes associated with Gauduchon metrics on compact complex manifolds. Bull. Soc. Math. France 143(4), 763–800 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Popovici, D.: Sufficient bigness criterion for differences of two nef classes. Math. Ann. 364(1–2), 649–655 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Popovici, D.: Holomorphic deformations of balanced Calabi-Yau \(\partial {\overline{\partial }}\)-manifolds. arXiv:1304.0331v1
  44. 44.
    Popovici, D.: Volume and self-intersection of differences of two nef classes. arXiv:1505.03457v1
  45. 45.
    Popovici, D., Ugarte, L.: The sGG classes of compact complex manifolds. arXiv:1407.5070v1
  46. 46.
    Rao, S., Wan, X., Zhao, Q.: Power series proofs for local stabilities of Kähler and balanced structures with mild \(\partial \bar{\partial }\)-lemma. arXiv:1609.05637v1
  47. 47.
    Rao, S., Zhao, Q.: Several special complex structures and their deformation properties. arXiv: 1604.05396v1 (2015)
  48. 48.
    Schweitzer, M.: Autour de la cohomologie de Bott-Chern. arXiv:0709.3528v1
  49. 49.
    Sullivan, D.: Cycles for the dynamical study of foliated manifolds and complex manifolds. Invent. Math. 36, 225–255 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Sakane, Y.: On compact complex parallelizable solvmanifolds. Osaka J. Math. 13(1), 187–212 (1976)MathSciNetzbMATHGoogle Scholar
  51. 51.
    Siu, Y.-T.: Analycity of sets associated to Lelong numbers and the extension of closed positive currents. Invent. Math. 27, 53–156 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Sun, X.: Deformation of canonical metrics I. Asian J. Math. 16(1), 141–155 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Sun, X., Yau, S.-T.: Deformation of Kähler-Einstein metrics, surveys in geometric analysis and relativity. Adv. Lect. Math. 20, 467–489 (2011)zbMATHGoogle Scholar
  54. 54.
    Tian, G.: Smoothness of the universal deformation space of compact Calabi-Yau manifolds and its Petersson-Weil metric, Mathematical aspects of string theory (San Diego, Calif., 1986). Adv. Ser. Math. Phys. 1, 629–646 (1987)Google Scholar
  55. 55.
    Todorov, A.: The Weil-Petersson geometry of the moduli space of \(\mathbb{SU}(n\ge 3)\) (Calabi-Yau) manifolds I. Commun. Math. Phys. 126(2), 325–346 (1989)CrossRefzbMATHGoogle Scholar
  56. 56.
    Toma, M.: A note of the cone of mobile curves. C. R. Math. Acad. Sci. Paris 348, 71–73 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Wavrik, J.: Deforming cohomology classes. Trans. Am. Math. Soc. 181, 341–350 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    Wu, D., Yau, S.-T., Zheng, F.: A degenerate Monge-Ampère equation and the boundary classes of Kähler cones. Math. Res. Lett. 16(2), 365–374 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Xiao, J.: Weak transcendental holomorphic Morse inequalities on compact Kähler manifolds. Ann. Inst. Fourier 65(3), 1367–1379 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    Xiao, J.: Characterizing volume via cone duality. Math. Ann. (2016).
  61. 61.
    Ye, X.: The jumping phenomenon of Hodge numbers. Pac. J. Math. 235(2), 379–398 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  62. 62.
    Yau, S.-T.: On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation, I. Commun. Pure. Appl. Math 31, 339–411 (1978)CrossRefzbMATHGoogle Scholar
  63. 63.
    Zhao, Q., Rao, S.: Applications of deformation formula of holomorphic one-forms. Pac. J. Math. 266(1), 221–255 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  64. 64.
    Zhao, Q., Rao, S.: Extension formulas and deformation invariance of Hodge numbers. C. R. Math. Acad. Sci. Paris 353(11), 979–984 (2015)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2017

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsWuhan UniversityWuhanChina
  2. 2.Department of MathematicsUniversity of California at Los AngelesLos AngelesUSA
  3. 3.School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical SciencesCentral China Normal UniversityWuhanPeople’s Republic of China

Personalised recommendations