Abstract
Using Hodge theory and Banach fixed point theorem, Liu and Zhu developed a global method to deal with various problems in deformation theory. In this note, the authors generalize Liu-Zhu’s method to treat two deformation problems for non-Kähler manifolds. They apply the \(\partial \overline \partial \)-Hodge theory to construct a deformation formula for (p, q)-forms of compact complex manifold under deformations, which can be used to study the Hodge number of complex manifold under deformations. In the second part of this note, by using the \(\partial \overline \partial \)-Hodge theory, they provide a simple proof of the unobstructed deformation theorem for the non-Kähler Calabi-Yau \(\partial \overline \partial \)-manifolds.
References
Angella, D. and Tomassini, A., On the \(\partial \overline \partial \)-lemma and Bott-Chern cohomology, Invent. Math., 192(1), 2013, 71–81.
Clemens, H., Geometry of formal Kuranishi theory, Adv. Math., 198, 2005, 311–365.
Kodaira, K., Nirenberg, L. and Spencer, D. C., On the existence of deformations of complex analytic structures, Ann. Math., 68, 1958, 450–459.
Kodaira, K. and Spencer, D. C., On deformations of complex analytic structures, III, Stability theorems for complex structures, Ann. Math., 71, 1960, 43–76.
Liu, K., Rao, S. and Wan, X., Geometry of logarithmic forms and deformations of complex structures, J. Algebraic Geom., 28(4), 2019, 773–815.
Liu, K., Rao, S. and Yang, X., Quasi-isometry and deformations of Calabi-Yau manifolds, Invent. Math., 199(2), 2015, 423–553.
Liu, K. and Zhu, S., Solving equations with Hodge theory, arXiv: 1803.01272.
Liu, K. and Zhu, S., Global Methods of Solving Equations on Manifolds, Survey in Differential Geometry, 23, Int. Press, Boston, MA, 2020.
Morrow, J. and Kodaira, K., Complex Manifolds, Hlt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1971.
Popovici, D., Holomorphic deformations of Balanced Calabi-Yau \(\partial \overline \partial \)-manifold, 2013, ArXiv: 1304.0331.
Popovici, D., Aeppli cohomology classes associated with Gauduchon metrics on compact complex manifolds, 2013, ArXiv: 1310.3685.
Rao, S., Wan, X. and Zhao, Q., On local stabilities of p-Kähler structures, Compositio Math., 155, 2019, 455–483.
Rao, S. and Zhang, R., On extension of closed complex (basic) differential forms: Hodge numbers and (transversely) p-Kähler structures, 2022, arXiv: 2204.06870v1.
Rao, S. and Zhao, Q., Several special complex structure and their deformation properties, J. Geom. Anal., 28(4), 2018, 2984–3047.
Rao, S. and Zou, Y., On Tian-Todorov lemma and its applications to deformation of CR-structures, Math. Z., 297(1–2), 2021, 943–960.
Tian, G., Smoothness of the universal deformation space of compact Calabi-Yau manifolds and its Petersson-Weil metric, Math. Aspects of String Theory, Adv. Ser. Math. Phy., 1, Worlds Scientific Publishing, Singapore, 1987, 629–646.
Todorov, A., The Weil-Petersson geometry of the moduli space of SU(n ≥ 3) (Calabi-Yau) manifolds I, Commun. Math. Phys., 126, 1989, 325–346.
Acknowledgement
The authors would like to thank Professor Kefeng Liu for many inspired discussions on the global methods in deformation theory.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflicts of interest The authors declare no conflicts of interest.
Additional information
This work was supported by the National Natural Science Foundation of China (No. 12061014).
Rights and permissions
About this article
Cite this article
Wei, D., Zhu, S. Two Applications of the \(\partial \overline \partial \)-Hodge Theory. Chin. Ann. Math. Ser. B 45, 137–150 (2024). https://doi.org/10.1007/s11401-024-0007-7
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11401-024-0007-7